Operator norm of family matrix products

Question

Consider the following problem:
Let $A(\xi),B(\xi),C(\xi)$ be three families of matrices indexed by $\xi\in\mathbb{N}$, such that $$\sup_{\xi\in\mathbb{N}}\|B(\xi)\|_{op}<+\infty,\quad \sup_{\xi\in\mathbb{N}}\|C(\xi)A(\xi)\|<+\infty$$ and every $A(\xi)$ is self-adjoint, diagonal and uniformly bounded. Is it true that $$\sup_{\xi\in\mathbb{N}}\|A(\xi)B(\xi)C(\xi)\|_{op}<+\infty? $$ I have tried many different matrix-norm inequalities and identities, such as the ones in this paper: https://arxiv.org/pdf/2005.08299.pdf
But they don't seem to work as I can't use that $A(\xi)^{-1}$ is uniformly bounded...

Answer 1

I believe this is false, I found the following counter-example: \begin{align*} A(\xi) = \begin{pmatrix} 1&0 \\ 0&\xi^{-1} \end{pmatrix},B(\xi) = \begin{pmatrix} 0&1 \\ 0&0 \end{pmatrix}, C(\xi) = \begin{pmatrix} 1&1 \\ 0&\xi \end{pmatrix} \end{align*}