---
tag: EM8Q0636YT
kind: definition
name: Schatten class
msc: [47B10, 47L20]
defines: 153N5DW55Z
uses: [HM2HHE3J51, M2NEP1K5AE, QSJKPKPM7H]
sources:
  - url: https://en.wikipedia.org/wiki/Schatten_class_operator
---

## Definition

Let $H$ and $K$ be complex [[Hilbert spaces|HM2HHE3J51]], and let $1 \le p < \infty$.
A [[compact operator|M2NEP1K5AE]] $T \colon H \to K$ with [[singular values|QSJKPKPM7H]] $s_1(T) \ge s_2(T) \ge \cdots$ belongs to the *Schatten $p$-class* $S_p(H, K)$ if
$$\norm{T}_p = \Big( \sum_{n=1}^{\infty} s_n(T)^p \Big)^{1/p} < \infty.$$
The number $\norm{T}_p$ is the Schatten $p$-norm of $T$.

## Remarks

Example: on $\ell^2$, the diagonal operator $T e_n = e_n / n$ has $s_n(T) = 1/n$, so $T \in S_p$ for every $p > 1$.
Non-example: the same operator is not in $S_1$, because $\sum 1/n$ diverges.
$S_1$ is the trace class, and $S_2$ is the class of Hilbert–Schmidt operators, with $\norm{T}_2^2 = \sum_{i} \norm{T e_i}^2$ for any orthonormal basis $(e_i)$ of $H$.
For $p \le q$, $S_p \subseteq S_q$ and $\norm{T}_q \le \norm{T}_p$.
Each $S_p(H) = S_p(H, H)$ is a two-sided ideal of the algebra of bounded operators on $H$, and a Banach space under $\norm{\cdot}_p$.
The finite-rank operators are dense in each $S_p$, and the operator norm satisfies $\norm{T} \le \norm{T}_p$.
Some authors also allow $0 < p < 1$, where $\norm{\cdot}_p$ is only a quasi-norm.