Holomorphic functional calculus

Summary

Let $a$ be an element of a unital Banach algebra, and let $f$ be a holomorphic function on a neighbourhood of the spectrum of $a$. The holomorphic functional calculus gives a meaning to $f(a)$. It is defined by a Cauchy integral of $f$ against the resolvent of $a$. The map $f \mapsto f(a)$ is an algebra homomorphism that extends the evaluation of polynomials, and it satisfies the spectral mapping theorem $\sigma(f(a)) = f(\sigma(a))$. It works for all bounded operators on a Banach space, not only normal ones, and it gives spectral projections, logarithms and square roots of operators.