An extremal problem for operators

Abstract

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 e A whose spectrum σ(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 e A, σ(a) ⊂ F, |a| ⩽ 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| ⩽ 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.