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Published online by Cambridge University Press: 25 June 2025
We discuss some recent achievements in function theory and operator theory on the Dirichlet space, paying particular attention to invariant subspaces, interpolation and Hankel operators.
Introduction
In recent years the Dirichlet space has received a lot of attention from mathematicians in the areas of modern analysis, probability and statistical analysis. We intend to discuss some recent achievements in function theory and operator theory on the Dirichlet space. The key references are [Richter and Shields 1988; Richter and Sundberg 1992; Aleman 1992; Marshall and Sundberg 1993; Rochberg and Wu 1993; Wu 1993]. In this introductory section we state the basic results. Proofs will be discussed in the succeeding sections.
1. Invariant Subspaces
The codimension-one property for invariant subspaces of the Dirichlet space is related to the cellular indecomposibility of the operator (Mz,D). This concept was first introduced and studied by Olin and Thomson [1984] for more general Hubert spaces. Later Bourdon [1986] proved in that if the operator Mz is cellular indecomposable on a Hubert space Hof analytic functions on 𝔻 with certain properties, then every nonzero invariant subspace for (Mz, H) has the codimension-one property.
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