On the eigenvalues of certain Hermitian operators

As for general kernels p(x), little has been proved but much has been suspected. Bellman and Latter [1] have obtained upper and lower bounds for the largest eigenvalue of (1.1) for quite general p, these bounds, however, being rather weak for most interesting kernels. More to the point is a conjecture of Kac, Murdock, and Szego [5]. Assume p(x) > 0 end f0(1 +x2)p(x)dx< c. Denote the positive eigenvalues of (1.1) by