Positive integral operators in unbounded domains
The author considers a generalization to the case of the entire real line (\(I={\mathbb R}\)) of well-known results on the representation of positive integral operators with continuous kernel \(k(x,y)\) acting in \(L^2(I)\), \({\mathcal K} \phi=\int_I k(x,y) \phi(y) dy\), where \(I\) is a compact subinterval of \({\mathbb R}\). The main results are as follows: If \({\mathcal K}\) is positive and \(k(x,x)\in L^1({\mathbb R})\), then \({\mathcal K}\) is Hilbert-Schmidt and therefore compact. If, in addition, \(k(x,x)\rightarrow 0\) as \(| x| \rightarrow \infty\), then \(k\) is represented by an absolutely and uniformly convergent bilinear series of continuous eigenfunctions and \({\mathcal K}\) is trace class (Theorem 1). Replacing the first assumption by the stronger one \(k^{1/2} (x,x)\in L^1({\mathbb R})\), one gets that then \(k\in L^1({\mathbb R})\) and the bilinear series also converges in \(L^1({\mathbb R})\) (Theorem 2). Sharp norm bounds are given and Mercer's theorem is derived as a special case.
positive integral operator with continuous kernel
scientific article; zbMATH DE number 2093829