Journal of Operator Theory

Volume 96, Issue 1, Summer 2026 pp. 121-143.

On the convergence of the normalized power sequence of spectral operators

Authors: Soumyashant Nayak (1), Renu Shekhawat (2)
Author institution: (1) Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bengaluru, Karnataka - 560 059, India
(2) Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bengaluru, Karnataka - 560 059, India

Summary: 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 $|T|:= (T^*T)^{{1}/{2}}$. For $A \in \mathscr{B}(\mathscr{H})$, we refer to the sequence, $\{ |A^n|^{{1}/{n}} \}_{n \in \mathbb{N}}$, as the {\it 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 $\mathrm{SOT}$-convergence of the normalized power sequence of an operator in a II$_1$ factor.

DOI: http://dx.doi.org/10.7900/jot.2025jan17.2511
Keywords: spectral radius formula, Yamamoto's theorem, spectral operators, normalized power sequences, Haagerup--Schultz theorem

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