Cesàro means in local Dirichlet spaces

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.