Strongly continuous semigroup

Depends on

• The operator norm as the least bound and as the unit-sphere or unit-ball supremum
• Banach space
• Real and complex scalar conventions for normed spaces
• A bounded linear operator between normed spaces
• The spaces \(\mathcal B(X,Y)\) and \(\mathcal B(X)\) of bounded linear operators

Used by

• A semigroup with unbounded generator is not norm continuous at zero Corollary
• Restriction to a closed invariant subspace is a C0-semigroup and its generator is the part Corollary
• Strong continuity does not imply operator-norm continuity Counterexample
• The translation semigroup is not analytic Counterexample
• The translation semigroup is not strongly continuous on L-infinity Counterexample
• Classical, strong and mild abstract Cauchy solutions Definition
• Complex sector and bounded analytic semigroup Definition
• Infinitesimal generator of a C0-semigroup Definition
• A multiplication semigroup with an unbounded generator Example
• The analytic semigroup generated by a bounded operator Example
• The Dirichlet Laplacian generates the heat semigroup Example
• The exponential of a bounded operator is a uniformly continuous semigroup Example
• The right-translation semigroup on Lp has the weak derivative as generator Example
• The sectorial multiplication operator Example
• A semigroup with continuity at zero is uniformly bounded on every compact time interval Lemma
• Continuity at time zero implies continuity of every orbit Lemma
• Laplace uniqueness identifies two exponentially bounded semigroups Lemma
• The generator commutes with the semigroup on its domain Lemma
• The variation-of-constants integral is continuous for integrable forcing Lemma
• Time integrals of semigroup orbits lie in the generator domain Lemma
• An orbit is right differentiable at zero exactly on the generator domain Theorem
• Bounded Yosida semigroups converge to the generated semigroup Theorem
• Exponential bound for a C0-semigroup Theorem
• Hille-Yosida generation theorem Theorem
• Lumer-Phillips generation theorem Theorem
• Sectorial resolvent characterisation of bounded analytic semigroups Theorem
• The generator is closed and densely defined Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Klaus-Jochen Engel and Rainer Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194 (complete author-hosted monograph) (standard reference, not scraped)
• Roland Schnaubelt, Evolution Equations, Karlsruhe Institute of Technology (2023/24 course, complete lecture notes) (standard reference, not scraped)