Generalized Ces\`aro operators on Dirichlet-type spaces

TL;DR

This paper introduces generalized Cesàro operators on Dirichlet-type spaces, characterizes the measures for boundedness and compactness, and advances understanding of operator behavior in these function spaces.

Contribution

It defines a new class of Cesàro operators on Dirichlet-type spaces and characterizes the measures ensuring their boundedness and compactness.

Findings

Characterization of measures for boundedness of _{} operators

Criteria for compactness of _{} operators

Extension of operator theory in Dirichlet-type spaces

Abstract

In this note, we introduce and study a new kind of generalized Ces\`aro operators $\mathcal{C}_{\mu}$, induced by a positive Borel measure $\mu$ on $[0, 1)$, between the Dirichlet-type spaces. We characterize the measures $\mu$ for which $\mathcal{C}_{\mu}$ is bounded (compact) from one Dirichlet-type space $\mathcal{D}_{\alpha}$ into another one $\mathcal{D}_{\beta}$.