Integral operator

Summary

An integral operator is a linear map between spaces of functions that acts by integration against a kernel: $(Tf)(x) = \int K(x, y) f(y)\, dy$. Integral operators are the infinite-dimensional analogue of matrices. They appear as integral transforms, as solution operators of differential equations, and in Volterra integral equations and Fredholm integral equations. With a square-integrable kernel they are compact operators, which gives them a spectral theory close to that of matrices.