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 ∥T∥_p² ≤ ∥A∥_p² + 2^1-2/p ∥B∥ _p², for all p ≥ 2; and that this inequality is sharp. When 1 ≤ p ≤ 2 this inequality is reversed. For p = 1, we prove the sharper inequality ∥T∥₁² ≥ ∥A∥ ₁² + ∥B∥₁². 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 inequalities are also proved in the paper.