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On the reflexivity of operators on function spaces. (English) Zbl 0806.47026

Summary: Let \(\Omega\) be a bounded plane domain. Sufficient conditions are given so that an operator \(T\) in the Cowen-Douglas class \({\mathcal B}_ n(\Omega)\) is reflexive. The operator \(M_ z\) of multiplication by \(z\) on a Hilbert space of functions analytic on a finitely connected domain \(\Omega\) is shown to be reflexive whenever \(\sigma(M_ z)= \overline\Omega\) is a spectral set.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
47A25 Spectral sets of linear operators
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