Schatten class operator

The space of "p"th Schatten class operators is the subspace consisting of the operators $Tin\; B(X)$ with a finite "p"th 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 "p"th Schatten class operators is an ideal in "B(X)" and that the Schatten norm satisfies a type of Hölder inequality:

: $|\; S\; T|\; \_\{S\_1\}\; leq\; |\; S|\; \_\{S\_p\}\; |\; T|\; \_\{S\_q\}\; mbox\{if\}\; S\; in\; S\_p\; ,\; Tin\; S\_q\; mbox\{\; and\; \}\; 1/p+1/q=1$

If we define $S\_infty\; :=\; B(X)$ this holds for $p\; in\; [1,infty]$. From this it follows that $phi\; :\; S\_p\; ightarrow\; S\_q\; \text{'}$, $T\; mapsto\; mathrm\{tr\}(Tcdot\; )$ is a welldefined contraction. Here the prime denotes (topological) dual.