A conjecture on hyponormality for the Ces\`aro matrix of positive integer order

TL;DR

This paper investigates the hyponormality of Cesàro matrices of positive integer orders, establishing results for orders three and four, and proposing a conjecture for higher orders using elementary calculus and posinormality techniques.

Contribution

It extends known hyponormality results from orders one and two to orders three and four, and formulates a conjecture for all higher orders.

Findings

Cesàro matrices of orders three and four are hyponormal.

The proofs use posinormality and elementary calculus techniques.

A conjecture is proposed for matrices of order greater than four.

Abstract

It is already known that the Ces\`{a}ro matrices of orders one and two are coposinormal, hyponormal operators on $\ell^2$. Here it is shown that the Ces\`{a}ro matrices of order three and four are also coposinormal, hyponormal; the proofs employ posinormality, achieved by means of a diagonal interrupter, and elementary computational techniques from calculus. A conjecture is then propounded for the Ces\`{a}ro matrix of positive integer order greater than four.