In [[mathematics]], specifically [[functional analysis]], a '''''p''th Schatten-class operator''' is a [[bounded linear operator]] on a [[Hilbert space]] with finite ''p''th [[Schatten norm]]. The space of ''p''th Schatten-class operators is a [[Banach space]] with respect to the Schatten norm.<ref>{{Harvard citation|Conway|2000|p=93}}</ref>
Via [[polar decomposition]], one can prove that the space of ''p''th Schatten class operators is an ideal in ''B(H)''. Furthermore, the [[Schatten norm]] satisfies a type of [[Hölder inequality]]:
: <math> \| 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. </math>
If we denote by <math> S_\infty</math> the Banach space of [[compact operator]]s on ''H'' with respect to the [[operator norm]], the above Hölder-type inequality even holds for <math> p \in [1,\infty] </math>. From this it follows that <math> \phi : S_p \rightarrow S_q '</math>, <math> T \mapsto \mathrm{tr}(T\cdot ) </math> is a well-defined contraction. (Here the prime denotes (topological) dual.)
Observe that the ''2''nd Schatten class is in fact the Hilbert space of [[Hilbert–Schmidt operator]]s. Moreover, the ''1''st Schatten class is the space of [[trace class]] operators.
== References == {{Reflist}} * {{cite book |last=Schatten |first=Robert |author-link=Robert Schatten |title=Norm Ideals of Completely Continuous Operators |date=1960 |publisher=Springer-Verlag |series=Ergebnisse der Mathematik und ihrer Grenzgebiete |location=Berlin}} * {{cite book |last=Conway |first=John B. |author-link=John B. Conway |title=A Course in Operator Theory |date=2000 |publisher=American Mathematical Society |isbn=978-0-8218-2065-0 |series=Graduate Studies in Mathematics |volume=21 |location=Providence, R.I.}}
[[Category:Operator theory]]