Compact operators whose real and imaginary parts are positive

Abstract

Let T be a compact operator on a Hilbert space such that the operators A=½ (T+T*) and B=1/2i(T-T*) are positive. Let {s_j} be the singular values of T and {α_j} {β_j} the eigenvalues of A,B, all enumerated in decreasing order. We show that the sequence {s²_j} is majorised by {α²j+β²j}. An important consequence is that, when p≥2, ||T||²_p is less than or equal to ||A||²_p+||B||²_p, and when 1≤p≤2, this inequality is reversed.