A Superclass of the Posinormal Operators

TL;DR

This paper introduces a new superclass of posinormal operators, shows that all injective unilateral weighted shifts belong to it, and explores conditions for hyponormality within this class, linking it to lower triangular matrices.

Contribution

It defines a new superclass of posinormal operators, expanding understanding of hyponormality and its relation to weighted shifts and matrix factorizations.

Findings

All injective unilateral weighted shifts are in the superclass.

Sufficient conditions for hyponormality are established.

Connections to lower triangular factorable matrices are demonstrated.

Abstract

The starting place is a brief proof of a well-known result, the hyponormality of $C_k$ (the generalized Ces\`{a}ro operator of order one) for $k \geq 1$. This leads to the definition of a superclass of the posinormal operators. It is shown that all the injective unilateral weighted shifts belong to this superclass. Sufficient conditions are determined for an operator in this superclass to be posinormal and hyponormal. A connection is established between this superclass and some recently-published sufficient conditions for a lower triangular factorable matrix to be a hyponormal bounded linear operator on $\ell^2$.