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. They form the Schatten class of exponent $2$ on $L^2$, and they are the natural setting for the $L^2$ theory of integral equations.