---
tag: W3BX66Y47G
kind: concept
name: Hilbert–Schmidt operator
msc: [47B10, 45P05, 47G10]
specializes: [M2NEP1K5AE]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator
same_as:
  wikidata: Q1518047
  wikipedia: Hilbert–Schmidt operator
---

## Summary

A Hilbert–Schmidt operator is a bounded operator between [[Hilbert spaces|HM2HHE3J51]] whose images of an orthonormal basis have a finite sum of squared norms.
These operators are [[compact|M2NEP1K5AE]], and they form a Hilbert space themselves, with the Hilbert–Schmidt norm.
On $L^2$ spaces they are exactly the [[integral operators with square-integrable kernels|CB5G212Q03]].
This makes them the natural class for the Fredholm theory of integral equations.

## Explanations

### L1 human

Think of an operator as an infinite matrix.
It is Hilbert–Schmidt when the sum of the squares of the absolute values of all the entries is finite.
This is the infinite version of the Frobenius norm of a matrix.
For example, the diagonal operator on sequences that multiplies the $n$-th entry by $1/n$ is Hilbert–Schmidt, since $\sum 1/n^2$ is finite.