Cesàro-type operators acting on Dirichlet spaces
Abstract: If $(\eta )={ \eta_n} {n=0}\infty $ is a sequence of complex numbers, the Ces`aro-type operator $\mathcal C{(\eta )}$ is formally defined in the space of analytic funtions in the unit disc $\mathbb D$ as follows: If $f$ is an analytic function in $\mathbb D$, $f(z)=\sum_{n=0}\infty a_nzn$ ($z\in \mathbb D$), then $\mathcal C_{(\eta )}(f)$ is formally defined by $$\mathcal C_{(\eta )}(f)(z)=\mathcal C_{{\eta_n}}(f)(z)=\sum_{n=0}\infty \eta n\left (\sum{k=0}na_k\right )zn.$$ The operator $\mathcal C_{(\eta )}$ is a natural generalization of the Ces`{a}ro operator. In this paper we give a complete characterization of the sequences of complex numbers $(\eta )$ for which the operator $\mathcal C_{(\eta )}$ is bounded (compact) from $\mathcal D2_\alpha $ into $\mathcal D2_\beta $ for any $\alpha , \beta >-1$ and, also, a characterization of those ones for which $\mathcal C_{(\eta )}$ is bounded (compact) from $S2$ into itself. The space $S2$ consists of those $f$ which are analytic in $\mathbb D$ such that $f\prime \in H2$.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.