Supraposinormality and hyponormality for the generalized Ces\`aro matrices of order two

TL;DR

This paper investigates the properties of generalized Cesàro matrices of order two, demonstrating they are posinormal, coposinormal, and mostly hyponormal, using the concept of supraposinormality, and proposes a conjecture to extend hyponormality results.

Contribution

It introduces supraposinormality to analyze generalized Cesàro matrices of order two, establishing their posinormality, coposinormality, and hyponormality, and suggests a conjecture for broader hyponormality.

Findings

Generalized Cesàro matrices of order two are posinormal and coposinormal.

Most of these matrices are hyponormal.

A conjecture is proposed to extend hyponormality results.

Abstract

It is well known that the generalized Ces\`aro matrices of order one are hyponormal operators on $\ell^2$, and it has recently been shown that the Ces\`aro matrix of order two is also hyponormal. Here the relatively new concept of supraposinormality is used to show that the generalized Ces\`aro matrices of order two are both posinormal and coposinormal, and that "most" of them are also hyponormal. A conjecture is propounded that would extend the hyponormality result.