Ahlfors–David Regular Sets, Point Spectrum and Dirichlet Spaces.

Let E be a closed subset of the unit circle T , and let α ∈ (0 , 1) . Nikolski's result states that if the Hausdorff dimension of E is strictly greater than α , then for any operator T on a separable Hilbert space such that the point spectrum σ p (T) of T contains E, the series ∑ 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 α strictly greater than the upper box dimension of E, an operator T on a separable Hilbert space such that σ p (T) contains E and 1 n ∑ 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 α strictly greater than the Hausdorff dimension of E there exists an operator T on a separable Hilbert space such that σ p (T) contains E and ‖ T n ‖ 2 ≍ n α . [ABSTRACT FROM AUTHOR]