Here's a simple question for which I can't find an answer. Let $W$ be a square real matrix with eigenvalues all real and positive ($W$ is not necessarily symmetric nor diagonalizable) and $A$ a real matrix of same size (no assumptions are made on $A$). Is it true that $$ \operatorname{Tr}(AW) \leq \vert \vert\vert A \vert \vert\vert \operatorname{Tr}(W) $$ where $ \vert \vert\vert A \vert \vert\vert = \sup_{\Vert x \Vert = 1} \Vert A x \Vert$? (the norm $\Vert . \Vert$ is the Euclidean norm)
Thank you for your help.
Take $$W=\begin{pmatrix}1&a\\0&1\end{pmatrix}$$ then $\mathrm{tr}W=2$. Take $$A=\begin{pmatrix}2&0\\1&2\end{pmatrix}$$ then $$AW=\begin{pmatrix}2&2a\\1&a+2\end{pmatrix}$$ and $\mathrm{tr}AW=4+a$. Now, set $\sigma=\|A\|_{\mathrm{op}}$. You have $$4+a\leq 2\sigma$$ where $\sigma$ is independent of $a$. This is clearly false for $a$ large enough.