Hilbert-Schmidt Integral Operator

Hilbert-Schmidt Kernel

A Hilbert-Schmidt kernel is a function \(k: \Omega \times \Omega \rightarrow \mathbb{C}\) such that \(k\) is a square integrable function, written \(k \in L^2(\Omega \times \Omega, \mathbb{C}). This means that\)k \(\)\int_{\Omega} \int_{\Omega} |k(x, y)|^2 dx dx < \infty$$.

Hilbert-Schmidt Integral Operator

For a given kernel \(k\), we can define an associated following linear operator \(T_k: L^2(\Omega, \mathbb{C}) \rightarrow L^2(\Omega, \mathbb{C})\):

\[(T_k u)(x) = \int_{\Omega} k(x, y) u(y) dy\]

To be clear, the input function is \(u \in L^2(\Omega, \mathbb{C})\) and the output function is \((T_k u) \in L^2(\Omega, \mathbb{C})\); the above sentence tells us how to evaluate the function for a given \(x \in \Omega\).