Ahlfors–David Regular Sets, Point Spectrum and Dirichlet Spaces

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发表日期：

2024年1月20日

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O. El-Fallah,
Y. Elmadani,
I. Labghail

摘要

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