H²(μ) spaces and bounded point evaluations

Tavan Thomas Trent

Vol. 80 (1979), No. 1, 279–292

DOI: 10.2140/pjm.1979.80.279

Abstract

Let H²(μ) denote the closure of the polynomials in L²(μ), where μ is a positive finite compactly supported Borel measure carried by the closed unit disc D. For λ ∈D, define E(λ) = sup{|p(λ)|∕∥p∥_μ}, where the suprenum is taken over all polynomials whose L²(μ) norm is not zero. If E(λ) < ∞ we say that μ has a bounded point evaluation at λ, abbreviated b.p.e. at λ. Whenever E(λ) < ∞ we may fix the value of f ∈ H²(μ) at λ. We determine lhe set on which all functions in H²(μ) have (fixed) analytic values in terms of the parts of the spectrum of a certain operator.

Mathematical Subject Classification 2000

Primary: 30D55

Secondary: 30G99, 46J15

Milestones

Received: 13 March 1978

Revised: 21 July 1978

Published: 1 January 1979