Journal of Operator Theory

Volume 94, Issue 2, Autumn 2025 pp. 339-402.

Non-commutative weighted shifts, joint similarity, and function theory in several variables

Authors: Gelu Popescu
Author institution: Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, U.S.A.

Summary: The goal of this paper is to study the structure of non-commutative weighted shifts, their properties, and to understand their role as models (up to similarity) for $n$-tuples of operators on Hilbert spaces as well as their implications to function theory on non-commutative (respectively commutative) Reinhardt domains. We obtain a Rota type similarity result concerning the joint similarity of $n$-tuples of operators to parts of non-commutative weighted multi-shifts and provide a non-commutative multivariable analogue of Foia\c s-Pearcy model for quasi-nilpotent operators. We also develop a functional calculus for arbitrary $n$-tuples of operators on a Hilbert space.

DOI: http://dx.doi.org/10.7900/jot.2023nov03.2473
Keywords: non-commutative weighted shifts, joint similarity, quasi-nilpotent tuple, Hardy space, von Neumann inequality, functional calculus, function theory in several variables, $C^*$-algebra

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