The Ces\`{a}ro operator on local Dirichlet spaces

TL;DR

This paper studies Cesàro operators on local Dirichlet spaces, analyzing their approximation properties and asymptotic norms, revealing thresholds for effective approximation and providing precise estimates.

Contribution

It extends understanding of Cesàro operators by deriving asymptotic norms and approximation thresholds on local Dirichlet spaces, improving previous results.

Findings

Cesàro operators provide linear approximation when α > 1/2

Asymptotic norms are precisely characterized for α ≤ 1/2

Upper and lower norm estimates are established for α > 1/2

Abstract

The family of Ces\`{a}ro operators $\sigma_n^\alpha$, $n \geq 0$ and $\alpha \in [0,1]$, consists of finite rank operators on Banach spaces of analytic functions on the open unit disc. In this work, we investigate these operators as they act on the local Dirichlet spaces $\mathcal{D}_\zeta$. It is well-established that they provide a linear approximation scheme when $\alpha > \frac{1}{2}$, with the threshold value $\alpha = \frac{1}{2}$ being optimal. We strengthen this result by deriving precise asymptotic values for the norm of these operators when $\alpha \leq \frac{1}{2}$, corresponding to the breakdown of approximation schemes. Additionally, we establish upper and lower estimates for the norm when $\alpha > \frac{1}{2}$.