Norm inequalities for operators with positive real part

Let T = A + iB 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 parallel toTparallel to(p)(2) less than or equal to parallel toAparallel to(p)(2) + 2(1-2/p) parallel toBparallel to(p)(2), for all p greater than or equal to 2; and that this inequality is sharp. When 1 less than or equal to p less than or equal to 2 this inequality is reversed. For p = 1, we prove the sharper inequality parallel toTparallel to(1)(2) greater than or equal to parallel toAparallel to(1)(2) + parallel toBparallel to(1)(2). Such inequalities axe useful in studying the geometry of Schatten spaces, and our results include and improve upon earlier results proved in this context. Some related inequalities are also proved in the paper.