---
tag: CB5G212Q03
kind: concept
name: Hilbert–Schmidt integral operator
msc: [47G10, 47B10, 45P05]
specializes: [M2NEP1K5AE]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_integral_operator
same_as:
  wikidata: Q5761249
  wikipedia: Hilbert–Schmidt integral operator
---

## Summary

A Hilbert–Schmidt integral operator is an operator $f \mapsto \int K(\cdot, y) f(y) \, dy$ on a space $L^2$ whose kernel $K$ is square-integrable in both variables together.
Such operators are bounded by the $L^2$ norm of the kernel and are [[compact|M2NEP1K5AE]].
They form the [[Schatten class|153N5DW55Z]] of exponent $2$ on $L^2$, and they are the natural setting for the $L^2$ theory of integral equations.

## Explanations

### L1 human

The kernel $K(x, y)$ is like an infinite matrix, with rows indexed by $x$ and columns indexed by $y$.
The Hilbert–Schmidt condition asks that the sum of the squares of all entries be finite, here as a double integral.
This makes the operator behave much like a matrix.
For example, $K(x, y) = \min(x, y)$ on $[0, 1]$ is bounded, so its square has a finite integral, and its operator is Hilbert–Schmidt.