Spectral mapping theorem - Imported Knowlpedia Content

Statement

Let $A$ be a complex unital Banach algebra, let $a\in A$ , and let $f$ be holomorphic on a neighborhood of the spectrum $\sigma_A(a)$ . For $f(a)$ defined by the holomorphic functional calculus, the spectral mapping theorem states

\sigma_A(f(a))=f(\sigma_A(a)).

If $A$ is a unital $C^*$ -algebra, $a$ is normal, and $f$ is continuous on $\sigma_A(a)$ , the same identity holds for the continuous functional calculus. Thus functional calculus transports spectral values exactly; it neither adds nor loses them.

Proof mechanism

For the holomorphic calculus, if $\mu\notin f(\sigma_A(a))$ , then

g(z)=\frac{1}{f(z)-\mu}

is holomorphic near $\sigma_A(a)$ , and the composition rule gives $(f(a)-\mu 1_A)g(a)=1_A$ . This proves one inclusion. For the reverse inclusion, factor $f(z)-f(\lambda)$ by $z-\lambda$ near a chosen $\lambda\in\sigma_A(a)$ ; invertibility of $f(a)-f(\lambda)1_A$ would force invertibility of $a-\lambda1_A$ . See Murphy, §1.3.

Consequences

Polynomial spectral mapping is the special case $\sigma_A(p(a))=p(\sigma_A(a))$ . The theorem also detects invertibility: $f(a)$ is invertible exactly when $0\notin f(\sigma_A(a))$ . For a normal element of a $C^*$ -algebra, it combines with the norm formula to yield

\lVert f(a)\rVert=\max_{\lambda\in\sigma_A(a)}|f(\lambda)|.

Conventions and scope

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §1.3 on holomorphic functional calculus and spectral mapping, and Theorem 2.1.10 on continuous functional calculus.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. DOI record. Relevant: the chapters on spectra and analytic functional calculus in Banach algebras.