Compact operators whose real and imaginary parts are positive

Let $T$ be a compact operator on a Hilbert space such that the operators $A = \frac{1}{2} (T + T^{*})$ and $B = \frac{1}{2i}(T-T^{*})$ 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 $\{ s^{2}_{j}\}$is majorised by $\{ \alpha ^{2}_{j} + \beta ^{2}_{j}\}$. An important consequence is that, when $p \ge 2, ~\Vert T\Vert ^{2}_{p}$ is less than or equal to $\Vert A\Vert ^{2}_{p} + \Vert B\Vert ^{2}_{p}$, and when $1\le p \le 2,$ this inequality is reversed.