Classes of Operators Related to Subnormal Operators

Professor Raúl Curto

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. 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 a 2020 research paper with S.H. Lee and J. Yoon, we proved that if an operator T is subnormal, left-invertible, and such that T^2 is quasinormal, then T is quasinormal. In subsequent work, P.Pietrzycki and J. Stochel improved this result by removing the assumption of left invertibility.

Here, we consider suitable analogs of this result for 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.

The talk is based on joint work with Thankarajan Prasad (University of Calicut, Kerala, India), and recently pub-lished as: R.E. Curto and T. Prasad, Classes of operators related to subnormal operators, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM (2026) 120, 41(2026); 15 pages.

To participate in this event via Zoom, go to https://uiowa.zoom.us/j/95316149275.