Trace Class: Relativeness

Question

Given a Hilbert space $\mathcal{H}$.

Consider a selfadjoint: $$H\in\mathcal{B}(\mathcal{H}):\quad H=H^*$$

Denote trace class: $$\mathcal{B}_1(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):\mathrm{Tr}|A|<\infty\}$$

Does it hold true: $$0\leq A\leq B:\quad BH\in\mathcal{B}_1(\mathcal{H})\implies AH\in\mathcal{B}_1(\mathcal{H})$$ $$0\leq A\leq B:\quad HB\in\mathcal{B}_1(\mathcal{H})\implies HA\in\mathcal{B}_1(\mathcal{H})$$

Answer 1

this is not really an answer.

Claim: $X $ is trace-class if and only if $X^* $ is trace-class.

Proof of the claim.

\begin{align} \text {Tr}(|X|)&=\sum \{\lambda:\ \lambda\in\sigma ((X^*X)^{1/2})\} =\sum \{\lambda^{1/2}:\ \lambda\in\sigma (X^*X)\}\\ \ \\ &=\sum \{\lambda^{1/2}:\ \lambda\in\sigma (XX^*)\} =\sum \{\lambda:\ \lambda\in\sigma ((XX^*)^{1/2})\}\\ \ \\ &=\text {Tr}(|X^*|). \end{align}