Spectrum and resolvent set in a Banach algebra

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.