Generalizations of Recent Singular Value Inequalities of Commutators

Abstract

Given four $n\times n$ complex matrices $A, B, X, Y$, finding possible bounds for the singular values of the new matrix $AX+YB$ has been of interest.  In this paper, we discuss this interest and prove some new bounds that sharpen recently found bounds in the literature. Applications of the obtained results include bounds for unitarily invariant bounds and bounds for the real part of certain matrix forms.