Partial Taylor sums on local Dirichlet spaces

Abstract

The partial Taylor sums $S_n$, $n \ge 0$, are finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical setting of disc algebra ${\mathcal {A}}$, the precise value of $\Vert S_n\Vert _{{\mathcal {A}} \rightarrow {\mathcal {A}}}$ is not known. These numbers are referred as the Lebesgue constants and they grow like $\log n$, modulo a multiplicative constant, when n tends to infinity. In this note, we study $\Vert S_n\Vert$ when it acts on the local Dirichlet space ${\mathcal {D}}_\zeta$. There are several distinguished ways to put a norm on ${\mathcal {D}}_\zeta$ and each choice naturally leads to a different operator norm for $S_n$, as an operator on ${\mathcal {D}}_\zeta$. We consider three different norms on ${\mathcal {D}}_\zeta$ and, in each case, evaluate the precise value of $\Vert S_n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D}}_\zeta }$. In all cases, we also show that the maximizing function is unique. These formulas indicate that $\Vert S_n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D}}_\zeta } \asymp \sqrt{n}$ as n grows. Hence, in the light of uniform boundedness principle, there is a function $f \in {\mathcal {D}}_\zeta$ such that the local sequence $\Vert S_nf\Vert _{{\mathcal {D}}_{\zeta }}$, $n \ge 1$, is unbounded. We provide two explicit constructions.