  The Carleman operator is defined as integral operator with kernel
$(t+s)^{-1}$ in the space $L^2 ({\Bbb R}_{+}) $. This is the simplest example
of a Hankel operator which can be explicitly diagonalized. Here we study a
class of self-adjoint Hankel operators (we call them quasi-Carleman operators)
generalizing the Carleman operator in various directions. We find explicit
formulas for the total number of negative eigenvalues of quasi-Carleman
operators and, in particular, necessary and sufficient conditions for their
positivity. Our approach relies on the concepts of the sigma-function and of
the quasi-diagonalization of Hankel operators introduced in the preceding paper
of the author.
