Cyclicity and invariant subspaces in the Dirichlet spaces

2014/11/18 by Omar El-Fallah, El-Fallah, Omar, Y. Elmadani +3 · 4 citations
Mathematics · #Holomorphic and Operator Theory #Analytic and geometric function theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1411.4977

Abstract

Let μ be a positive finite measure on the unit circle and D (μ) the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function f ∈ D (μ) is cyclic if and only if c_μ(Z (f))= 0, where c_μ is the capacity associated with D (μ) and Z(f) is the zero set of f. In this paper we prove that this conjecture is true for measures with countable support. We also give in this case a complete and explicit characterization of invariant subspaces.

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