On the convergence of the normalized power sequence of spectral operators on Hilbert space

Abstract

Let $\mathcal{H}$ be a complex Hilbert space, and let $\mathcal{B}(\mathcal{H})$ denote the set of all bounded operators on $\mathcal{H}$ . For an operator $T\in \mathcal{B}(\mathcal{H})$, let $|T|:=(T^{\ast }T{)}^{\frac{1}{2}}$. For $A$ in $\mathcal{B}(\mathcal{H})$, we refer to the sequence, $\{|A^n{|}^{\frac{1}{n}}{\}}_{n\in \mathbb{N}}$, as the $\text{normalized power sequence}$ of $A$. As our main result, we prove that the normalized power sequence of a spectral operator in $\mathcal{B}(\mathcal{H})$ converges in norm, and provide an explicit description of the limit in terms of its idempotent-valued spectral resolution. Our approach substantially generalizes the corresponding result by the first-named author in the case of matrices in $M_m(\mathbb{C})$, and supplements the Haagerup-Schultz theorem on SOT-convergence of the normalized power sequence of an operator in a $I{I}_1$ factor.

21 pages. Title changed based on referee comments + a few minor changes. Accepted for publication in JOT