Does Hilbert-Schmidt operators which are also trace class, satisfy Holders inequality? That is, we have two Hilbert Schmidt operators $A$ and $B$. Is the following true?
$$\langle A, B \rangle \leq \lVert A \rVert_p \lVert B \rVert_q $$

such that $1/p + 1/q = 1$ Specifically, is it true for $p=1, q=\infty$?