# Schatten class operator

$\| S T\| _{S_1} \leq \| S\| _{S_p} \| T\| _{S_q} \ \mbox{if} \ S \in S_p , \ T\in S_q \mbox{ and } 1/p+1/q=1.$
If we denote by $S_\infty$ the Banach space of compact operators on H with respect to the operator norm, the above Hölder-type inequality even holds for $p \in [1,\infty]$. From this it follows that $\phi : S_p \rightarrow S_q '$, $T \mapsto \mathrm{tr}(T\cdot )$ is a well-defined contraction. (Here the prime denotes (topological) dual.)