Vol. 80, No. 1, 1979

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H2(μ) spaces and bounded point evaluations

Tavan Thomas Trent

Vol. 80 (1979), No. 1, 279–292

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)|pμ}, 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
Received: 13 March 1978
Revised: 21 July 1978
Published: 1 January 1979
Tavan Thomas Trent