## 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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### References:

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