Journal of Operator Theory

Volume 50, Issue 1, Summer 2003 pp. 67-76.

Norm inequalities for operators with positive real part

Authors: Rajendra Bhatia (1) and Xingzhi Zhan (2)
Author institution: (1) Indian Statistical Institute, New Delhi--110 016, India
(2) Department of Mathematics, East China Normal University, Shanghai 200062, P.R. China

Summary: Let $T = A+ {\rm i}B$ with $A$ positive semidefinite and $B$ Hermitian. We derive a majorisation relation involving the singular values of $T,A$, and $B$. As a corollary, we show that $\|T\|^2_p\le\|A\|^2_p + 2^{1-2/p} \|B\|^2_p$, for all $p \ge 2$; and that this inequality is sharp. When $1\le p \le 2$ this inequality is reversed. For $p=1$, we prove the sharper inequality $\|T\|^2_1 \ge \|A\|^2_1 + \| B\|^2_1$. Such inequalities are useful in studying the geometry of Schatten spaces, and our results include and improve upon earlier results proved in this context. Some related in equalities are also proved in the paper.

Keywords: Positive operators, singular values, majorisation, Schatten $p$-norms, inequalities

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