I have a complex-valued function of two real variables - let's denote it $m(x,y)$. I am told that it should have a spectral decomposition of the form: $$ m(x,y)=\sum_n\lambda_n\chi_n^*(x)\chi_n(y) $$ It is true that $m(x,y)=m(y,x)$. This is in a physics context, so 'nice' properties like continuity and differentiability can be assumed as necessary. On physics grounds, I am reasonably sure that the $\chi_n$ should be orthogonal to each other with respect to the usual inner product, and somewhat sure that we should have $\lambda_n\in\mathbb{R}$.
I want to know why such a decomposition should exist, and if it is unique. If it's not unique, are there any obvious further constraints that would enforce uniqueness?
I found a similar problem here, but that only dealt with a real-valued function and didn't mention uniqueness of the decomposition.
My relevant math background is limited to one course in real analysis and just enough complex analysis to calculate a contour integral, so further references would be handy!