Classes of operators related to subnormal operators

TL;DR

This paper explores the relationships among classes of operators extending subnormal operators, establishing inclusion relations, providing examples, and analyzing properties like weight sequence periodicity in weighted shifts.

Contribution

It introduces the concepts of $n$--subnormal and sub-$n$--normal operators, compares their strengths, and extends known results about subnormal operators to these new classes.

Findings

Sub-$n$--normality is stronger than $n$--subnormality.

Constructed a 3--subnormal operator not sub-2--normal.

Proved that the weight sequence of an $n$--quasinormal unilateral weighted shift is periodic with period at most $n$.

Abstract

In this paper we attempt to lay the foundations for a theory encompassing some natural extensions of the class of subnormal operators, namely the $n$--subnormal operators and the sub-$n$--normal operators. 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 \cite{CU1}, R.E. Curto, S.H. Lee and J. Yoon proved that if an operator $T$ is subnormal, left-invertible, and such that $T^n$ is quasinormal for some $n \le 2$, then $T$ is quasinormal. in \cite{JS}, 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 $n$.