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Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. (English) Zbl 0939.47020

The composition operators acting between weighted Banach spaces of analytic functions in open domains are considered. It is established that, in particular, every weakly compact composition operator is compact. Some corollaries for a disk are presented. Lower and upper estimates of the essential norm of bounded continuous operators are given, as well as the norms of the point evaluation functionals on the above spaces. Let \(H\) be Hilbert space and let \(F\) denote the class of bounded linear operators on \(H\) with Fuglede’s property. It is proved that if \(T\) from \(F\) is a contraction then the positive square root of the strong limit of \(T^nT^{*n}\) is a projection operator from \(H\) onto the maximal reducing subspace of \(T\) where its restriction is unitary.

MSC:

47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
30D55 \(H^p\)-classes (MSC2000)
47B38 Linear operators on function spaces (general)
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