---
tag: 153N5DW55Z
kind: concept
name: Schatten class
msc: [47B10, 47L20]
sources:
  - url: https://en.wikipedia.org/wiki/Schatten_class_operator
same_as:
  wikidata: Q2231600
  wikipedia: Schatten class operator
---

## Summary

For $1 \le p < \infty$, the Schatten $p$-class consists of the compact operators on a Hilbert space whose sequence of singular values is $p$-summable.
These classes are the operator analogues of the sequence spaces $\ell^p$.
The case $p = 1$ is the trace class and the case $p = 2$ is the Hilbert–Schmidt class, the two most used ideals of operators.

## Explanations

### L1 human

A compact operator has a sequence of singular values that tends to zero, like the stretch factors of a matrix.
The Schatten classes sort compact operators by how fast these numbers decrease.
An operator lies in the $p$-class when the sum of the $p$-th powers of its singular values is finite.
For example, the diagonal operator with entries $1/n$ lies in the $p$-class exactly when $p > 1$, because $\sum 1/n^p$ converges exactly then.