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Strictly cyclic algebra of operators acting on Banach spaces \(H^p(\beta)\). (English) Zbl 1049.47033

Summary: Let \(\{\beta (n)\}^{\infty }_{n=0}\) be a sequence of positive numbers and \(1 \leq p < \infty \). We consider the space \(H^{p}(\beta )\) of all power series \(f(z)=\sum ^{\infty }_{n=0}\hat {f}(n)z^{n}\) such that \(\sum ^{\infty }_{n=0}| \hat {f}(n)| ^{p}\beta (n)^{p} < \infty \). We investigate strict cyclicity of \(H^{\infty }_{p}(\beta)\), the weakly closed algebra generated by the operator of multiplication by \(z\) acting on  \(H^{p}(\beta)\), and determine the maximal ideal space, the dual space and the reflexivity of the algebra \(H^{\infty }_{p}(\beta)\). We also give a necessary condition for a composition operator to be bounded on \(H^{p}(\beta)\) when \(H^{\infty }_{p}(\beta)\) is strictly cyclic.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A25 Spectral sets of linear operators
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