Refinements of recent inequalities via the numerical radii of matrices

Abstract

In this paper, we employ the numerical radii of matrices as a tool to refine and strengthen recent results in matrix inequalities. Our results not only refine previous findings but also provide generalizations of several known results. Among other results, we prove that if A and B are complex matrices of size n×n, then

‖A+B‖≤√(‖A^{∗}A+B^{∗}B‖+2√((1/2)‖A^{∗}B‖²+(1/2)w(A^{∗}B)‖A^{∗}B‖)),

which presents a refinement of the triangle inequality.