Let $A,B \in \mathbb{R^{n\times n}_{\geq 0}}$ be random matrices where A is symmetric and B is both symmetric and PSD.
Suppose $\|A-\mathbb{E}A\|_{\text{op}}\leq \epsilon_1$ and $\|B-\mathbb{E}B\|_{\text{op}}\leq \epsilon_2$ both with high probability. Is this possible to prove a concentration as follows:
$\| (B+\lambda I)^{1/2} A (B+\lambda I)^{1/2} - \mathbb{E}[(B+\lambda I)^{1/2}] \mathbb{E}[A] \mathbb{E}[(B+\lambda I)^{1/2}]\|_{\text{op}}\leq \epsilon_3$ with high probability?
,where $\lambda \geq 0$ is a scalar and $I$ is the identity matrix.
(I tried to add and subtract $(B+\lambda I)^{1/2} \mathbb{E}[A] (B+\lambda I)^{1/2}$ and use triangle inequality but got stuck to prove the concentration of the second term.)