Let $A$ be an $n \times m$ matrix with non-negative entries and $B \in \mathbb{R^{n\times n}_{\geq 0}}, C \in \mathbb{R^{m\times m}_{\geq 0}}$ be random matrices where B and C are both symmetric and PSD.
Suppose $\|A-\mathbb{E}A\|_{\text{op}}\leq \epsilon_1$, $\|B-\mathbb{E}B\|_{\text{op}}\leq \epsilon_2$ , $\|C-\mathbb{E}C\|_{\text{op}}\leq \epsilon_3 $ all with high probability. Is this possible to prove a concentration as follows:
$\| (B+\lambda I)^{1/2} A (C+\lambda I)^{1/2} - \mathbb{E}[(B+\lambda I)^{1/2}] \mathbb{E}[A] \mathbb{E}[(C+\lambda I)^{1/2}]\|_{\text{op}}\leq \delta$ with high probability? ,where $\lambda \geq 0$ is a scalar and $I$ is the identity matrix.
If this is hard to prove, a proof of a special case where $A$ is an $n \times n $ symmetric matrix and $B=C$ is also very appreciated.
(For the symmetric case, 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.)
