Knowledge on weighted integral operators?

Question

There are tons of books and a huge literature on the properties of the following integral operator: \begin{equation} T(f) = \int_{\mathcal{X}} K(x,\cdot)f(x)dx, \end{equation} where $K(x,z)$ is, say, a Mercer kernel.

I am wondering if people have systematically studied the properties (spectrum for example) of the following weighted integral operator: \begin{equation} T^*(f) = \int_{\mathcal{X}} K(x,\cdot)f(x)d\mu(x), \end{equation} where $\mu$ is a non-Lebesgue measure on $\mathcal{X}$. And I am imagining that if $\mu$ is closer to the Lebesgue measure, $T$ and $T^*$ should be more similar to each other. Are you all aware of anything on this topic? Thanks!

Answer 1

Here's an important theorem which relates kernels to Hilbert-Schmidt operators:

If $T:L^2(X_1,\mu_1)\rightarrow L^2(X_2,\mu_2)$ is a Hilbert-Schmidt operator, then there exist $K\in L^2(X_1\times X_2, \mu_1\times \mu_2)$ so that $$(Tu,v)_{L^2}=\int\int K(x_1,x_2)u (x_1)\overline{v(x_2)}d\mu_1(x_1)d\mu_2(x_2).$$

The converse also holds: given $K\in L^2(X_1\times X_2, \mu_1\times \mu_2)$, then $T$ as written above defines a Hilbert-Schmidt operator, and it satisfies $\|T\|_{\text{HS}}=\|K\|_{L^2}.$

Hilbert-Schmidt operators are, in particular, compact operators. If $K(x,y)=\overline{K(y,x)},$ then $T$ is both compact and self-adjoint, which gives you nice spectral-theoretic results.