Cesàro means in local Dirichlet spaces

Abstract

The Cesàro means of Taylor polynomials $\sigma _n,$ $n \ge 0,$ are finite rank operators on any Banach space of analytic functions on the open unit disc. They are particularly exploited when the Taylor polynomials do not constitute a valid linear polynomial approximation scheme (LPAS). Notably, in local Dirichlet spaces ${\mathcal {D}}_\zeta ,$ they serve as a proper LPAS. The primary objective of this note is to accurately determine the norm of $\sigma _n$ when it is considered as an operator on ${\mathcal {D}}_\zeta .$ There exist several practical methods to impose a norm on ${\mathcal {D}}_\zeta ,$ and each norm results in a distinct operator norm for $\sigma _n.$ In this context, we explore three different norms on ${\mathcal {D}}_\zeta$ and, for each norm, precisely compute the value of $\Vert \sigma _n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D}}_\zeta }.$ Furthermore, in all instances, we identify the maximizing functions and demonstrate their uniqueness.