Banach Algebras Spectrum and Holomorphic Functional Calculus

A unital complex Banach algebra is a nonzero complex vector space $A$ equipped with

• an associative bilinear multiplication $A\times A\to A$, $(a,b)\mapsto ab$, and
• a norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ under which $A$ is a Banach space (Banach space),

such that

• the norm is submultiplicative: $\mathrm{\parallel }ab\mathrm{\parallel }\le \mathrm{\parallel }a\mathrm{\parallel }\mathrm{\parallel }b\mathrm{\parallel }$ for all $a,b\in A$, and
• there is a unit $1\in A$ with $1a=a1=a$ for every $a\in A$, normalized by $\mathrm{\parallel }1\mathrm{\parallel }=1$.

The phrase nonzero is part of the definition: the identity is required to exist, and the zero algebra $\{0\}$ has no element satisfying $1\mathrm{\ne }0$. In every unital Banach algebra the unit is unique, because if $1^{\mathrm{\prime }}$ is a second element acting as an identity then $1^{\mathrm{\prime }}=1^{\mathrm{\prime }}1=1$; from now on $1$ denotes that element. The norm condition $\mathrm{\parallel }1\mathrm{\parallel }=1$ is a normalization rather than a consequence of submultiplicativity, which would give only $\mathrm{\parallel }1\mathrm{\parallel }\ge 1$ for a nonzero unit; the two conventions $\mathrm{\parallel }1\mathrm{\parallel }=1$ and $\mathrm{\parallel }1\mathrm{\parallel }\le 1$ agree on every nonzero unital algebra, since submultiplicativity turns the latter into an equality.

• The scalar field is complex and fixed. Every spectrum, resolvent and holomorphic-calculus statement on this page is about complex unital Banach algebras. Real Banach algebras are not silently complexified: the one place where a real structure enters, the spectrum of a real operator, is defined through a specified complexification in Complexification and spectrum of a real operator.

• Multiplication is bilinear and associative, and nothing more. No commutativity, involution, or approximate unit is assumed. The algebra $\mathcal{B}(X)$ of bounded operators on a nonzero complex Banach space (ex-bounded-operators-form-a-noncommutative-banach-algebra) is the motivating noncommutative example, and $C(K,\mathbb{C})$ for compact Hausdorff $K$ (ex-continuous-functions-form-a-commutative-banach-algebra) the motivating commutative one.

• Completeness is with respect to the submultiplicative norm. A complete normed algebra whose norm is merely equivalent to a submultiplicative one is not thereby a unital Banach algebra in this sense; rescaling a norm to $\lambda \mathrm{\parallel }\cdot \mathrm{\parallel }$ with $\lambda >1$ preserves completeness and submultiplicativity but destroys the normalization $\mathrm{\parallel }1\mathrm{\parallel }=1$.

• The unit is not a separate structure. It is determined by the multiplication, so an algebra homomorphism between unital Banach algebras that preserves multiplication and the unit is exactly a multiplicative linear map sending $1$ to $1$; this is the convention used for characters on the following page of this track.

• Reading order. The example items named by ID above are homed on later pages of the plan, so they are named rather than hyperlinked: a body link to later material must be declared as a forward reference, and Step-5b closure removes every such declaration. Rehoming those items to an earlier page (an owner-only reading-order change) would make the citations backward and restore the links.

Let $A$ be a unital complex Banach algebra (Unital Banach algebra). An element $a\in A$ is invertible when there is $b\in A$ with

$ab=ba=1.$

Such an element $b$ is then unique: if $b$ and $b^{\mathrm{\prime }}$ both satisfy the two equations, then

$b=b1=b(a{b}^{\mathrm{\prime }})=(ba)b^{\mathrm{\prime }}=1b^{\mathrm{\prime }}=b^{\mathrm{\prime }},$

using associativity and the unit law. The unique $b$ is called the inverse of $a$ and is written $a^{-1}$. The set of all invertible elements of $A$ is denoted

$A^{\times }:=\{a\in A:a\text{ is invertible}\}$

and is called the general linear group of $A$. It is a group under multiplication:

• $1\in A^{\times }$ with $1^{-1}=1$;
• if $a,b\in A^{\times }$ then $ab\in A^{\times }$ with $(ab{)}^{-1}=b^{-1}{a}^{-1}$, since $(ab)(b^{-1}{a}^{-1})=a(b{b}^{-1})a^{-1}=a{a}^{-1}=1$ and symmetrically;
• $(a^{-1}{)}^{-1}=a$ by symmetry of the defining equations.

The map $a\mapsto a^{-1}$ is the inversion map of $A^{\times }$.

• Two-sided inverses are required, and one-sided inverses do not suffice. If $ab=1$ and $ba\mathrm{\ne }1$, then $a$ is not invertible by definition, and this situation really occurs in a unital complex Banach algebra: on ${\mathrm{\ell }}^2({\mathbb{N}}_0)$ the right shift $T{e}_n=e_{n-1}$ for $n\ge 1$, $T{e}_0=0$, and the left shift $S{e}_n=e_{n+1}$ satisfy $TS=1$ while $ST=1-P$, where $P$ is the orthogonal projection onto $\mathbb{C}{e}_0$, so $T$ has a right inverse and is not invertible (ex-spectrum-of-the-unilateral-shift). Thus $ab=1$ alone does not force $ba=1$ in a general unital Banach algebra; both inverses are always verified explicitly below.

• The group need not be dense or connected, but it is open. In a Banach algebra $A^{\times }$ is an open subset of $A$ and inversion is continuous there; this is Invertible group is open and inversion is continuous. Openness is what makes the resolvent set of an element open and hence makes the spectrum closed Spectrum and resolvent set in a Banach algebra.

• Nonunital algebras are not covered here. In a Banach algebra without a unit there is no element $1$ to compare with, so invertibility is not defined by this definition. The companion examples introduce the unitization $A\oplus \mathbb{C}1$ and declare that spectra of elements of a nonunital algebra are always computed in that named unitization (ex-unitization-of-a-nonunital-banach-algebra).

• Reading order. The example items named by ID above are homed on later pages of the plan, so they are named rather than hyperlinked: a body link to later material must be declared as a forward reference, and Step-5b closure removes every such declaration. Rehoming those items to an earlier page (an owner-only reading-order change) would make the citations backward and restore the links.

Let $A$ be a unital complex Banach algebra, let $a\in A$ with $\mathrm{\parallel }a\mathrm{\parallel }<1$, and for $N\in \mathbb{N}$ write $S_N:={\sum }_{n=0}^N{a}^n$, a finite sum with $a^0:=1$ (Unital Banach algebra). Then

1. the series ${\sum }_{n\ge 0}{a}^n$ converges in $A$, and its sum $S:={\lim }_{N\to \mathrm{\infty }}{S}_N$ satisfies $(1-a)S=S(1-a)=1$; in particular $1-a$ is invertible with $(1-a{)}^{-1}=S$ (Invertible element and general linear group of a Banach algebra);
2. for every $N\in \mathbb{N}$ the tail estimate $\parallel (1-a{)}^{-1}-S_N\parallel \le \frac{\mathrm{\parallel }a{\mathrm{\parallel }}^{N+1}}{1-\mathrm{\parallel }a\mathrm{\parallel }}$ holds. The hypothesis $\mathrm{\parallel }a\mathrm{\parallel }<1$ is not symmetric: the estimate is in terms of $\mathrm{\parallel }a\mathrm{\parallel }$, and $1-a$ is invertible whenever $a$ lies in the open unit ball.

Let $A$ be a unital complex Banach algebra (Unital Banach algebra) and let $a\in A^{\times }$ be invertible with inverse $a^{-1}$ (Invertible element and general linear group of a Banach algebra). Then:

1. every $b\in A$ with $\mathrm{\parallel }{a}^{-1}\mathrm{\parallel }\mathrm{\parallel }b-a\mathrm{\parallel }<1$ is invertible, with $b^{-1}=(1-a^{-1}(a-b){)}^{-1}{a}^{-1}\text{and}\mathrm{\parallel }{b}^{-1}\mathrm{\parallel }\le \frac{\mathrm{\parallel }{a}^{-1}\mathrm{\parallel }}{1-\mathrm{\parallel }{a}^{-1}\mathrm{\parallel }\mathrm{\parallel }b-a\mathrm{\parallel }};$
2. $A^{\times }$ is an open subset of $A$;
3. inversion $A^{\times }\to A^{\times }$, $a\mapsto a^{-1}$, is continuous at every point of $A^{\times }$ (with the relative topology on $A^{\times }$).

Let $A$ be a unital complex Banach algebra (Unital Banach algebra) and let $a\in A$. The spectrum of $a$ in $A$ is the set

${\sigma }_A(a):=\{z\in \mathbb{C}:z1-a\mathrm{\notin }{A}^{\times }\},$

where $A^{\times }$ is the group of invertible elements (Invertible element and general linear group of a Banach algebra). The complement

${\rho }_A(a):=\mathbb{C}\setminus {\sigma }_A(a)=\{z\in \mathbb{C}:z1-a\in A^{\times }\}$

is the resolvent set of $a$, and for $z\in {\rho }_A(a)$ the element

$R(z,a):=(z1-a{)}^{-1}\in A$

is the resolvent of $a$ at $z$. Thus $R(z,a)$ is characterized by the two equations

$(z1-a)R(z,a)=R(z,a)(z1-a)=1,$

and it is the unique element with these properties. The subscript in ${\sigma }_A(a)$ and ${\rho }_A(a)$ records the ambient algebra: if $B\subseteq A$ is a closed unital subalgebra containing $a$ and having the same unit, then $B$ is itself a unital complex Banach algebra in the inherited norm, so both spectra are defined. They can differ, and the spectrum in the smaller algebra contains the spectrum in the larger one: ${\sigma }_B(a)\supseteq {\sigma }_A(a)$.

The zero algebra is excluded by the unital-algebra convention. For $a=0$, ${\sigma }_A(0)=\{0\}$: $z1$ has inverse $z^{-1}1$ for $z\mathrm{\ne }0$, whereas $0$ cannot have a two-sided inverse since $1\mathrm{\ne }0$.

• The ambient algebra is always part of the data. Every use of a spectrum below states the algebra in which it is computed. For $a\in A$ and a closed unital subalgebra $B\subseteq A$ with $a\in B$ and the same unit, containment of the invertible groups $B^{\times }\subseteq A^{\times }$ gives ${\sigma }_B(a)\supseteq {\sigma }_A(a)$, and the companion page exhibits a case where the inclusion is strict (cex-spectrum-can-shrink-in-a-larger-banach-algebra). The complex spectrum of an element of a nonunital Banach algebra is not defined by this formula; it is taken in the unitization, as recorded in ex-unitization-of-a-nonunital-banach-algebra.

• Operator spectra are the special case $A=\mathcal{B}(X)$. For a nonzero complex Banach space $X$ and $T\in \mathcal{B}(X)$ one has $z1-T\in \mathcal{B}(X)$ and ${\sigma }_{\mathcal{B}(X)}(T)$ agrees with the spectrum of $T$ computed in any unital Banach subalgebra of $\mathcal{B}(X)$ that contains $T$ and the identity and is closed under inverses of its elements. The closed unital algebra generated by $T$ need not be inverse-closed, so no agreement with that algebra is asserted (ex-bounded-operators-form-a-noncommutative-banach-algebra).

• The spectrum is closed and bounded; under AC it is nonempty. The map $z\mapsto z1-a$ is continuous and $A^{\times }$ is open (Invertible group is open and inversion is continuous), so ${\rho }_A(a)$ is open and ${\sigma }_A(a)$ is closed. The Neumann series gives $\{z:\mathrm{\mid }z\mathrm{\mid }>\mathrm{\parallel }a\mathrm{\parallel }\}\subseteq {\rho }_A(a)$ and hence bounds the spectrum by $\mathrm{\parallel }a\mathrm{\parallel }$; under the Axiom of Choice (The Axiom of Choice), the nonempty-spectrum conclusion is the substance of Spectrum is nonempty compact and norm bounded. Nothing in the present definition assumes either conclusion.

• The resolvent convention fixed here is $R(z,a)=(z1-a{)}^{-1}$. With this order of the factors the resolvent identity reads $R(z,a)-R(w,a)=(w-z)R(z,a)R(w,a)$ and the derivative of the resolvent map is $-R(z,a{)}^2$ (Resolvent identity, Resolvent is Banach-valued holomorphic). For the opposite convention $\tilde{R}(z,a)=(a-z1{)}^{-1}=-R(z,a)$, the identity has factor $z-w$ and the derivative is $+\tilde{R}(z,a{)}^2$.

• Reading order. The example items named by ID above are homed on later pages of the plan, so they are named rather than hyperlinked: a body link to later material must be declared as a forward reference, and Step-5b closure removes every such declaration. Rehoming those items to an earlier page (an owner-only reading-order change) would make the citations backward and restore the links.

Let $A$ be a unital complex Banach algebra and let $a,b\in A$. With the resolvent $R(z,a)=(z1-a{)}^{-1}$ of Spectrum and resolvent set in a Banach algebra:

1. for all $z,w\in {\rho }_A(a)$, $R(z,a)-R(w,a)=(w-z)R(z,a)R(w,a);$ in particular $R(z,a)$ and $R(w,a)$ commute;
2. for all $z\in {\rho }_A(a)\cap {\rho }_A(b)$, $R(z,a)-R(z,b)=R(z,a)(a-b)R(z,b)\mathrm{.}$

Both identities are equalities of two-sided products; no commutativity of $A$ is assumed, and the factor order shown is the one that is used later.

Let $A$ be a unital complex Banach algebra and let $a\in A$ with resolvent set ${\rho }_A(a)$ and resolvent $R(z,a)=(z1-a{)}^{-1}$ (Spectrum and resolvent set in a Banach algebra). Then:

1. ${\rho }_A(a)$ is an open subset of $\mathbb{C}$, so ${\sigma }_A(a)$ is closed;
2. for every $z_0\in {\rho }_A(a)$ and every $h\in \mathbb{C}$ with $\mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }R(z_0,a)\mathrm{\parallel }<1$ the Neumann expansion $R(z_0+h,a)={\sum }_{n\ge 0}(-h{)}^{n}R(z_0,a{)}^{n+1}$ converges in $A$ and exhibits $z_0+h\in {\rho }_A(a)$;
3. the map $z\mapsto R(z,a)$ is holomorphic on ${\rho }_A(a)$ in the norm sense, with derivative $R^{\mathrm{\prime }}(z,a)=-R(z,a{)}^2(z\in {\rho }_A(a)),$ and in particular it is norm continuous there, with the local estimate $\mathrm{\parallel }R(z_0+h,a)-R(z_0,a)\mathrm{\parallel }\le \mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }R(z_0,a){\mathrm{\parallel }}^2\mathrm{/}(1-\mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }R(z_0,a)\mathrm{\parallel })$ for $\mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }R(z_0,a)\mathrm{\parallel }<1$.

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a nonzero unital complex Banach algebra and let $a\in A$, with spectrum ${\sigma }_A(a)$ and resolvent $R(z,a)=(z1-a{)}^{-1}$ as in Spectrum and resolvent set in a Banach algebra. Then

1. ${\sigma }_A(a)$ is a compact subset of the closed disc $\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le \mathrm{\parallel }a\mathrm{\parallel }\}$;
2. ${\sigma }_A(a)\mathrm{\ne }\mathrm{\varnothing }$.

The Axiom of Choice is used exactly once, in the form of the Hahn–Banach separation supplied by The dual space separates points of a normed space; the closedness, boundedness and nonemptiness arguments are otherwise choice-free.