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The commutant of a certain compression

Author: William T. Ross
Journal: Proc. Amer. Math. Soc. 118 (1993), 831-837
MSC: Primary 47B38; Secondary 47A20, 47B35
MathSciNet review: 1145951
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Abstract: Let $ G$ be any bounded region in the complex plane and $ K \subset G$ be a simple compact arc of class $ {C^1}$. Let $ {A^2}(G\backslash K)$ (resp. $ {A^2}(G)$) be the Bergman space on $ G\backslash K$ (resp. $ G$). Let $ S$ be the operator multiplication by $ z$ on $ {A^2}(G\backslash K)$ and $ C = {P_\mathcal{N}}S{\vert _\mathcal{N}}$ be the compression of $ S$ to the semi-invariant subspace $ \mathcal{N} = {A^2}(G\backslash K) \ominus {A^2}(G)$. We show that the commutant of $ {C^{\ast}}$ is the set of all operators of the form $ {A^{ - 1}}{M_h}A$, where $ h$ is a multiplier on a certain Sobolev space of functions on $ K$ and $ (Af)(w) = \int_G {f(z){{(\overline z - \overline w )}^{ - 1}}dA(z)(w \in K)} $. We also use multiplier theory in fractional order Sobolev spaces to obtain further information about $ C$.

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Keywords: Bergman spaces, multiplication operators, Sobolev spaces, multipliers
Article copyright: © Copyright 1993 American Mathematical Society

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