On positive Hilbert-Schmidt operators
The aim of this paper is to provide sufficient conditions on a class of integral self-adjoint Hilbert-Schmidt operators to be positive definite. Namely, given a compact subset \(G\subset \mathbb R^N\) and \(F(x,y)\) a continuous function on \(G\times G\) satisfying \(F(x,y)=\overline{F(y,x)}\), the author considers integral operators of the form \[ Tf(x)=\int_G\exp(F(x,y))f(y)\, dy \] for \(f\in L^2(G,dx)\). Several conditions on \(F\) are provided to guarantee positivity of \(T\) (in the positive definite sense). The research of these questions is motivated by problems in the area of integrable \(N\)-particle systems of elliptic Calogero-Moser type.
scientific article; zbMATH DE number 6153946