Holomorphic functional calculus in a Banach algebra

Let $A$ be a complex unital Banach algebra, let $a\in A$ , and let $f$ be holomorphic on an open neighborhood of the spectrum $\sigma_A(a)$ . Choose a positively oriented contour $\Gamma$ in the resolvent set that winds once around $\sigma_A(a)$ and zero times around points outside the chosen neighborhood. The holomorphic functional calculus defines

f(a)=\frac{1}{2\pi i}\int_\Gamma f(z)(z1_A-a)^{-1}\,dz.

The Banach-valued contour integral is independent of every admissible choice of $\Gamma$ , so $f(a)$ depends only on $a$ and the germ of $f$ near its spectrum.

Algebraic and spectral properties

For a fixed neighborhood of $\sigma_A(a)$ , the assignment $f\mapsto f(a)$ is a continuous unital algebra homomorphism. It agrees with polynomial evaluation and with rational evaluation when the poles avoid the spectrum. It also satisfies the spectral mapping theorem

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

If $g$ is holomorphic near $f(\sigma_A(a))$ , then $(g\mathbin{\circ}f)(a)=g(f(a))$ . These properties follow from the resolvent identity and the Cauchy integral formula.

Spectral projections and comparison

Suppose $\sigma_A(a)$ is the disjoint union of two compact subsets separated by open neighborhoods. A locally constant holomorphic function that is one near the first subset and zero near the second produces an idempotent $p=f(a)$ , called a Riesz spectral projection. Thus the calculus can isolate spectral components even when $a$ is not normal.

When $A$ is a $C^*$ -algebra and $a$ is normal, this construction agrees with the continuous functional calculus on functions that are holomorphic near the spectrum.

Conventions and scope

References

Gerard J. Murphy, C-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §1.3 on the holomorphic functional calculus and spectral mapping.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. Publisher record. Relevant: Chapter III on spectra and analytic functional calculus in Banach algebras.