Extreme operators on H∞

Abstract

Let A1 and A2 be sup-norm algebras, each containing the constant functions. Let P(A1, A2) denote the set of bounded linear operators from A1 to A2 which carry 1 into 1 and have norm 1. Several authors have considered the problem of describing the extreme points of P(A1, A2). In the case where A1 is the algebra of continuous complex functions on some compact Hausdorff space, and A2 is the algebra of complex scalars, Arens and Kelley proved that the extreme operators in P(A1, A2) are exactly the multiplicative ones (see [1]). It was shown by Phelps in [6] that if A1 is self-adjoint, then every extreme point of P(A1, A1) is multiplicative. In [4], Lindenstrauss, Phelps, and Ryff exhibited non-multiplicative extreme points of P(A, A) and P(H∞, H∞), where A and H∞ are, respectively, the disk algebra, and the algebra of bounded analytic functions on the open unit disk D. The extreme multiplicative operators in P(A, A) were described in [6]. Rochberg proved in [8] that, if T is a member of P(A, A) which carries the identity on D into an extreme point of the unit ball of A, then T is multiplicative and is an extreme point of P(A, A). Rochberg's paper [9] is a study of certain extremal subsets of P(A, A), namely, those of the form K(F, G) = {T∊P(A, A):TF=G}, where F and G are inner functions in A. We proved in [5] that, if F is non-constant, then K(F, G) contains an extreme point of P(A, A).