On Some Inequalities for the Generalized Euclidean Operator Radius

Abstract

In the literature, there are many criteria to generalize the concept of a numerical radius; one of the most recent and interesting generalizations is the so-called generalized Euclidean operator radius, which reads: ωpT1,⋯,Tn:=supx=1∑i=1nTix,xp1/p,p≥1, for all Hilbert space operators T1,⋯,Tn. Simply put, it is the numerical radius of multivariable operators. This study establishes a number of new inequalities, extensions, and generalizations for this type of numerical radius. More precisely, by utilizing the mixed Schwarz inequality and the extension of Furuta’s inequality, some new refinement inequalities are obtained for the numerical radius of multivariable Hilbert space operators. In the case of n=1, the resulting inequalities could be considered extensions and generalizations of the classical numerical radius.