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Operators acting on certain Banach spaces of analytic functions. (English) Zbl 0821.47022

Summary: Let \(\mathcal X\) be a reflexive Banach space of functions analytic on a plane domain \(\Omega\) such that for every \(\lambda\) in \(\Omega\) the functional of evaluation at \(\lambda\) is bounded. Assume further that \(\mathcal X\) contains the constants and \(M_ z\), multiplication by the independent variable \(z\), is a bounded operator on \(\mathcal X\). We give sufficient conditions for \(M_ z\) to be reflexive. In particular, we prove that the operators \(M_ z\) on \(E^ p(\Omega)\) and certain \(H^ p_ a(\beta)\) are reflexive. We also prove that the algebra of multiplication operators on Bergman spaces is reflexive, giving a simpler proof of a result of Eschmeier.

MSC:

47B38 Linear operators on function spaces (general)
47L10 Algebras of operators on Banach spaces and other topological linear spaces
47A25 Spectral sets of linear operators
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