Partial isometries and norm equalities for operators

Abstract

Let $H$ be a Hilbert space and $B(H)$ the algebraof all bounded linear operators on $H$. In this paper we shallshow that if $AınB(H)$ is a nonzero closed rangeoperator, then the injective norm $\Vert A^{\ast }\otimes A^{+}+A^{+}\otimes A^{\ast }{\Vert }_{\lambda }$ attains its minimal value 2if and only if $A/\Vert A\Vert$ is a partial isometry.Also we shall give some characterizations of partial isometriesand normal partial isometries in terms of norm equalities foroperators. These characterizations extend previous ones obtainedby A. Seddik in [On the injective norm and characterization ofsome subclasses of normal operators by inequalities or equalities,J. Math. Anal. Appl. 351 (2009), 277–284], and by M. Khosravi in[A characterization of the class of partial isometries, LinearAlgebra Appl. 437 (2012)].