Some generaliztions of numerical radius inequalities including the off-diagonal parts of block matrices

Abstract

In this paper, we introduce several numerical radius inequalities involving off-diagonal part of 2 × 2 positive semidefinite block matrices and their diagonal blocks. It is shown that if A, B,C ∈ Mn(C) are such that [A B* B C] ≥ 0, then w2r(B) ≤ 1/2 √||A4rα + A4r(1−α)|| ||C4rα + C4r(1−α)|| , and w2r(B) ≤ ||αA r/α + (1 − α)C r/1−α , for 0 < α < 1, r ≥ 1. Moreover, we establish some numerical radius inequalities for products and sums of matrices.