Weak subnormality of operators

Abstract

We consider the gap between weak subnormality and 2-hyponormality for Toeplitz operators. In addition, we study the spectrum of the minimal partially normal extension of a weakly subnormal operator, and the inverse of an invertible weakly subnormal operator. Introduction.In [15], the notion of weak subnormality of an operator was introduced as a generalization of subnormality, with an aim at providing a model for 2-hyponormal operators. Weak subnormality was conceived as a notion at least as strong as hyponormality, and as a tool to understand the gap between hyponormality and 2-hyponormality; however, it remains open whether every 2-hyponormal operator is weakly subnormal. In this paper we explore weak subnormality of operators.In Section 1, we consider the gap between weak subnormality and 2-hyponormality for Toeplitz operators. In Section 2, we consider the passage from the spectrum of a weakly subnormal operator whose self-commutator has closed range to the spectrum of its minimal partially normal extension. In Section 3, we provide an example of an invertible subnormal operator whose inverse is neither 2-hyponormal nor weakly subnormal.

A note on Putinar's matricial models

“…which will be called the n-th minimal partially normal extension of T. It was ( [10], [11], [13]) shown that 2-hyponormal =⇒ weakly subnormal =⇒ hyponormal and the converses of both implications are not true in general. It was [13]…”