Compact operators whose real and imaginary parts are positive

摘要

Let T be a compact operator on a Hilbert space such that the operators A = 1/2 (T + T*) and B = 1/2i (T - T*) are positive. Let {s(j)} be the singular values of T and {alpha (j)}, {beta (j)} the eigenvalues of A, B, all enumerated in decreasing order. We show that the sequence {s(j)(2)} is majorised by {alpha (2)(j) + beta (2)(j)}. An important consequence is that, when p greater than or equal to 2, parallel toT parallel to (2)(p) is less than or equal to parallel toA parallel to (2)(p) + parallel toB parallel to (2)(p), and when 1 < p less than or equal to 2, this inequality is reversed. [References: 7]