The Cesaro operator in growth Banach spaces of analytic functions

TL;DR

This paper analyzes the Cesaro operator's behavior in classical growth Banach spaces of analytic functions, determining its norms, spectral properties, ergodic behavior, and optimal domain spaces.

Contribution

It provides explicit norms, spectral analysis, and characterizes the ergodic properties of the Cesaro operator in these growth spaces, also identifying the largest domain spaces for the operator.

Findings

Exact norms of the Cesaro operator in growth Banach spaces are computed.

Spectral properties of the operator are characterized.

Largest Banach domain spaces for the operator are identified.

Abstract

The Cesaro operator $\mathsf{C}$, when acting in the classical growth Banach spaces $A^{-\gamma}$ and $A_0^{-\gamma}$, for $\gamma > 0 $, of analytic functions on $\mathbb{D}$, is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is then possible to characterize the mean ergodic and related properties of $\mathsf{C}$ acting in these spaces. In addition, we determine the largest Banach space of analytic functions on $\mathbb{D}$ which $\mathsf{C}$ maps into $A^{-\gamma}$ (resp. into $A_0^{-\gamma}$); this optimal domain space always contains $A^{-\gamma}$ (resp. $A_0^{-\gamma}$) as a proper subspace.