Holders inequality for Hilbert Schmidt operators which are also trace class

Question

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 $p=1, q=\infty$?

Answer 1

(I'll assume you are talking about the Schatten norms)

Yes, it's true. It follows from the fact that you can express the trace-norms in terms of singular values.

First, since $\sigma_n(B^*A)\leq \|B\|\,\sigma_n(A)$,
\begin{align} \operatorname{Tr}(B^*A) &=\sum_n\sigma_n(B^*A)\leq\|B\|\,\sum_n\sigma_n(A)=\|A\|_1\,\|B\|. \end{align}

Now, \begin{align} |\operatorname{Tr}(B^*A)| &\leq \|I\|\,\|B^*A\|_1=\|B^*A\|_1=\sum_n\sigma_n(B^*A)\\ \ \\ &\leq \sum_n\sigma_n(A)\sigma_n(B) \leq \left(\sum_n\sigma_n(A)^p\right)^{1/p}\left(\sum_n\sigma_n(B)^q\right)^{1/q}\\ \ \\ &=\|A\|_p\,\|B\|_q. \end{align} The last inequality above is the usual Hölder; the singular value inequality appears as Corollary 4.1 in Gohberg-Krein's Introduction to the Theory of Linear Nonselfadjoint Operators