Schur multipliers and Pellers theorem

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1 Introduction

The present monograph is devoted to a thorough study of means for Hilbert space operators, especially comparison of (unitarily invariant) norms of oper- ator means and their convergence properties in various aspects. The Hadamard product (or Schur product) A B of two matrices A [aij ], B [bij ] means their entry-wise product [aij bij ]. This notion is a com- mon and powerful technique in investigation of general matrix (and/or opera- tor) norm inequalities, and particularly so in that of perturbation inequalities and commutator estimates. Assume that n n matrices H, K, X Mn (C) are given with H, K 0 and diagonalizations

    H  U diag(s1 , s2 , . . . , sn )U  and K  V diag(t1 , t2 , . . . , tn )V  .

In our previous work [39], to a...

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