Extreme operators on H_∞

Extract

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).