Schatten class operator

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The space of pth Schatten class operators is the subspace consisting of the operators $T\in B(X)$ with a finite pth Schatten norm, here X denotes a Hilbert space. This is a Banach space in the Schatten norm.

One can via polar decomposition easily prove that the space of pth Schatten class operators is an ideal in B(X) and that the Schatten norm satisfies a type of Hölder inequality:

$\| S T\| _{S_1} \leq \| S\| _{S_p} \| T\| _{S_q} \ \mbox{if} \ S \in S_p , \ T\in S_q \mbox{ and } 1/p+1/q=1$

If we define $S_\infty := B(X)$ this 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 welldefined contraction. Here the prime denotes (topological) dual.