Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Strongly Continuous Semigroups and Hille Yosida
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Areas of Elementary Plane Figures
- Banach Algebras Spectrum and Holomorphic Functional Calculus
- Banach Valued Integration and the Radon Nikodym Property
- Banach-Space Differential Calculus and Banach Manifolds
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Differentiability and the Cauchy–Riemann Equations
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Unbounded Self Adjoint Operators and Stones Theorem
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
This page develops the theory of strongly continuous () semigroups on Banach spaces from the definition through the generation theorems, with the sign convention and the resolvent normalised as . It begins with the semigroup axioms, local uniform boundedness, the equivalence of strong continuity with continuity at zero, and the exponential bound; the Bochner-calculus prerequisites (linearity of the Bochner integral, average convergence, the mean value inequality and the Banach-valued fundamental theorem of calculus) are proved alongside so that every integral manipulation has an exact supplier. The infinitesimal generator is defined by one-sided difference quotients and developed: domain invariance and commutation with orbits, integrated orbits, closedness of the generator and density of its domain, the resolvent of the closed operator, the resolvent identity, the Laplace-transform formula, and the resolvent power estimates.
The generation theory follows: the exponential series of a bounded operator, the Yosida approximants and the strong convergence of the resolvent and of the approximants on the domain, the convergence of the bounded Yosida semigroups to the generated semigroup, and the Hille-Yosida theorem with all resolvent powers together with its contraction shortcut. Dissipative operators, the Lumer-Phillips theorem and the maximal-dissipativity statement are proved in the norm form, and the classical, strong, mild and integral solutions of the abstract Cauchy problem are defined and related by the variation-of-constants theorem and the well-posedness theorem. A closing remark records the sign, resolvent and contraction dictionaries. The generation and automatic boundedness results assume Dependent Choice, which supplies the uniform boundedness and closed graph arguments and their Countable Choice consequences. Pure resolvent algebra instead uses bounded inverses in the resolvent definition; the equivalence with bijectivity alone is stated under DC. The norm-duality description of dissipativity and the Laplace uniqueness argument state HB separately. Classical upgrades of the inhomogeneous problem use forcing or its explicit Bochner-primitive extension; Lipschitz continuity alone on an arbitrary Banach space does not supply that extension.
3 · Logical flowchart
4 · Definitions, theorems and proofs
Strongly continuous semigroup
Definition
Let be a Banach space over (Banach space, Real and complex scalar conventions for normed spaces) and let be the bounded linear operators on (A bounded linear operator between normed spaces, The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators). A family is a strongly continuous one-parameter semigroup (or -semigroup) if (i) ; (ii) for all ; (iii) for every the orbit map is continuous from into . Property (iii) says that is continuous for the strong operator topology on ; no continuity in the operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum) is assumed or implied, and the operators need be neither isometries nor contractions. A strongly continuous group is defined analogously with in place of and the functional equation holding for all .
A semigroup with continuity at zero is uniformly bounded on every compact time interval
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a Banach space and let satisfy , for and for every (in particular every strongly continuous semigroup satisfies these hypotheses, Strongly continuous semigroup). Then for every , .
Facts & Assumptions
Given: A Banach space and a family with , for all , and for every . These are the hypotheses of Strongly continuous semigroup with continuity at every time weakened to continuity at ; the item assumes Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain), carried by the cited uniform boundedness principle, and step 1.1 selects a sequence, which uses Countable Choice, a consequence of DC.
The operator norm satisfies for all (Composition satisfies |ST|\le|S|,|T|), and is the operator norm on (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces).
Under DC, a pointwise bounded family of bounded linear operators between a Banach space and a normed space is norm bounded (Uniform boundedness principle).
For every one has as ; this is hypothesis (iii) of Strongly continuous semigroup at the point , where .
Proof
There are and with . Otherwise for every , so for each one may select with ; this selection is the only use of Countable Choice, available because DC implies .
For every the sequence converges to , because and ; hence the family is pointwise bounded on the Banach space .
By the uniform boundedness principle [F2] the family is norm bounded, that is , contradicting ; hence the assumed unboundedness of every right neighbourhood of is impossible, proving [step 1.1].
Put . For and write with and . The functional equation gives , by induction on from .
Therefore by [F1] and [step 1.1], where ; hence .
Since was arbitrary, for every , as required.
Continuity at time zero implies continuity of every orbit
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a Banach space and satisfy , for and for every . Then is a strongly continuous semigroup (Strongly continuous semigroup): every orbit map is continuous on .
Facts & Assumptions
Given: Dependent Choice; A Banach space and a family with , for and for every .
Local boundedness: for every there is with for all ; this uses DC through the uniform boundedness principle (A semigroup with continuity at zero is uniformly bounded on every compact time interval).
For the operator norm satisfies for all , and (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Composition satisfies |ST|\le|S|,|T|).
The hypotheses are those of a strongly continuous semigroup with continuity required only at (Strongly continuous semigroup): , the functional equation holds, and as for every .
Proof
Fix and let be a bound for on , which exists by [F1]; fix also .
Right continuity at : for , the functional equation gives , whose norm is at most as by [F2] and [F3].
Left continuity at : for one has , hence and, since , by [F1], [F2] and [F3].
The two one-sided limits at both equal , so the orbit is continuous at every ; as was arbitrary, all orbits are continuous on , and the family is a strongly continuous semigroup as defined in [F3].
Exponential bound for a C0-semigroup
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a strongly continuous semigroup on a Banach space (Strongly continuous semigroup). Then there exist and with for all . For one may take and ; for take , .
Facts & Assumptions
Given: Dependent Choice; A strongly continuous semigroup on a Banach space (Strongly continuous semigroup).
Local boundedness: ; the proof of the lemma uses DC through the uniform boundedness principle (A semigroup with continuity at zero is uniformly bounded on every compact time interval).
The operator norm is submultiplicative: , and for every (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Composition satisfies |ST|\le|S|,|T|, A bounded linear operator between normed spaces).
The semigroup law for and (Strongly continuous semigroup).
The real exponential satisfies , for all real , is a bijection, and for (The real exponential function and the number by a power series, The exponential addition formula , The exponential is a continuous bijection from onto ).
Proof
If take and : then for every . Otherwise , so and [F1] gives with ; set , which exists by [F4] because .
For write with and . By the semigroup law, , hence by [F2] and the choice of .
Since and , [F4] gives .
Combining [step 2.1] and [step 3.1], for every , with and ; in the nonzero case the displayed and are the explicit choices, and in the zero-space case the bound holds trivially for , .
Note. The constant need not be optimal: any larger also works, since is nondecreasing in for ; the growth bound is not needed here.
Linearity of the Bochner integral
Statement
Let be a measure space, let be a real or complex Banach space, and let be Bochner integrable (Bochner-integrable function) with scalars. Then is Bochner integrable and ; for a measurable set the same identity holds with replaced by . The integral is therefore additive and homogeneous, and in particular well defined on differences.
Facts & Assumptions
Given: A measure space , a real or complex Banach space , Bochner integrable functions , scalars , and a measurable set .
By the definition of Bochner integrability (Bochner-integrable function) there are sequences , of integrable -valued simple functions with , , and , .
The integral of an integrable Banach-valued simple function is independent of its representation, is linear, and satisfies for every measurable (The Banach-valued simple integral is well defined).
A strongly measurable is Bochner integrable if and only if ; for such and any defining approximating sequence of integrable simple functions, the integral is the limit of the simple integrals (Bochner integrability criterion, Bochner-integrable function).
Addition in and scalar multiplication are continuous (Vector addition and scalar multiplication are continuous in a normed space).
Proof
The functions are strongly measurable by [F1]. Let be their measurable simple approximations converging pointwise outside measurable null sets ; these need not be the defining approximations . The simple functions converge to off by [F4], proving strong measurability.
Norm estimate: for every , pointwise, hence after integration ; in particular , since by [F3].
By [step 1.1], [step 1.2] and the integrability criterion [F3], the function is Bochner integrable, and is a defining sequence of integrable simple functions for it, so .
Linearity of the simple integral [F2] gives for every , whose right-hand side converges to by [F1] and continuity of the vector operations [F4]; combining with [step 2.1] yields .
Restricted form: the functions and are Bochner integrable, because they are strongly measurable and dominated in norm by and respectively, and pointwise; applying [step 3.1] to the pair gives .
The integral is thus additive and homogeneous on the Bochner integrable functions: taking gives additivity, with arbitrary gives homogeneity, and shows the difference is Bochner integrable with , so the integral is well defined on differences.
Average convergence for a continuous Banach-valued function
Statement
Assume Countable Choice (The Axiom of Countable Choice ()) for the Lebesgue-measure interfaces. Let be a real or complex Banach space, let and let be continuous. Then is Bochner integrable, and for and with one has as ; analogously as for . The same one-sided limits hold for vector-valued curves that are merely continuous at provided they are Bochner integrable on some neighbourhood of .
Facts & Assumptions
Given: Countable Choice; A real or complex Banach space , real numbers , and a continuous .
is Bochner integrable when there are integrable -valued simple functions with , and then ; integrals over subintervals are defined through the indicators , and constant functions have the expected integrals (Bochner-integrable function, Linearity of the Bochner integral).
The Bochner integral is linear: for Bochner integrable and scalars the function is Bochner integrable with (Linearity of the Bochner integral).
Norm inequality: for every Bochner integrable and measurable (Bochner integral norm inequality).
Continuity of at a point means: for every there is with , , implying (Continuity of a map between metric spaces, at a point and globally, in the - form).
The interval is a compact metric space (Heine-Borel in : with the Euclidean metric a subset of is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), and a continuous map from a compact metric space to a metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous): for every there is such that whenever and .
Proof
By [F5], is uniformly continuous on . Its oscillation over pairs at distance at most therefore tends to zero as .
For each , set , for , and , with the last interval including . These are integrable simple functions and converge uniformly, hence pointwise, to .
The norm error is at most the oscillation from step 1.1, so . Together with the pointwise simple approximation, [F1] proves Bochner integrability of .
For and , linearity gives , since the constant function has integral .
By the norm inequality, the norm of this difference is at most .
Continuity at makes the last supremum tend to zero as : given , take below a continuity radius for at . Hence the forward averages converge to .
For and , the same linearity and norm estimates give . This proves the backward form directly.
If instead is only continuous at and Bochner integrable on a neighbourhood of , the forward and backward estimates above still apply: local integrability supplies the integrals and continuity at makes their norm errors vanish. Thus the stated general one-sided limits also hold.
Mean value inequality for a differentiable Banach-valued curve
Statement
Let be a Banach space, , and let be continuous on and differentiable on in the sense of Fréchet derivative between Banach spaces. If there is with for all , then . In particular, if is continuous, differentiable on and there, then is constant.
Facts & Assumptions
Given: A real or complex Banach space (Banach space), read as a real vector space for differentiation so that the real-variable Fréchet derivative of Fréchet derivative between Banach spaces applies (the complex case uses the underlying real structure, and complex differentiability is a special case); real numbers , a curve continuous on and differentiable on , and a constant with for all .
At each the Fréchet derivative is bounded linear and there is a remainder with as (Fréchet derivative between Banach spaces).
The norm function is continuous on and satisfies the reverse triangle inequality (The reverse triangle inequality in a normed space).
Vector addition and scalar multiplication on are continuous (Vector addition and scalar multiplication are continuous in a normed space), so limits of sums and scalar multiples may be taken termwise.
Proof
Fix and , and put for . By [F2] and [F3] the function is continuous, so is a nonempty closed subset of , since ; being nonempty and bounded above it has a supremum , which is therefore its maximum.
If then : if , then [F1] at (legitimate because ) gives with , so by [F2] for all sufficiently small , contradicting the maximality of ; hence , where is differentiable.
If , then differentiability at gives with , so for small [F2] gives , since ; then , contradicting the maximality of . Hence .
At the defining inequality of reads , that is .
Letting along a sequence: by continuity and [F3], so [F2] gives , and ; hence .
Since was arbitrary, .
If in addition on , take in [step 6.1]; then for every the same argument applied to the restriction of to gives , and , so is constant on .
Fundamental theorem of calculus for Banach-valued continuous curves
Statement
Assume Countable Choice (The Axiom of Countable Choice ()) for the Lebesgue-measure interfaces. Let be a real or complex Banach space, let , and let be continuous. Differentiation uses the underlying real structure. Then is differentiable on and has the corresponding one-sided derivatives at , with , and this derivative extends continuously to . Consequently, if is continuous, differentiable on with continuous on and extendable to a continuous -valued function on , then . The Bochner integral here is the one of Bochner-integrable function; the identities also hold for continuous curves on restricted to compact subintervals.
Facts & Assumptions
Given: Countable Choice; A Banach space , real numbers , a continuous , the primitive for , and a continuous differentiable on whose derivative extends to a continuous -valued function on .
Average convergence (Average convergence for a continuous Banach-valued function): a continuous is Bochner integrable, and for , , , , with the analogous backward limit for ; the same one-sided limits hold for curves continuous at and Bochner integrable near .
The Bochner integral is linear, so for the difference of primitives is (Linearity of the Bochner integral, Bochner-integrable function).
Differentiability on means Fréchet differentiability at every point of the open interval (Fréchet derivative between Banach spaces); at the endpoints only the relevant one-sided difference quotients are considered.
Mean value inequality (Mean value inequality for a differentiable Banach-valued curve): a curve continuous on an interval, differentiable inside with derivative bounded by , changes by at most times the length; in particular a curve with vanishing interior derivative is constant.
Proof
By [F1] the continuous is Bochner integrable on , so is defined for every ; by [F2] for .
Difference quotients of : for and with , by [F1]; similarly for . Hence is differentiable on with , and has the one-sided derivatives at the endpoints.
is continuous on , and the existence of the one-sided derivative at and at makes continuous there from the appropriate side; at interior points is continuous by differentiability.
Since extends to a continuous -valued function on , denote the extension again by and put for . By [step 2.1] the primitive of is differentiable on with derivative , and is differentiable on ; by linearity of the derivative and of the integral, on , and .
is continuous on and differentiable on with there; by [F5] (constant case) is constant on , so , that is .
Therefore . The same computation, applied to each compact subinterval after restricting a continuous curve on , gives the stated identity in that setting.
Infinitesimal generator of a C0-semigroup
Definition
For a Banach space , the operator vocabulary of Unbounded linear operators: domain, graph and extension and Densely defined, closed and closable operators, and cores is extended as follows: an operator is a linear map on a linear subspace of , its graph is , it is closed when this graph is closed, and densely defined when is dense in . Use on and the graph norm . These norms are equivalent to the square-sum norms in the Hilbert suppliers. The product is Banach because its two coordinate Cauchy sequences converge in ; the closed graph is therefore Banach, and is an isometry of the graph-norm domain onto it. Under Countable Choice, sequential closedness is equivalent to closedness: for any point in a closure, choose graph points within and pass to their limit.
Let be a strongly continuous semigroup on a Banach space (Strongly continuous semigroup). Its infinitesimal generator is the linear operator with domain and for . The limit is a one-sided limit at the boundary point , and is a linear subspace of (Normed subspace); the operator is recorded as the pair in the sense of Unbounded linear operators: domain, graph and extension and Densely defined, closed and closable operators, and cores. Neither boundedness nor closedness of , nor density of , is assumed in the definition; under its stated choice hypothesis, The generator is closed and densely defined proves that the graph of is closed and is dense in . The domain need not be closed in the norm of .
The one-sided limit. For the difference quotient is defined for every , and the defining limit is taken along only; no two-sided limit at the boundary point of is considered, and the vector is the limit when it exists. The value is unique because is a metric space (Banach space).
The domain is a linear subspace. The zero vector lies in and . If and are scalars, then by linearity of each (A bounded linear operator between normed spaces) the difference quotient of equals for every ; as this converges to , because vector addition and scalar multiplication are continuous and scalar multiplication by the fixed scalars is continuous. Hence with , so is a linear subspace of (Normed subspace) and is linear on it. The operator is recorded as the pair in the vocabulary of Unbounded linear operators: domain, graph and extension and Densely defined, closed and closable operators, and cores.
What is not assumed. Boundedness and graph closedness of , and density of in , are not defining assumptions. Graph closedness and domain density are conclusions of the later theorem under its stated choice hypothesis; they do not assert that is closed in the norm of . The semigroup axioms used here are those of Strongly continuous semigroup, in particular is everywhere defined and bounded for every .
The generator commutes with the semigroup on its domain
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a strongly continuous semigroup on a Banach space with generator (Infinitesimal generator of a C0-semigroup). If , then and for every ; moreover the orbit is differentiable on with , and right differentiable at with right derivative .
Facts & Assumptions
Given: Dependent Choice; A strongly continuous semigroup on a Banach space with generator (Strongly continuous semigroup, Infinitesimal generator of a C0-semigroup) and a vector .
means that as (Infinitesimal generator of a C0-semigroup).
The semigroup law holds for all , and operator norms satisfy (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Composition satisfies |ST|\le|S|,|T|).
The family is locally bounded and the orbit of every vector is continuous on (A semigroup with continuity at zero is uniformly bounded on every compact time interval, Strongly continuous semigroup): for there is with for , and for every as .
Proof
For and , the semigroup law gives .
Since by [F1] and is a fixed bounded operator, the right difference quotient in [step 1.1] converges to as . Hence the right derivative of the orbit at exists and equals ; taking shows that the right derivative at is and, for general , that with .
Left derivative at : for one has . Let and use the local bound on from [F3]: , because and strongly as by [F3]. Hence the left derivative at also equals .
Combining [step 2.1] and [step 3.1], for every and every the orbit satisfies , , and is differentiable on with derivative , with right derivative at .
Time integrals of semigroup orbits lie in the generator domain
Statement
Assume Countable Choice (The Axiom of Countable Choice ()) for the Lebesgue-measure interfaces. Let be a strongly continuous semigroup on a Banach space with generator (Infinitesimal generator of a C0-semigroup). For every and every the Bochner integral belongs to , and . The integral is taken in the sense of Bochner-integrable function; the integrand is continuous, hence Bochner integrable on .
Facts & Assumptions
Given: Countable Choice; A strongly continuous semigroup on a Banach space with generator (Strongly continuous semigroup, Infinitesimal generator of a C0-semigroup), and , .
The generator is defined by and that limit (Infinitesimal generator of a C0-semigroup).
Every orbit is continuous on , is linear and bounded, and for (Strongly continuous semigroup, A bounded linear operator between normed spaces).
Average convergence (Average convergence for a continuous Banach-valued function): a continuous curve on a compact interval is Bochner integrable, and its forward and backward averages converge to its value at the point.
Linearity of the Bochner integral over measurable sets (Linearity of the Bochner integral, Bochner-integrable function), including additivity for adjacent subintervals via indicators.
The Bochner integral is defined through integral-norm limits of integrable simple functions, and a strongly measurable function is Bochner integrable exactly when the integral of its norm is finite (Bochner-integrable function, Bochner integrability criterion); the norm inequality holds (Bochner integral norm inequality).
Bounded linear maps commute with Bochner integration: if and is Bochner integrable, then (Bounded linear maps commute with Bochner integration).
Lebesgue measure and measurability are translation invariant (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).
Proof
The orbit is continuous on the compact interval by [F2], hence Bochner integrable there by [F3]; therefore is a well-defined element of .
By [F6] applied to and [F2], .
Shift identity for continuous integrands: if is continuous on , then . For an integrable simple function the identity holds termwise, since translating back by gives intersected with , a set of the same measure by [F7]; for a nonnegative measurable function it follows by taking the supremum of the pairings of dominated simple functions, and for a Bochner integrable it follows by applying the scalar case to the nonnegative integrable and the simple case to along a defining approximation with [F5]. A continuous on the compact interval is Bochner integrable by [F3].
Adding the identity of [F4], [steps 1.2 and 1.3] give .
Dividing by and applying the average-convergence limits of [F3] to the continuous orbit at the points and (where ) yields .
By the definition of the generator [F1], the convergence of these right difference quotients means exactly that and .
Since and were arbitrary, for every and every the integral lies in and ; at both sides are .
The generator is closed and densely defined
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a strongly continuous semigroup on a Banach space with generator (Infinitesimal generator of a C0-semigroup). Then is a closed linear operator and is dense in (Densely defined, closed and closable operators, and cores).
Facts & Assumptions
Given: Dependent Choice; A strongly continuous semigroup on a Banach space with generator (Strongly continuous semigroup, Infinitesimal generator of a C0-semigroup).
Time integrals of orbits lie in the generator domain: for every and the Bochner integral satisfies and (Time integrals of semigroup orbits lie in the generator domain, Bochner-integrable function).
Average convergence (Average convergence for a continuous Banach-valued function): a continuous curve on a compact interval is Bochner integrable, and as for every ; hence for every .
The semigroup is locally bounded and all orbits are continuous, and as for every (A semigroup with continuity at zero is uniformly bounded on every compact time interval, Continuity at time zero implies continuity of every orbit); the norm inequality for Bochner integrals bounds . For the orbit is differentiable with derivative (The generator commutes with the semigroup on its domain), so the fundamental theorem of calculus for Banach-valued continuous curves gives (Fundamental theorem of calculus for Banach-valued continuous curves).
Closedness and density of a linear operator are the graph and domain conditions of Densely defined, closed and closable operators, and cores with the operator vocabulary of Unbounded linear operators: domain, graph and extension.
Proof
Density: for and , [F1] gives , and by [F2] as ; hence lies in the closure of . Since was arbitrary, is dense in .
Closedness: suppose with and in . For fixed and every , the orbit of is differentiable with derivative , so [F3] gives .
As , the left-hand side tends to , because is bounded and the orbit of is continuous; the right-hand side tends to , because by the local bound [F3]. Hence for every .
Dividing by and using the average-convergence limit of [F2] for the continuous curve gives as . By the definition of the generator, and ; hence the graph of contains the limits of all convergent graph sequences, and under DC (hence Countable Choice) the closure-sequence criterion in Infinitesimal generator of a C0-semigroup makes the graph closed.
Together with [step 1.1], the generator of a strongly continuous semigroup is a closed and densely defined linear operator, in the sense of [F4].
Resolvent and spectrum of a closed operator on a Banach space
Definition
Let be a Banach space over (Banach space, Real and complex scalar conventions for normed spaces) and let be a closed linear operator: is a linear subspace and is closed in with norm . This extends the Hilbert-space vocabulary of Unbounded linear operators: domain, graph and extension and Densely defined, closed and closable operators, and cores to Banach spaces. A scalar belongs to the resolvent set if is bijective and its inverse is bounded on . Under Dependent Choice, boundedness of the inverse follows from bijectivity by Closed graph theorem; thus under DC this is equivalent to the bijectivity-only convention. For the bounded operator is the resolvent, and is the spectrum. One has , for , and for ; in particular . This is the Banach-space form of the Hilbert-space vocabulary Resolvent and spectrum of an unbounded operator; the shift convention is the same, with the closed graph theorem replacing the Hilbert-space bounded-inverse convention.
Boundedness under DC. Assume Dependent Choice. If and is bijective, then is a closed operator: its graph is the image of the graph (Unbounded linear operators: domain, graph and extension) under the homeomorphism of with inverse . Hence has closed graph and is everywhere defined, so the closed graph theorem (Closed graph theorem, which assumes DC) makes it bounded; this is the one place where an axiom beyond ZF enters, and it is the DC carried by the cited theorem. The Banach-space vocabulary is Banach space with the scalar convention of Real and complex scalar conventions for normed spaces; boundedness and the space are those of A bounded linear operator between normed spaces.
Elementary identities. Let and . Since is the inverse of the bijection , its range is : . It satisfies The first identity says and the second says . Reading the second identity as and rearranging gives that is as everywhere-defined operators ; in particular even though itself need not be bounded. The same identities hold for every , and collects the scalars for which fails to be bijective or has an unbounded inverse. Under DC, the latter possibility is excluded by the closed graph argument above. When is a Hilbert space this is the Banach-space form of Resolvent and spectrum of an unbounded operator, with the same shift convention ; the closed graph theorem replaces the Hilbert-space convention that the resolvent is bounded by definition.
Resolvent identity for closed operators
Statement
Let be a closed linear operator on a Banach space , with resolvent on (Resolvent and spectrum of a closed operator on a Banach space). For all , and consequently .
Facts & Assumptions
Given: A closed linear operator on a Banach space , scalars , and the resolvents (Resolvent and spectrum of a closed operator on a Banach space).
satisfies , for and for ; the same holds with in place of (Resolvent and spectrum of a closed operator on a Banach space). Ranges of resolvents lie in , and composition with the bounded maps is associative and bilinear wherever defined (A bounded linear operator between normed spaces, Unbounded linear operators: domain, graph and extension).
Proof
For the vector lies in by [F1], so and are defined and differ by ; applying the bounded linear map and using linearity gives .
The first term on the right equals , because for and ; the second equals , because on . Hence , that is .
Since was arbitrary, .
Interchanging and gives ; adding the two identities yields . For this gives , and for the equality is trivial.
Laplace transform formula for the resolvent
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a strongly continuous semigroup on a Banach space with generator , and let , satisfy for all (Exponential bound for a C0-semigroup). Then for every real : ; for every the improper Bochner integral converges in ; and In particular .
Facts & Assumptions
Given: Dependent Choice; A strongly continuous semigroup on a Banach space with generator , constants , with , and a real ; for , .
The exponential bound and the Bochner framework: continuous curves on compact intervals are Bochner integrable, , and is complete (Exponential bound for a C0-semigroup, Bochner-integrable function, Bochner integrability criterion, Bochner integral norm inequality).
The generator is closed and densely defined (The generator is closed and densely defined), for the orbit is differentiable with and (The generator commutes with the semigroup on its domain), and the fundamental theorem of calculus holds for continuous curves with continuous derivative (Fundamental theorem of calculus for Banach-valued continuous curves, Infinitesimal generator of a C0-semigroup).
Linearity of the Bochner integral (Linearity of the Bochner integral), and average convergence for continuous curves (Average convergence for a continuous Banach-valued function).
Resolvent vocabulary: consists of the scalars with bijective and bounded inverse, and then (Resolvent and spectrum of a closed operator on a Banach space).
Proof
Convergence and bound: for the curves are continuous, hence Bochner integrable on compacts, and [F1] gives as . Thus is Cauchy for every , so exists, and the same estimate at gives ; the map is linear.
For the curve is differentiable with by [F2]. The pair curve is continuous with values in the closed graph , so its Bochner integral lies in : approximate the pair uniformly on by step functions sampled at partition points. Each simple integral is a finite linear combination of graph vectors, hence lies in the graph; the coordinate integrals converge by the norm inequality, and closedness retains their limit. Thus consequently and .
Hence, for , ; more explicitly by the fundamental theorem of calculus [F2], and as .
Passing to the limit in the identity of [step 2.1]: and ; since is closed (hence is closed), the pair limit gives and for every .
Likewise for every , by the same computation as [step 2.1]; note that is a fixed vector to which the definition of applies.
is injective: if for , then by [step 3.2]. It is also surjective: for choose with (density, [F2]); then converges to and , so closedness of gives and .
Therefore is bijective with inverse , which is bounded with ; hence and as an improper Bochner integral, with . As was arbitrary, .
Resolvent power estimates for semigroup generators
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a strongly continuous semigroup on a Banach space with generator and let , satisfy (Exponential bound for a C0-semigroup). Then for every real and every integer , the integral converging absolutely, and
Facts & Assumptions
Given: Dependent Choice; A strongly continuous semigroup on a Banach space with generator and (Exponential bound for a C0-semigroup); real and an integer .
Laplace formula: for real , and for every , with (Laplace transform formula for the resolvent).
Resolvent identity: (Resolvent identity for closed operators).
Bochner-integral toolkit: linearity, the norm inequality, and the fact that continuous curves on compact intervals are Bochner integrable; improper integrals of -dominated curves converge by the usual Cauchy estimate (Linearity of the Bochner integral, Bochner integral norm inequality, Bochner-integrable function).
Proof
For the integrand is dominated in norm by ; for a fixed the tails satisfy as , which is the uniform-in- domination used below.
The function is differentiable on with derivative : by [F2], , and in operator norm as because and is bounded near by [F1].
The same derivative computed from the integral is : the difference quotient is , whose integrands converge pointwise to and are dominated by for small and ; splitting the integral at and using uniform convergence on for the mean-value estimate and the tail estimate of [step 1.1] passes the limit through the improper integral.
Comparing [step 1.2] and [step 2.1]: for every , and the integral converges absolutely.
Induction on gives : the case is [F1] and the case is [step 3.1]. Assume the formula for ; since the resolvents commute, by [step 1.2], while differentiating the integral representation in (the same tail-splitting argument as [step 2.1], with domination with and ) gives . Equating the two expressions yields the formula for .
Norm bound: by [F3] and , , the scalar integral is : integration by parts on gives after , with and vanishing polynomial-exponential boundary terms. For all nearby difference quotients use a strictly smaller parameter ; polynomial times is integrable by the same recurrence. Thus taking the supremum over gives .
The exponential series of a bounded operator
Statement
Let be a real or complex Banach space and let (A bounded linear operator between normed spaces). For define . Then the series converges absolutely in the operator norm of , uniformly for in compact subsets of ; with ; and for all ; is of class in the operator norm with and , so that in operator norm as . In particular is a uniformly continuous (hence strongly continuous) group of bounded operators on whose generator is the bounded operator .
Facts & Assumptions
Given: A real or complex Banach space , an operator , and for and the partial sums and the series .
The operator norm is a norm on (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, The operator norm is a norm on the space of bounded linear operators, A bounded linear operator between normed spaces), and is complete for it because is a Banach space (If (Y) is Banach then (\mathcal B(X,Y)) is Banach).
Composition in is associative and bilinear, is its identity, and for all (Composition satisfies |ST|\le|S|,|T|); consequently for every . Completeness of for the operator norm [F1] together with these facts is all the structure used below; no separate Banach-algebra packaging is needed, and the estimates are identical over and .
In a Banach space a series converges whenever it converges absolutely, i.e. whenever the series of norms converges; its partial sums are then Cauchy (Series criterion for Banach spaces, An absolutely convergent series has Cauchy partial sums, Series and absolute convergence in a normed space).
For real the exponential series satisfies , and (The real exponential function and the number by a power series, The exponential is a continuous bijection from onto ); by for , its tail obeys .
Proof
The series converges absolutely for every real : by [F2] the general term obeys , so with the comparison series is the scalar exponential of [F4] and converges.
Cauchy-product step. If and converge absolutely in and , then converges absolutely with sum . Indeed by [F2], so converges by [F3]. Writing , , , the product differs from only by the terms with or , so by [F2] and absolute convergence; since by continuity of the product, the subsequence converges to that product, and a subsequence of a convergent sequence has the same limit, so .
Hence is defined for every by [F3], the family of series is dominated by the convergent scalar series on every compact interval , so the convergence is uniform there and in particular is continuous in operator norm; and .
Applying [step 1.2] to and , whose series converge absolutely by [step 1.1], gives , where and the binomial theorem in the commutative subalgebra generated by were used.
For and , the binomial expansion gives ; subtracting the two absolutely convergent series and using , the difference quotient obeys , which tends to as ; the rearrangement of the nonnegative double series is legitimate and the tail estimate is [F4].
At all terms with vanish, so ; and since multiplication is continuous in the operator norm by [F1] and [F2], the product of the partial sums converges, which is what the next steps quantify.
The quadratic remainder is by [F2] and the tail bound of [F4], since for .
Therefore is differentiable on with ; since commutes with every power , continuity of multiplication and [step 2.1] give as well. Iterating, if is times differentiable with , then is differentiable with derivative because is bounded; hence for all , that is is with .
Taking in [step 3.2] gives , so the difference quotients of at converge to in operator norm; consequently is a uniformly continuous group: by [step 2.2] and , so each is invertible with inverse , and is norm continuous on by [step 2.1], hence strongly continuous, and the difference-quotient limit at identifies its generator with the bounded operator .
Notes. The same estimates give and show the series converges in operator norm uniformly on compact -intervals; nothing here uses a choice principle, and the zero space is included through the estimates , .
The Yosida resolvent converges strongly to the identity
Statement
Let be closed and densely defined on a Banach space , and let , be such that and for all real and every (Resolvent and spectrum of a closed operator on a Banach space). Then, as along the reals, for every , and for every .
Facts & Assumptions
Given: A closed densely defined operator on a Banach space with and for all real and (Resolvent and spectrum of a closed operator on a Banach space).
Resolvent identities: , for , and for ; in particular for , since (Resolvent and spectrum of a closed operator on a Banach space).
First power estimate: for (Resolvent power estimates for semigroup generators is the source of this estimate in the semigroup case; here it is assumed directly).
is dense in , and operators of the form are bounded (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).
Proof
For and , [F1] gives , hence by [F2] as .
The operators are uniformly norm bounded for large : for when , and for and .
For arbitrary : given , choose with by density [F3]; by [step 1.1] choose with for ; then for such , . Hence for every .
The second statement is [step 2.1] applied to the vector : ; by [F1] and [F2] this is the same as for .
Yosida approximants
Definition
Let be closed and densely defined on a Banach space with , and let (Resolvent and spectrum of a closed operator on a Banach space). For real the Yosida approximant of at is The equality with uses ; in particular is a bounded everywhere defined operator. Distinct approximants commute, , and each commutes with .
The two formulae agree. Since , the resolvent identity of Resolvent and spectrum of a closed operator on a Banach space gives and the right-hand side is a sum of bounded everywhere defined operators (A bounded linear operator between normed spaces). Hence each is bounded with , and it is meant as a bounded approximation of the possibly unbounded ; the sense in which for is a theorem proved on this page, not part of the definition.
Commutativity. For real the resolvents commute, , by Resolvent identity for closed operators. Therefore is symmetric in and : the only mixed term that is not obviously symmetric is , and that product is symmetric by the resolvent identity. Hence distinct approximants commute, and for the same reason equals , since is the product of with bounded operators and resolvents at and commute. The closedness and density of are hypotheses, not conclusions drawn from a generator theorem. The resolvent vocabulary is Resolvent and spectrum of a closed operator on a Banach space, and the definition itself asserts no approximation property.
Yosida approximants are bounded and converge on the domain
Statement
Under the hypotheses of The Yosida resolvent converges strongly to the identity, let be the Yosida approximants (Yosida approximants). Then for , and as for every .
Facts & Assumptions
Given: The hypotheses of The Yosida resolvent converges strongly to the identity on the closed densely defined operator , and the Yosida approximants (Yosida approximants).
and for all , for (the bound is a hypothesis of The Yosida resolvent converges strongly to the identity, which proves both convergence conclusions).
For one has : both equal by the resolvent identity (Resolvent and spectrum of a closed operator on a Banach space).
Proof
By [F1] and [F2], for , which is the asserted bound.
For , by [F3]; by [F2] the right-hand side converges to as . Hence for every .
Both conclusions hold for every , and no uniform convergence on all of or on bounded sets is claimed.
Bounded Yosida semigroups converge to the generated semigroup
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be closed and densely defined on a Banach space and let , satisfy and for all real , . Let be the Yosida approximants (Yosida approximants) and let be the bounded-operator exponentials of The exponential series of a bounded operator. Then for every the limit exists, uniformly for in compact subsets of , and is a strongly continuous semigroup on with and generator .
Facts & Assumptions
Given: Dependent Choice; A closed densely defined operator on a Banach space with and for all real and ; the Yosida approximants and the exponentials of The exponential series of a bounded operator (Yosida approximants, Resolvent and spectrum of a closed operator on a Banach space).
Exponential series: converges in operator norm, , , , is with , so for every by the fundamental theorem of calculus (The exponential series of a bounded operator, Fundamental theorem of calculus for Banach-valued continuous curves).
The approximants satisfy the norm bound of Yosida approximants are bounded and converge on the domain and for every ; distinct approximants commute and so do the exponentials (Yosida approximants, Composition satisfies |ST|\le|S|,|T|).
is closed and is dense by hypothesis; since and commutes with , the binomial Cauchy-product argument in the exponential-series proof, applied to the commuting bounded operators and , gives , so the resolvent power estimates yield for . [F1, F2]
Average convergence and strong continuity: a continuous curve is Bochner integrable and its forward averages converge to its value (Average convergence for a continuous Banach-valued function); continuity at plus the semigroup law gives continuity of every orbit (Continuity at time zero implies continuity of every orbit, Strongly continuous semigroup).
Laplace formula: a strongly continuous semigroup with has (Laplace transform formula for the resolvent).
Proof
Uniform bound. For and , [F3] gives . For every , uniformly on . Thus the displayed majorants converge uniformly to there; they are uniformly bounded for large , and for each fixed , regardless of the sign of .
Cauchy estimate on . For and , the exponentials commute and the FTC gives ; hence with finite by [step 1.1]. Since by [F2], the family is Cauchy, uniformly for in compact intervals.
The limit and its bound. For and close to , , and the first two terms are small uniformly in and in compacts by [step 1.1] while the last is small by [step 2.1]; density [F3] gives convergence uniformly on compact -intervals for every . The limit orbit is continuous on each compact interval: for any point, bound its increment by the two uniform approximation errors and the increment of one continuous approximating orbit. Define ; then is linear and bounded with by [step 1.1].
Semigroup law. For and , by [step 3.1] and the uniform bound on compacts, while ; hence , and .
Strong continuity. For and in a compact interval, , and the two integrals tend to and uniformly, because strongly uniformly on the interval by [step 3.1] and ; hence as by average convergence [F4]. With the local bound of [step 3.1] this extends from the dense domain to all , so as ; by the semigroup law and [F4] every orbit is continuous, so is a strongly continuous semigroup with bound .
The generator contains . The identity of [step 4.2] shows for ; dividing by and using average convergence for the continuous curve gives . Hence and for , where is the generator of .
. By [F5] applied to and its bound, every real lies in ; it also lies in by hypothesis, and on . Thus and are both bijections agreeing on : given , for some , and since while is injective, ; hence and .
Hille-Yosida generation theorem
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a closed and densely defined linear operator on a Banach space and let , . Then generates a strongly continuous semigroup with for all if and only if both: (i) , and (ii) for every real and every (Resolvent and spectrum of a closed operator on a Banach space). All resolvent powers are required in general; the first power alone guarantees all power estimates when , the exponentially rescaled contraction case, where all higher power estimates follow from the single one by submultiplicativity (treated later on this page).
Facts & Assumptions
Given: Dependent Choice; A closed densely defined linear operator on a Banach space and constants , (Resolvent and spectrum of a closed operator on a Banach space, The generator is closed and densely defined).
Sufficiency: under (i) and (ii) for all real , , the operator generates a strongly continuous semigroup with (Bounded Yosida semigroups converge to the generated semigroup).
Necessity of the location of the spectrum and of the first estimate: if is a strongly continuous semigroup with generator and , then is closed and densely defined, , and (The generator is closed and densely defined, Laplace transform formula for the resolvent, Strongly continuous semigroup).
Necessity of all powers: under the hypotheses of [F2], and for every (Resolvent power estimates for semigroup generators).
Proof
(Sufficiency.) Assume (i) and (ii). Then all hypotheses of [F1] hold, so generates a strongly continuous semigroup with for all .
(Necessity, domain and spectrum.) Assume conversely that generates with . Then is closed with dense domain by [F2]; the Laplace-transform formula for the resolvent gives and for every real ; in particular , which is (i).
(Necessity, all powers.) Under the same hypothesis, [F3] gives the integral representation of every power and the estimate for all and real , which is (ii).
Combining [step 1.1] with [steps 1.2-1.3]: generates a strongly continuous semigroup with if and only if (i) and (ii) hold. The general theorem retains all power estimates; the case where the first estimate alone suffices (, ) is isolated as the next corollary.
Contraction Hille-Yosida theorem
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be closed and densely defined on a Banach space . Then generates a strongly continuous semigroup of contractions ( for all ) if and only if and for all , equivalently for all . In this case the first-power estimate implies all the power estimates by submultiplicativity, so no separate power condition is needed; when is complex, every with belongs to and for every .
Facts & Assumptions
Given: Dependent Choice; A closed densely defined operator on a Banach space (Hille-Yosida generation theorem).
Hille-Yosida with , : generates a strongly continuous semigroup with for all if and only if and for every real and every (Hille-Yosida generation theorem).
The operator norm is submultiplicative, (Composition satisfies |ST|\le|S|,|T|), and is a norm on (The operator norm is a norm on the space of bounded linear operators).
The complex exponential satisfies and (, , and , The complex exponential is entire and its complex derivative is itself). The Laplace inverse argument and resolvent differentiation for real parameters are proved in Laplace transform formula for the resolvent and Resolvent power estimates for semigroup generators; their complex extension is derived in step 2.1.
Proof
Suppose generates a contraction semigroup, . Then [F1] with , gives and for all ; in particular the first-power estimate , equivalently , holds for all .
Conversely, suppose and , i.e. , for all . By submultiplicativity [F2], for every ; hence the power conditions of [F1] hold with , , and generates a strongly continuous semigroup of contractions.
Let be complex, , and . By [F3] its tail norm is at most . For , integration of the derivative of , with the closed-graph integration argument of the Laplace theorem, gives ; density and closedness extend the first identity to every , exactly as in that theorem, so and . For small real , the resolvent identity gives in operator norm. Differentiating the integral along this real increment is justified by splitting off its tail and dominating by for each needed order; induction gives . The scalar integral of the norm majorant is , by the integration-by-parts recurrence of the power-estimate proof. Hence , the precise complex half-plane estimate.
Therefore contractivity of the generated semigroup is equivalent to and for all , and in this case the single estimate forces all the resolvent power bounds.
Dissipative operator
Definition
Let be a Banach space over and let be a linear operator with domain (Unbounded linear operators: domain, graph and extension). is dissipative if equivalently for all and . A dissipative operator has injective for every and on the range of ; no surjectivity, closedness or density is implied. If is a Hilbert space (Hilbert space), then is dissipative if and only if for every : from the norm dissipativity inequality, squaring gives for every , so letting yields . Conversely, if this real-part inequality holds, expanding gives the norm dissipativity inequality. Finally, under the Hahn-Banach extension principle HB (The real dominated-extension principle as an additional hypothesis over ZF) dissipativity is equivalent to the norm-duality form: for every there exists with , and ; the equivalence is the two-dimensional argument of [T] Lemma 11.19, where HB produces the norming functionals and supplies the extension from (Relative dual norming, point separation, and recovery of the norm), and the extraction of the limit uses compactness of the finite-dimensional dual unit ball (For every bounded sequence in has a convergent subsequence).
Two normalisations of the defining inequality. Putting shows that the displayed inequality is equivalent to for all and . Since , the inequality is in turn equivalent to the one-sided estimate used below.
Injectivity and the inverse bound. If for some and , then , so : each is injective. If lies in the range, then , so the inverse defined on the range satisfies . No surjectivity onto , no closedness of and no density of is asserted, and none is implied.
The real-part form on a Hilbert space. Let be a Hilbert space. If is dissipative and , then for every the expansion gives , that is ; letting yields . Conversely, if for all , the same expansion gives for every , so is dissipative. This real-part form is choice-free; the Hilbert-space vocabulary comes from Hilbert space.
The norm-duality form under HB. Assume the Hahn-Banach extension principle (The real dominated-extension principle as an additional hypothesis over ZF) and let . We claim that is dissipative if and only if for every there is with , and .
Sufficiency. If such is given and , then For this is ; for it is trivial. Hence is dissipative.
Necessity. Fix ; for take , so assume and put , a subspace of finite dimension at most two over the scalar field . For each positive integer the vector is nonzero, because is injective. Construct a sequence without simultaneously choosing functionals on . Fix a basis of the finite-dimensional space . The coordinate vectors of functionals of norm at most one form a closed bounded subset of a finite real coordinate space: for every is an intersection of closed conditions, and each basis evaluation is bounded. For each , intersect this set with . The intersection is nonempty by Relative dual norming, point separation, and recovery of the norm applied to , and compact by Heine-Borel in : with the Euclidean metric a subset of is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. Select its lexicographically least coordinate vector by minimizing its real coordinates successively; each minimum exists because the corresponding projected compact set is nonempty. This finite deterministic procedure defines for all , with norm one and the required norming identity, without Countable Choice. Write , , below. Then and likewise , where and ; hence .
Passing to a subsequence. The restrictions lie in the unit ball of the dual of the finite-dimensional space , which is sequentially compact: after choosing coordinates for , the coordinates of a functional amount to a bounded sequence in a Euclidean space, and For every bounded sequence in has a convergent subsequence extracts a convergent subsequence. Take along a subsequence on which converges to some . Then , (a closed condition), and ; consequently because with .
Extension to . The real part is a real-linear functional on the real vector space with for ; here the real structure of is the one underlying the complex case as well. Apply HB, with the sublinear functional , to extend to a real-linear satisfying for all ; then by applying the inequality to . In the real case set . In the complex case set ; then ; together with real linearity this proves complex linearity, and its real part is . For put . If , take , so and is real. Thus ; if the same bound is immediate. Thus in either case, and satisfies because , having real part and modulus at most , equals the positive real number . Finally , as required. The definition and both elementary forms are choice-free; only the norm-duality form uses HB and the finite-dimensional compactness above.
Lumer-Phillips generation theorem
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a densely defined dissipative operator on a Banach space (Dissipative operator). Then the following are equivalent: (a) generates a strongly continuous semigroup of contractions; (b) for some ; (c) for every . In that case is closed, , for all , and is maximal dissipative (it has no proper dissipative extension).
Facts & Assumptions
Given: Dependent Choice; A densely defined dissipative operator on a Banach space (Dissipative operator, Strongly continuous semigroup).
Dissipativity means for all and ; hence each is injective and on the range of (Dissipative operator).
Contraction Hille-Yosida: a closed densely defined operator with and for all generates a strongly continuous semigroup of contractions; conversely the generator of a contraction semigroup has and (Contraction Hille-Yosida theorem).
Resolvent identity: (Resolvent identity for closed operators). Consequently, for the series converges in and its sum is the inverse of : writing and , one has and, on , . Thus is a two-sided inverse and has range in . In the Neumann series , the operators commute with by taking limits of polynomials, giving the displayed series. Indeed, for , is complete for the operator norm (If (Y) is Banach then (\mathcal B(X,Y)) is Banach) with submultiplicative composition (Composition satisfies |ST|\le|S|,|T|, The operator norm as the least bound and as the unit-sphere or unit-ball supremum), so the Neumann-series computation applies — for complex through Neumann series and the unital Banach-algebra structure (Unital Banach algebra), and for real by the identical telescoping computation. An operator with a bounded everywhere-defined inverse is closed: the inverse graph is the zero set of the continuous map , and swapping graph coordinates gives the graph of the original operator. A scalar shift of its graph is a homeomorphism. Hence is closed as soon as has such an inverse; a closed bijective operator with bounded inverse lies in the resolvent set (Resolvent and spectrum of a closed operator on a Banach space).
Operators of the form with are closed when is closed, and is closed as soon as some has a bounded everywhere-defined inverse; generators are closed and densely defined (The generator is closed and densely defined, Resolvent and spectrum of a closed operator on a Banach space).
Proof
If then and all claims hold for the unique zero operator and semigroup; hence assume . (b) closedness, . Assume for some . By [F1] is injective with for all ; thus its inverse is a bounded everywhere-defined operator and is bijective, so and is closed by [F4].
(a)(b),(c). If generates a contraction semigroup, [F2] gives with ; therefore every , , is bijective onto , which is (c) and, taking e.g. , also (b). Trivially (c)(b).
Propagation to . With and , the series of [F3] converges for and represents ; hence . Dissipativity now gives for every by [F1], in particular on . Replacing by any , the same argument gives with ; iterating with the explicit points , each inside the previous interval , gives for all . Since , this yields and for all . In particular is surjective for every , so (c) holds.
(b)(a). is closed and densely defined by hypothesis and [step 1.1]; [step 2.1] supplies and ; hence [F2] makes the generator of a strongly continuous semigroup of contractions.
Maximal dissipativity. Let be a dissipative extension and fix . By [step 2.1], maps onto ; given , choose with . Since agrees with on , , and injectivity of by [F1] gives . Hence and : has no proper dissipative extension.
Combining the implications: (a), (b) and (c) are equivalent for a densely defined dissipative operator, and in that case is closed, , , and is maximal dissipative.
The variation-of-constants integral is continuous for integrable forcing
Statement
Assume Countable Choice (The Axiom of Countable Choice ()) for Lebesgue time integration. Let be a strongly continuous semigroup on a Banach space with constants , and (Exponential bound for a C0-semigroup). Let and let be Bochner integrable with (Bochner-integrable function). Then is a well-defined element of , the map is continuous on , and for a constant depending only on and the local bound of on .
Facts & Assumptions
Given: Countable Choice; A strongly continuous semigroup on a Banach space with for some , (Exponential bound for a C0-semigroup); ; a Bochner integrable with ; and .
The exponential bound makes finite, since . [thm-exponential-bound-for-a-c-zero-semigroup]
Bochner integrability of supplies integrable simple functions with arbitrarily small and a strong-measurability approximation (Bochner-integrable function, Strongly measurable Banach-valued function); a strongly measurable with is Bochner integrable (Bochner integrability criterion).
Linearity of the Bochner integral and the norm inequality (Linearity of the Bochner integral, Bochner integral norm inequality, Bochner-integrable function).
Absolute continuity of the scalar integral: for every there is with whenever (Absolute continuity of the integral).
The orbit map of every vector is continuous on and strongly: for every (Strongly continuous semigroup).
Proof
Fix . For each measurable simple approximation to , the map is strongly measurable: for each of its finitely many values , approximate the continuous curve uniformly by step functions on equal partitions of , then multiply by . Choose a partition size by its least integer giving error below on all finitely many curves. The resulting measurable simple function approximates uniformly within . As off a null set and , these approximants converge pointwise there to . Its norm is bounded by , so [F2] gives Bochner integrability and [F3] gives .
Claim: as . Given , choose an integrable simple function with by [F2]; then , and the finite sum tends to as because for each by [F5]. Hence , and was arbitrary.
Increment splitting: for , linearity [F3] and the semigroup law give , where in the last term .
Taking norms in [step 2.1] and using from [F1] and the norm inequality [F3]: as , the first term by absolute continuity [F4] and the second by [step 1.2]. The backward increment is bounded by by the same splitting with in place of , hence also tends to .
Continuity at the endpoints: by [F3], [F4], and at the backward bound of [step 3.1] applies; hence is continuous on the closed interval , and the estimate of [step 1.1] is the stated bound with .
The claims of the statement follow: is well defined, continuous on , and bounded by ; no compactness of the range of and no choice beyond the declared Bochner framework was used.
Classical, strong and mild abstract Cauchy solutions
Definition
Assume Countable Choice (The Axiom of Countable Choice ()) for the Lebesgue time integrals. Let be a linear operator on a real or complex Banach space , with the Banach graph conventions of Infinitesimal generator of a C0-semigroup. Let , and let be Bochner integrable (Bochner-integrable function). Consider the abstract Cauchy problem for , ; endpoint equations are imposed only when is continuously extended to . Derivatives use the underlying real structure, with one-sided derivatives at the endpoints. (1) A classical solution is a function with for every , , for and ; here membership in is with the graph norm and the derivative is the Fréchet derivative of a curve (Fréchet derivative between Banach spaces). (2) A strong solution is a continuous with for all , , and , such that for every . (3) When generates a strongly continuous semigroup satisfying an exponential norm bound, a mild solution (variation-of-constants solution) is the continuous function given by the convergent Bochner integral which is well defined by The variation-of-constants integral is continuous for integrable forcing. (4) An integral (integrated) solution is a continuous with and for all ; for this is the integrated form of the homogeneous problem. These are four formulations, with overlaps and equivalences under additional hypotheses: neither differentiability nor membership of in is asserted by the mild definition, and the definitions do not assign a derivative to a mild solution.
Here “strong solution” means the graph-continuous integral notion explicitly stated above; some sources use that term for an almost-everywhere differential notion instead. Engel–Nagel II.6.3 calls the homogeneous integral formulation “mild”, whereas the variation-of-constants terminology here follows Schnaubelt Definition 2.11. The mild and integral formulations coincide for generators by the variation-of-constants theorem; they are not asserted to be different classes. Under DC the exponential bound required for the mild formulation is automatic, by Exponential bound for a C0-semigroup.
Classical solution. A classical solution is on the closed interval and takes values in the domain of at every time, including the endpoints; the equation is required pointwise on , and the derivative is the Fréchet derivative of the curve (Fréchet derivative between Banach spaces). Since is generally unbounded, membership and continuity of are genuine restrictions. Together with continuity of , they mean graph-norm continuity in (Unbounded linear operators: domain, graph and extension).
Strong solution. A strong solution need not be differentiable; instead is continuous, and the equation is imposed in integrated form , which makes sense because is continuous and is Bochner integrable (Bochner-integrable function). If on extends continuously to , this identity gives , with in the interior and one-sided endpoint derivatives equal to the extension values, by Fundamental theorem of calculus for Banach-valued continuous curves. Thus strong and classical solutions coincide when extends continuously to ; continuity only on does not ensure endpoint derivatives.
Mild solution. The mild solution is the explicit variation-of-constants function which is a well-defined continuous -valued function on by The variation-of-constants integral is continuous for integrable forcing; here need only be Bochner integrable, and no differentiability, no membership of in and no pointwise equation are asserted.
Integral (integrated) solution. An integral solution replaces differentiability by a weaker regularity: the primitive lies in for every , and holds. For this is the integrated form of the homogeneous problem , ; its advantage is that it only evaluates on the primitive, which always lies in when is a mild solution of the homogeneous problem.
The definitions do not by themselves assert implications between the four notions beyond the elementary ones visible above, and they do not assign a derivative to a mild solution. The precise equivalence results under additional hypotheses, and the separation of the notions when those hypotheses fail, are proved as theorems and exhibited by counterexamples on the companion pages.
Well-posedness of the abstract Cauchy problem is equivalent to generation
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a closed linear operator on a Banach space and consider the homogeneous problem , . Let (EU) be the statement that for every there exists exactly one classical solution on (Classical, strong and mild abstract Cauchy solutions). Then the following conditions are equivalent: (a) generates a strongly continuous semigroup; (b) (EU) holds and ; (c) (EU) holds and there is a sequence with for every ; (d) (EU) holds, is dense, and for every sequence with one has uniformly for in compact subsets of . Condition (d) is the definition of well-posedness of the abstract Cauchy problem; it is existence plus uniqueness plus continuous dependence on the initial datum in the uniform topology on compact time intervals. If any (hence all) holds, then for the generated semigroup .
Facts & Assumptions
Given: A closed linear operator on a Banach space (Unbounded linear operators: domain, graph and extension, Densely defined, closed and closable operators, and cores), and the condition (EU) that for every there is exactly one classical solution of , on (Classical, strong and mild abstract Cauchy solutions). Write for the graph-norm space, with , and for a generator when it exists. The proof assumes Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain), carried by the closed graph theorem Closed graph theorem used in [F4]; the sequence selections in steps 1.3 and 1.4 are instances of Countable Choice, a consequence of DC.
Since is closed, its graph and the graph-norm space are Banach by the explicit Banach graph convention in Infinitesimal generator of a C0-semigroup. The operator is closed on : if and in , convergence in and closedness of give ; then and . This uses the assumed closedness, not a generator theorem.
For a generator the following hold: local boundedness on compact time intervals, orbit continuity, the Laplace formula for real whenever , hence , and is dense with closed (Exponential bound for a C0-semigroup, Laplace transform formula for the resolvent, The generator is closed and densely defined, A semigroup with continuity at zero is uniformly bounded on every compact time interval, Continuity at time zero implies continuity of every orbit).
For a strongly continuous semigroup generated by : , on , and (The generator commutes with the semigroup on its domain, Time integrals of semigroup orbits lie in the generator domain).
Closed graph theorem: an everywhere defined linear map between Banach spaces with closed graph is bounded (Closed graph theorem, under DC); consequently a closed operator whose graph-norm domain is complete has the properties used below.
Proof
Internal lemma (reduction to ), part 1: (EU) gives a semigroup on . Put for . Uniqueness makes linear and gives , ; the classical equation shows that is continuous into for each .
(a)(b),(c),(d). If generates a semigroup with : for the orbit is a classical solution and any classical solution satisfies by the rigidity computation , so (EU) holds; [F2] gives and for any in that half-line, so (c) holds and ; density and the local bound give, for in , , which is (d).
(d)(a). Define for ; uniqueness makes each linear, and the semigroup law holds on by uniqueness of solutions. The continuous-dependence hypothesis (d) transfers to a local bound: if no had for all with and all , then choosing with and would give a sequence with uniformly, contradicting (d); hence for all . Since is dense, each extends uniquely to a bounded operator on with the same bound, and the semigroup law and strong continuity extend by density, using on with from the semigroup law. The generator of the extension satisfies , because on the difference quotients are those of the classical solutions and converge to . The extension leaves invariant, so is a core of : for choose with ; then lies in the graph-norm closure of (the integrand is -valued and graph-norm continuous) and the integrated-orbits identity in graph norm as together with in graph norm shows . Since is closed, and is a core of , every graph limit from remains in , so .
Boundedness of . For any Banach space , is Banach in the supremum norm: a uniformly Cauchy sequence converges pointwise by completeness, uniformly by its common Cauchy estimates, and its uniform limit is continuous by the three-term increment estimate. Fix and consider , . Its graph is closed: if in and uniformly in , then the integral identity passes to the limit in and gives for ; the extension for , for , then solves (ACP) with initial value , so by uniqueness and . By [F4] is bounded on the Banach space , hence and is a strongly continuous semigroup on .
The generator of is . First for : the curve is differentiable with and satisfies (move inside the integral by the closed-graph argument in ), so by uniqueness and . Hence for the quotient converges to in and its -image converges to in ; that is, the convergence holds in , so . Conversely, if , then converges in and in ; closedness of gives , that is . Thus .
(b)(a). Let . For one has iff , and for : indeed and , and exactly when . Thus is a bounded isomorphism with bounded inverse , and with . By [step 3.1] and the internal lemma, generates on ; then is a strongly continuous semigroup on whose generator is , because the difference quotients of are those of conjugated by the bounded isomorphism .
(c)(b). By [step 3.1] the operator generates on (via [step 1.1] and [step 2.1]), so by [F2] its resolvent set contains a half-line ; choose with . If for some , then , so and ; since this forces . Hence is injective and, by hypothesis, surjective, so it is bijective; being closed it has bounded inverse by [F4], and .
All implications are established, so (a)-(d) are equivalent; and in each direction the solution is for the generated semigroup, as asserted.
Variation of constants for the inhomogeneous abstract Cauchy problem
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a strongly continuous semigroup on a Banach space with generator (Infinitesimal generator of a C0-semigroup), let , , and let be Bochner integrable with . Then: (1) (Duhamel rigidity) every classical solution of , on satisfies (2) The formula defines a continuous , which is the unique mild solution and the unique integral solution of the problem in the sense of Classical, strong and mild abstract Cauchy solutions: if is an integral solution, then . (3) (classical upgrade) If in addition and extends to either as a curve or in the form for some Bochner integrable , then is a classical solution: , for all , and pointwise. Mere continuity, or mere Lipschitz continuity on an arbitrary Banach space, is not asserted to give a classical solution. A Lipschitz curve is covered by (3) when it additionally has the displayed Bochner derivative representation.
Facts & Assumptions
Given: Dependent Choice; A strongly continuous semigroup on a Banach space with generator (Infinitesimal generator of a C0-semigroup); , , and a Bochner integrable with ; the continuous function (Classical, strong and mild abstract Cauchy solutions, The variation-of-constants integral is continuous for integrable forcing).
The formula defines a continuous -valued function on , and the norm inequality, linearity and Bochner framework of the integral apply; the exponential bound gives a local bound (The variation-of-constants integral is continuous for integrable forcing, Linearity of the Bochner integral, Bochner integral norm inequality, Bochner-integrable function).
For the orbit is differentiable with , , and the primitive of an orbit satisfies for every (The generator commutes with the semigroup on its domain, Time integrals of semigroup orbits lie in the generator domain); the fundamental theorem of calculus applies to continuous curves with continuous derivative (Fundamental theorem of calculus for Banach-valued continuous curves).
DC supplies the local operator bound by Exponential bound for a C0-semigroup. A continuous graph-valued curve has a graph-valued integral by sampled step approximation and closedness, as proved in Laplace transform formula for the resolvent. Uniform continuity of a continuous curve on a compact interval is Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous.
Proof
Duhamel rigidity. Let be a classical solution and fix . The curve on is differentiable: , where and [F2] was used. The classical conditions make continuous on , so agrees on the interior with this continuous extension. Its product with is continuous: an increment is bounded by , which tends to zero. Thus extends continuously to the endpoints, and by the fundamental theorem of calculus [F2], .
The formula is continuous and well defined. By [F1] is a well-defined continuous function on ; this is the mild solution of the problem in the sense of Classical, strong and mild abstract Cauchy solutions.
First take . Put . The needed exchange of vector integrals is justified directly: on the compact triangle , the curve is uniformly continuous. On a fine square grid approximate it uniformly by finitely valued functions sampled at points of the intersecting triangle cells, and multiply by . Scalar Fubini (Fubini's theorem for L^1 functions on a sigma-finite product) applies to each indicator coefficient. Both iterated integral errors are at most times the uniform approximation error by [F1], so exchange remains valid in the limit. Consequently . The latter integral lies in and its -image is : the pair is continuous by [F2], and sampled step approximations, multiplied by , have graph-valued integrals; closedness of retains the limiting pair. Thus . Finite linearity proves this for every integrable simple . For general , take defining simple with . The local bound gives , so both coordinates of the graph pair converge: and . Closedness proves the integral-solution identity for .
Uniqueness among integral solutions. Let be an integral solution of , , and put , which is continuous with and satisfies for all . For fixed define ; then is differentiable with by [F2] and the equation for , so is constant and (using and the primitive's value at ). Hence for every ; differentiating in with the fundamental theorem of calculus gives for all , so .
Assume and with Bochner integrable; the case is . Write (reflecting equal partitions under gives the same sampled sums, hence the same Bochner integral, for this continuous integrand). Substituting the primitive representation and exchanging the triangle integrals gives . This exchange follows by the same grid argument as step 1.3 for simple , and by approximation for general : both errors are at most . The integrand is continuous by [F1] applied to , so the FTC gives , continuously on . Since is by [F2], is .
Since is an integral solution by [step 1.3], subtraction gives for all ; at use the corresponding backward difference. Divide by : on the right, and by average convergence for the continuous , so the right side tends to ; on the left, by average convergence for the continuous . Since is closed, the limit pair lies in the graph of ; hence and , that is pointwise, and is a classical solution.
Claims (1), (2) and (3) are [step 1.1], [steps 1.2-1.4] and [steps 2.1, 3.1]; the classical upgrade holds for the stated or Bochner-primitive forcing.
Uniqueness of the scalar Laplace transform in the exponential-growth class
Statement
Assume Countable Choice (The Axiom of Countable Choice ()) for the Lebesgue-measure interfaces. Let and let be continuous with for some , and all . If the Laplace transform vanishes on a right half-line, then for every .
Facts & Assumptions
Given: Countable Choice; A real or complex-valued continuous with for some , and all , and for every real ; for the integrand is dominated by and the integral exists as a Lebesgue integral over .
If , is and injective with on a neighbourhood of , and the continuous function is defined on an interval containing , then (In one dimension the compact-Jordan formula is substitution over the unoriented image interval with the absolute derivative). This substitution is stated for Riemann integrals; on the compact intervals used below all its integrands are continuous, hence bounded and Riemann integrable, and A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral identifies those integrals with their Lebesgue integrals under Countable Choice.
Polynomials are uniformly dense in : for every continuous real on and there is a polynomial with (Polynomials are uniformly dense in ).
The Lebesgue integral is linear on and satisfies ; the integral over a measurable set is defined by restricting each real positive/negative and imaginary component, giving (The Lebesgue integral is linear on , The modulus of an integral is bounded by the integral of the modulus, The class of integrable functions, Integrable real and complex functions, and their integrals); Integral over a measurable subset alone supplies only the nonnegative convention.
Proof
It suffices to prove the theorem for real-valued : if is complex-valued, then and are continuous, satisfy the same bound , and by [F3] have and likewise for for every real .
Assume real. Fix and put and . Then is continuous with for , so ; moreover for every integer the number exceeds and .
Put for and . Then is continuous on : it is continuous on as a composition, and as because , matching ; also on , so and .
For and , [F1] applied on to and the continuous on gives .
Letting in [step 2.1]: the right-hand side tends to because its tail is bounded by ; the left-hand side tends to because the missing part satisfies ; by [step 1.2] the limits are , so for every integer .
Every continuous real on satisfies : fix and, by [F2], choose a polynomial with ; then by [F3], while by [step 3.1]; hence for every , so the integral vanishes.
The function vanishes identically on : otherwise for some with (or ), and by continuity there is an interval of positive length with on (respectively on ); choosing a continuous nonnegative bump supported in with gives , positive at and continuous, so (respectively ), contradicting [step 4.1].
Consequently for every , whence for every in the real case; the complex case follows by applying the real case to and as in [step 1.1].
Laplace uniqueness identifies two exponentially bounded semigroups
Statement
Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain) and the Hahn-Banach extension principle HB (The real dominated-extension principle as an additional hypothesis over ZF). Let and be strongly continuous semigroups on a Banach space with generators and and resolvents (Resolvent and spectrum of a closed operator on a Banach space), and suppose there are , with for all . If for every real , then for every . In particular two strongly continuous semigroups with the same generator coincide.
Facts & Assumptions
Given: Dependent Choice; The Hahn-Banach extension principle HB (The real dominated-extension principle as an additional hypothesis over ZF); strongly continuous semigroups , on a Banach space with generators and resolvents (Strongly continuous semigroup, Resolvent and spectrum of a closed operator on a Banach space); , with (Exponential bound for a C0-semigroup); and for every real .
Laplace formula: for real , lies in the resolvent sets of both generators and , (Laplace transform formula for the resolvent).
Bounded linear functionals and, more generally, bounded linear maps commute with Bochner integrals: (Bounded linear maps commute with Bochner integration, Bochner-integrable function).
Scalar Laplace uniqueness: a continuous scalar function with whose Laplace transform vanishes for every real is identically zero (Uniqueness of the scalar Laplace transform in the exponential-growth class).
Point separation and norming under HB: for every there is with and , so the dual separates points (Relative dual norming, point separation, and recovery of the norm).
Proof
Put for . For fixed and the scalar function is continuous and satisfies , because are strongly continuous and exponentially bounded.
For real , [F1] and [F2] give .
By scalar Laplace uniqueness [F3] applied with and , the continuous function vanishes identically: for every .
Since was arbitrary, the dual separates points of (using HB, [F4]), so for every and every ; that is, for all .
If moreover , then both semigroups have exponential bounds and, taking a common pair for the two bounds (for instance the maxima of the respective constants), their resolvents agree on because both are given by the Laplace formula for the same operator; [step 4.1] then gives .
Restriction to a closed invariant subspace is a C0-semigroup and its generator is the part
Statement
Assume Countable Choice (The Axiom of Countable Choice ()) for the Lebesgue-measure interfaces. Let be a strongly continuous semigroup on a Banach space with generator , and let be a closed linear subspace (Normed subspace, A closed subspace of a Banach space is Banach) such that for every . Then the restrictions form a strongly continuous semigroup on the Banach space , and its generator is the part of in : and . In particular the generator of the restricted semigroup is the restriction of to that domain.
Facts & Assumptions
Given: Countable Choice; A strongly continuous semigroup on a Banach space with generator (Strongly continuous semigroup, Infinitesimal generator of a C0-semigroup), and a closed linear subspace with for every .
A closed linear subspace of a Banach space is a Banach space for the restricted norm (A closed subspace of a Banach space is Banach, Normed subspace), and convergence in the norm of is the same as convergence in for vectors of .
The generator is defined by right difference quotients: exactly when converges as , and then the limit is (Infinitesimal generator of a C0-semigroup).
Time integrals of orbits lie in the generator domain: for and the Bochner integral satisfies and (Time integrals of semigroup orbits lie in the generator domain, Bochner-integrable function).
Proof
The restrictions are bounded linear maps of into itself by hypothesis, with and inherited from . For each the orbit is continuous into , because it is continuous into and the norm of is the restriction of the norm of by [F1]; hence is a strongly continuous semigroup on the Banach space .
Let denote the generator of . If and , then for all , so the difference quotients lie in and converge in to ; by [F1] they converge in to . Therefore and .
Conversely, if , then by definition in , hence also in by [F1]; the same vectors are the difference quotients of , so and . In particular , so .
The two inclusions give with , that is, the generator of is the part of in ; for reference, this domain is dense in , since for and the integral lies in with by [F3] and the -valued Bochner integral stays in the closed subspace , while as .
Semigroup sign and generator conventions
Statement
Generation and automatic exponential-bound results below are understood under the DC hypotheses of their supplier theorems.
This track consistently writes the abstract evolution equation as and defines the generator by ; the heat flow on a Dirichlet domain is therefore generated by , the Dirichlet Laplacian (the negative of the operator associated with the Dirichlet energy form), not by . Sources writing or use or generate : Brezis's maximal-monotone chapter is stated for with m-accretive, so its corresponds to here, and Pazy-type statements translate the same way. Resolvents are normalised as ; a source using has the opposite sign, and its resolvent equals at the same parameter, so the norms of all powers are unchanged. The contraction case corresponds to , ; boundedness () is not the same as contractivity, and the general generation theorem keeps all resolvent powers.
The convention of this track. The abstract evolution equation is written and the generator is defined by on its domain (Infinitesimal generator of a C0-semigroup). The resolvent is normalised as . If for all , then for every real the Laplace representation is (Resolvent and spectrum of a closed operator on a Banach space, Laplace transform formula for the resolvent); membership in alone does not guarantee convergence of this integral.
Translation dictionary. A source that writes the homogeneous equation as or is using the opposite sign: its equals in this track, and its solutions are in its own notation, that is here. In the same way a source whose resolvent is instead of has the opposite shift convention; at the same ; its th power is , so its norm is unchanged. Negating the operator itself is a separate change of generator sign.
Heat flow. With this convention the Dirichlet heat flow on a Dirichlet domain is generated by , the Dirichlet Laplacian (the negative of the operator associated with the Dirichlet energy form), and not by ; the sign of the generator is the sign of the spatial operator in the equation, not its negative. The dissipativity used by Lumer-Phillips is therefore the inequality on the domain of the Dirichlet Laplacian (Infinitesimal generator of a C0-semigroup names the generator whose behaviour is being discussed), and this is the identification used downstream when the heat semigroup is realised from the Laplacian.
Contraction versus boundedness. The contraction case is the pair , of the general generation theorem; a bound with alone does not imply that the semigroup is contractive, and the general theorem keeps all resolvent power estimates, the first estimate alone guaranteeing all powers when (with exponential rescaling if ). This dictionary is the one applied when Hille-Yosida generation theorem is specialised to and when variation of constants is written in the form .
5 · Examples, counterexamples and false statements
None yet.
Sources
- Klaus-Jochen Engel and Rainer Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194 (complete author-hosted monograph)
- Roland Schnaubelt, Evolution Equations, Karlsruhe Institute of Technology (2023/24 course, complete lecture notes)
- Gerald Teschl, Partial Differential Equations: From Classical to Modern (2026 author manuscript; complete 392-page archived text)
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer 2011 (complete 614-page text)
- Mathew A. Johnson, Math 951 Lecture Notes, Chapter 6: Introduction to Semigroup Methods, University of Kansas (complete 37-page chapter)