Ahlfors–David Regular Sets, Point Spectrum and Dirichlet Spaces

Abstract

Let E be a closed subset of the unit circle $\mathbb {T}$, and let $\alpha \in (0,1)$. Nikolski’s result states that if the Hausdorff dimension of E is strictly greater than $\alpha$, then for any operator T on a separable Hilbert space such that the point spectrum $\sigma _p(T)$ of T contains E, the series $\sum _{n}n^{\alpha -1}\Vert T^n\Vert ^{-2}$ converges. A partial converse of this result has been obtained by El-Fallah and Ransford. Namely they constructed, for any $\alpha$ strictly greater than the upper box dimension of E, an operator T on a separable Hilbert space such that $\sigma _p(T)$ contains E and $\frac{1}{n} \sum _{k=0}^{n-1}\left\| T^k\right\| ^2\lesssim n^{\alpha }$. In this paper, we improve on this latter result for regular sets. Indeed, for any Ahlfors–David regular set E and for any $\alpha$ strictly greater than the Hausdorff dimension of E there exists an operator T on a separable Hilbert space such that $\sigma _p(T)$ contains E and $\Vert T^n\Vert ^2\asymp n^{\alpha }$.