Compactness of resolvent operators generated by a class of composition semigroups on \(H^ p\)
Let \(H^ p\) be the Hardy space for the unit disk D, \(1\leq p<\infty\). A composition semigroup on \(H^ p\) is given by \(T_ tf=f\circ \vartheta_ t\quad (f\in H^ p,\quad t\geq 0),\) where \(\{\vartheta_ t:\) \(t\geq 0\}\) is a semigroup (under conposition) of analytic functions from D to D. Such objects were studied by E. Berkson and H. Porta. The author extends work of \textit{A. G. Siskakis} [J. Math. Anal. Appl. 127, 122-129 (1987; Zbl 0645.47032)]. Of concern is the resolvent \(R(\lambda,A_ p)\) of the generator \(A_ p\) of \(\{T_ t\}\). Criteria (and related examples) are given for the compactness of \(R(\lambda,A_ p)\).
scientific article; zbMATH DE number 4177947