Fresh Examples of Hyponormal Terraced Matrices with Transcendental Entries

TL;DR

This paper demonstrates that certain terraced matrices with transcendental entries are hyponormal operators on 2, using Maclaurin series and trigonometric identities, and also identifies a case that is not hyponormal.

Contribution

It provides new examples of hyponormal terraced matrices with transcendental entries and clarifies conditions under which these matrices are hyponormal or not.

Findings

Matrices generated by bc(1+1/(n+1)), bc(tan(1/(n+2))) and bc(sinh(1/(n+2))) are hyponormal on 2.

The matrix generated by bc(tan(1/(n+1))) is not hyponormal.

Uses Maclaurin series and trigonometric identities to establish hyponormality.

Abstract

This note calls attention to an alternative version of the main result from [4], which can be used together with Maclaurin series expansions and trigonometric identities to show that the terraced matrices generated by the sequences $\{\ln (1+1/(n+1)) : n \geq 0\}$, $\{\tan (1/(n+2)) : n \geq 0\}$, and $\{\sinh (1/(n+2)) : n \geq 0\}$ are hyponormal operators on $\ell^2$. Along the way it is also shown that the matrix generated by $\{\tan (1/(n+1)) : n \geq 0\}$ is not a hyponormal operator on $\ell^2$.