# Schatten class operator

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In [mathematics](/source/Mathematics), specifically [functional analysis](/source/Functional_analysis), a ***p*th Schatten-class operator** is a [bounded linear operator](/source/Bounded_linear_operator) on a [Hilbert space](/source/Hilbert_space) with finite *p*th [Schatten norm](/source/Schatten_norm). The space of *p*th Schatten-class operators is a [Banach space](/source/Banach_space) with respect to the Schatten norm.[1]

Via [polar decomposition](/source/Polar_decomposition), one can prove that the space of *p*th Schatten class operators is an ideal in *B(H)*. Furthermore, the [Schatten norm](/source/Schatten_norm) satisfies a type of [Hölder inequality](/source/H%C3%B6lder_inequality):

- ‖ S T ‖ S 1 ≤ ‖ S ‖ S p ‖ T ‖ S q if S ∈ S p , T ∈ S q and 1 / p + 1 / q = 1. {\displaystyle \|ST\|_{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 ∞ {\displaystyle S_{\infty }} the Banach space of [compact operators](/source/Compact_operator) on *H* with respect to the [operator norm](/source/Operator_norm), the above Hölder-type inequality even holds for p ∈ [ 1 , ∞ ] {\displaystyle p\in [1,\infty ]} . From this it follows that ϕ : S p → S q ′ {\displaystyle \phi :S_{p}\rightarrow S_{q}'} , T ↦ t r ( T ⋅ ) {\displaystyle T\mapsto \mathrm {tr} (T\cdot )} 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 operators](/source/Hilbert%E2%80%93Schmidt_operator). Moreover, the *1*st Schatten class is the space of [trace class](/source/Trace_class) operators.

## References

1. **[^](#cite_ref-1)** ([Conway 2000](#CITEREFConway2000), p. 93)

- [Schatten, Robert](/source/Robert_Schatten) (1960). *Norm Ideals of Completely Continuous Operators*. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin: Springer-Verlag.

- [Conway, John B.](/source/John_B._Conway) (2000). *A Course in Operator Theory*. Graduate Studies in Mathematics. Vol. 21. Providence, R.I.: American Mathematical Society. [ISBN](/source/ISBN_(identifier)) [978-0-8218-2065-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-2065-0).

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