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Classes of operators related to subnormal operators
Journal article   Open access   Peer reviewed

Classes of operators related to subnormal operators

Raul E. Curto and Thankarajan Prasad
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas, Vol.120(2), 41
04/01/2026
DOI: 10.1007/s13398-026-01830-8
url
https://doi.org/10.1007/s13398-026-01830-8View
Published (Version of record) Open Access

Abstract

In this paper we lay the foundations for a new theory encompassing two natural extensions of the class of subnormal operators, namely the n-subnormal operators and the sub-n-normal operators, where n is an element of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in \mathbb {N}$$\end{document}. We discuss inclusion relations among the above-mentioned classes and other related classes, e.g., n-quasinormal and quasi-n-normal operators. We show that sub-n-normality is stronger than n-subnormality, and produce a concrete example of a 3-subnormal operator which is not sub-2-normal. In (Curto et al., J. Funct. Anal. 278, 108342 2020), R.E. Curto, S.H. Lee and J. Yoon proved that if an operator T is subnormal, left-invertible, and such that T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T<^>2$$\end{document} is quasinormal, then T is quasinormal. In subsequent work, (Putnam, C.R., Proc. Amer. Math. Soc. 8, 768-769 1957), P. Pietrzycki and J. Stochel improved this result by removing the assumption of left invertibility. In this paper we consider suitable analogs of this result for the case of operators in the above-mentioned classes. In particular, we prove that the weight sequence of an n-quasinormal unilateral weighted shift must be periodic with period at most n.
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