Compact operators whose real and imaginary parts are positive

Let T T be a compact operator on a Hilbert space such that the operators A = 1 2 ( T + T ∗ ) A = \frac {1}{2} (T + T^{*}) and B = 1 2 i ( T − T ∗ ) B = \frac {1}{2i}(T-T^{*}) are positive. Let { s j } \{ s_{j}\} be the singular values of T T and { α j } , { β j } \{ \alpha _{j}\} , \{ \beta _{j}\} the eigenvalues of A , B A,B , all enumerated in decreasing order. We show that the sequence { s j 2 } \{ s^{2}_{j}\} is majorised by { α j 2 + β j 2 } \{ \alpha ^{2}_{j} + \beta ^{2}_{j}\} . An important consequence is that, when p ≥ 2 ,   ‖ T ‖ p 2 p \ge 2, ~\| T\| ^{2}_{p} is less than or equal to ‖ A ‖ p 2 + ‖ B ‖ p 2 \| A\| ^{2}_{p} + \| B\| ^{2}_{p} , and when 1 ≤ p ≤ 2 , 1\le p \le 2, this inequality is reversed.