A sufficient condition for the hyponormality of \(z\)-Cesàro operators (Q2762821)

The aim of the paper under review is to present a proof of the fact that the so-called \(z\)-Cesàro operator on \(\ell^2\) defined by the matrix NEWLINE\[NEWLINEM=\left(\begin{matrix} a_0& 0& 0&\dots\\ a_1&a_1& 0&\dots\\ a_2&a_2&a_2&\dots\\ \dots&\dots&\dots&\dots\end{matrix}\right)NEWLINE\]NEWLINE is hyponormal for \(a_n=1/(n+1)^z\) for every \(n\geq 0\), provided \(1\leq z<\infty\). NEWLINENEWLINENEWLINEWe note that the very last reasoning in the proof seems to be wrong. This remark also agrees with the results of \textit{H. C. Rhaly jun.} [Bull. Lond. Math. Soc. 21, No. 4, 399-406 (1989; Zbl 0695.47024)]. More precisely, Theorem 2.6 in this quoted paper shows that, if \(M\) is hyponormal and \(L=\lim_{n\to\infty}(n+1)\|a_n\|\), then \(\sum_{n=1}^{\infty}a_n^2\leq L^2\). In our case \(L=0\) for \(z>1\), so the last inequality would imply the contradiction \(a_1=a_2=\cdots=0\). Actually, a few lines above his Theorem 2.6, H.C. Rhaly Jr.\ himself explicitly notes (and proves) that ``nonzero \(M\) cannot be hyponormal when \(L=0\)''.