Norm inequalities involving a special class of functions for sector matrices

Abstract

In this paper, we present some unitarily invariant norm inequalities for sector matrices involving a special class of functions. In particular, if is a 2ntimes 2n matrix such that numerical range of Z is contained in a sector region S_{alpha } for some alpha in [0,frac{pi }{2} ) , then, for a submultiplicative function h of the class mathcal{C} and every unitarily invariant norm, we have∥h(|Zij|2)∥≤∥hr(sec(α)|Z11|)∥1r∥hs(sec(α)|Z22|)∥1s,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\bigl\\Vert h \\bigl( \\vert Z_{ij} \\vert ^{2} \\bigr) \\bigr\\Vert &\\leq \\bigl\\Vert h^{r} \\bigl( \\sec (\\alpha ) \\vert Z_{11} \\vert \\bigr) \\bigr\\Vert ^{\\frac{1}{r} } \\bigl\\Vert h^{s} \\bigl( \\sec (\\alpha ) \\vert Z_{22} \\vert \\bigr) \\bigr\\Vert ^{ \\frac{1}{s} }, \\end{aligned}$$ \\end{document} where r and s are positive real numbers with frac{1}{r}+frac{1}{s}=1 and i,j=1,2. We also extend some unitarily invariant norm inequalities for sector matrices.