The Commutant of Multiplication by z on the Closure of Polynomials in Lt(μ)

Abstract

In this work μ represents a positive, compactly supported Borel measure in the plane, C. For each t in [1, ∞), the space Pt(μ) consists of the functions in Lt(μ) that belong to the (norm) closure of the (analytic) polynomials. Multiplication by z on Pt(μ) is a bounded operator, which we denote by Sμ. Throughout, we tacitly assume that Sμ is irreducible. J. Thomson in [T] has shown that the set of bounded point evaluations, bpe μ, for Pt(μ) is a nonempty simple connect region G having the following properties: (a) The spectrum of Sμ, denoted σ(Sμ), equals G; (b) The essential spectrum of Sμ, denoted σe(Sμ), equals ∂G; (c) The commutant of Sμ, identified in the customary way with Pt(μ) ∩ L∞(μ), is isometrically isomorphic to the bounded analytic functions on G, denoted by H∞(G). We let ∼: f → f̃ signify this Banach algebra isometric isomorphism from H∞(G) to Pt(μ) ∩ L∞(μ). Let φ be a Riemann map from G to D where D denotes the open unit disc it follows that φ̃ is in Pt(μ) ∩ L∞(μ). Now define a measure, ν, by setting ν = μ ∘ φ̃−1. A routine argument shows that Sν acting on Pt(ν) is irreducible with bpe ν = D. If ψ is the inverse of φ, then we also have that ψ̃ is in Pt(ν) ∩ L∞(ν). In this paper, we prove: (1) The measure μ restricted to the boundary of G is absolutely continuous with respect to the harmonic measure on G; (2) The function ψ̃ is (almost) a one-to-one map from a carrier of ν|∂D to a carrier of μ|∂G; (3) The space P2(μ) C(sptμ) = A(G) where C(sptμ) denotes the continuous functions on sptμ and A(G) denotes those functions continuous an G that are analytic on G. We give examples to show that a carrier of μ|∂G need not be the "nice boundary part" of G, and that properties (1) and (2) are not sufficient to guarantee that σe(Sμ) = ∂G. We also show that each f ∈ Pt(μ) ∩ L∞(μ) has nontangential limits; that is, f(ω) → f(z) as ω → z nontangentially a.e. μ|∂G.