An extremal problem for operators (Q1819323)

Let A be a Banach algebra, F a compact set in the complex plane, and h a function holomorphic in some neighborhood of the set F. Thus h(a) is meaningful for each element \(a\in A\) whose spectrum \(\sigma\) (a) is contained in F, and it is possible to evaluate the norm \(| h(a)|\). Problem: Compute the supremum of the norms \(| h(a)|\) as a ranges over all elements of A with spectrum contained in F and whose norm does not exceed one; that is, compute sup\(\{| h(a)|\); \(a\in A\), \(\sigma\) (a)\(\subset F,| a| \leq 1\}\). This problem was first formulated and treated by the author in the particular case where A is the algebra of all linear operators on a finite-dimensional Hilbert space and F is the disc \(\{\) z;\(| z| \leq r\}\) for a given positive number \(r<1\). The paper discusses motivation, connections with complex function theory, convergence of iterative processes, critical exponents, and the infinite companion matrix.