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

2024/10/14 by Nayak, Soumyashant, Shekhawat, Renu
#15A18 #47A10 #47B40 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2410.16318

Abstract

Let \mathscrH be a complex Hilbert space, and let \mathscrB(\mathscrH) denote the set of all bounded operators on \mathscrH . For an operator T ∈ \mathscrB(\mathscrH), let |T| := (T^*T)^(1)/(2). For A in \mathscrB(\mathscrH), we refer to the sequence, \ |Aⁿ|^(1)/(n) \_{n ∈ ℕ} , as the normalized power sequence of A. As our main result, we prove that the normalized power sequence of a spectral operator in \mathscrB(\mathscrH) 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(ℂ), and supplements the Haagerup-Schultz theorem on SOT-convergence of the normalized power sequence of an operator in a II₁ factor.

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