Trace of an integral operator

Summary

The trace of an integral operator with a continuous kernel $K(s, t)$ on $[a, b]$ is the integral of the kernel along the diagonal, $\int_a^b K(s, s) \, ds$. It is the continuous analogue of the sum of the diagonal entries of a matrix. In Fredholm theory, minus the trace is the coefficient of $\lambda$ in the Fredholm determinant $D(\lambda) = 1 - \lambda \int_a^b K(s, s) \, ds + \cdots$. For a trace class operator with continuous kernel, it equals the operator trace and the sum of the eigenvalues.