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

TL;DR

This paper proves that the normalized power sequence of spectral operators on Hilbert space converges in norm and describes its limit explicitly, extending previous results from matrices and von Neumann algebras.

Contribution

It generalizes convergence results of normalized power sequences from matrices to spectral operators on Hilbert spaces, providing explicit limit descriptions.

Findings

Normalized power sequence of spectral operators converges in norm.

Explicit description of the limit in terms of spectral resolution.

Generalizes previous matrix and operator algebra results.

Abstract

Let $\mathscr{H}$ be a complex Hilbert space, and let $\mathscr{B}(\mathscr{H})$ denote the set of all bounded operators on $\mathscr{H}$ . For an operator $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. For $A$ in $\mathscr{B}(\mathscr{H})$, we refer to the sequence, $\{ |A^n|^{\frac{1}{n}} \}_{n \in \mathbb{N} }$, as the $\textit{normalized power sequence}$ of $A$. As our main result, we prove that the normalized power sequence of a spectral operator in $\mathscr{B}(\mathscr{H})$ 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 $II_1$ factor.