Xia spectrum for some class of operators (Q939144)

The authors introduce power pseudo-semihyponormal operators satisfying \(| T^{2}| \geq U | T^{2}| U^*\) for the polar decomposition of \(T = U | T| \). They show that a doubly commuting \(n\)-tuple of power pseudo-semihyponormal operators is a semi-hyponormal tuple. Thus the results on Xia spectrum and Putnam's inequality of such tuples are obtained because of [\textit{M.\,Ch\(\bar{o}\)}, Glasg. Math. J. 41, No.~1, 13--17 (1999; Zbl 0929.47012)] and [\textit{D.\,Xia}, Integral Equations Oper. Theory 6, 879--896 (1983; Zbl 0539.47015)]. In particular, the paper under review provides some concrete examples to show that there is no inclusion relation for power pseudo-semihyponormal operators and paranormal operators by using shift operators.