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Keywords:
the Banach space of formal power series associated with a sequence $\beta $; bounded point evaluation; strictly cyclic maximal ideal space; Schatten $p$-class; reflexive algebra; semisimple algebra; composition operator
the Banach space of formal power series associated with a sequence $\beta $; bounded point evaluation; strictly cyclic maximal ideal space; Schatten $p$-class; reflexive algebra; semisimple algebra; composition operator
Summary:
Let $\lbrace \beta (n)\rbrace ^{\infty }_{n=0}$ be a sequence of positive numbers and $1 \le 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.
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