Hilbert–Schmidt operator

Summary

A Hilbert–Schmidt operator is a bounded operator between Hilbert spaces whose images of an orthonormal basis have a finite sum of squared norms. These operators are compact, 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. This makes them the natural class for the Fredholm theory of integral equations.