w-hyponormal operators have scalar extensions (Q816935)

A bounded linear operator \(T:=U| T| \) on a complex Hilbert space is said to be \(w\)-hyponormal if \(| {\widetilde{T}}^*| \leq| T| \leq| \widetilde T| \). Here, \(T=U| T| \) is the polar decomposition of \(T\), and \(\widetilde T := | T| ^{1/2}U| T| ^{1/2}\) is the Aluthge transform of \(T\). The class of \(w\)-hyponormal operators includes hyponormal, \(p\)-hyponormal and log-hyponormal operators; see [Integral Equations Oper.\ Theory 36, No.~1, 1--10 (2000; Zbl 0938.47021)]. In [J.~Oper.\ Theory 12, 385--395 (1984; Zbl 0573.47016)], \textit{M.~Putinar} proved that hyponormal operators are subscalar of order \(2\). Several authors have exploited Putinar's ideas and generalized his result to the setting of \(p\)-hyponormal, log-hyponormal and \(w\)-hyponormal operators; see [Proc.\ Am.\ Math.\ Soc.\ 131, No.~9, 2753--2759 (2003; Zbl 1044.47016); Arch.\ Math.\ 80, No.~3, 284--293 (2003; Zbl 1048.47012); Stud.\ Math.\ 156, No.~2, 165--175 (2003; Zbl 1038.47017)]. While in [Integral Equations Oper.\ Theory 50, No.~2, 165--168 (2004; Zbl 1099.47023)], \textit{C.~Lin, Y.--B.\ Ruan} and \textit{Y.--K.\ Yan} proved firstly that if \(R\) and \(S\) are bounded linear complex Banach space operators, then \(RS\) is subscalar (resp., subdecomposable) if and only if \(SR\) is. Next, they took a bounded linear complex Hilbert space \(T=U| T| \) and applied this result, in particular, for \(R:=| T| ^{1/2}\) and \(S:=U| T| ^{1/2}\) to deduce that \(T\) is subscalar (resp., subdecomposable) if and only if \(\widetilde T\) is. This, together with Putinar's result and the fact that \(\widetilde{\widetilde T}\) is hyponormal for all \(w\)-hyponormal operators \(T\), allowed them finally to derive that all \(w\)-hyponormal operators are subscalar. The present author shows that \(w\)-hyponormal operators are subscalar of order \(4\).