## 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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