The N\"{o}rlund operator on $\ell^2$ generated by the sequence of positive integers is hyponormal

Abstract

First it is shown that the N örlund matrix associated with the sequence of positive integers is a coposinormal operator on ℓ 2 . This fact then turns out to be useful for showing that this operator is also posinormal and hyponormal. In contrast with the analogous weighted mean matrix result [6], the proof of hyponormality is accomplished without resorting to determinants or Sylvester's criterion.

Supraposinormality and hyponormality for the generalized Cesàro matrices of order two

“…Note again that these operators are in B(ℓ 2 ) for all α > −1 (not just for α ≥ 0). Comparison of the entries verifies that the Cesàro matrix (C (0) , 2) of order two (take α = 0) is precisely the matrix studied in [9], where it was shown that, as a result of the relationship…”

A conjecture on hyponormality for the Cesàro matrix of positive integer order

“…(See also [7].) The computations in [6] centered on coposinormality, and the diagonal form of P emerged somewhat serendipitously from those computations.…”