Criteria for Hankel operators to be sign-definite

We show that total multiplicities of negative and positive spectra of a self-adjoint Hankel operator $H$ with kernel $h(t)$ and of an operator of multiplication by some real function $s(x)$ coincide. In particular, $\pm H\geq 0$ if and only if $\pm s(x)\geq 0$. The kernel $h(t)$ and its "sign-function" $s(x)$ are related by an explicit formula. An expression of $h(t)$ in terms of $s(x)$ leads to an exponential representation of $h(t)$. Our approach directly applies to various classes of Hankel operators. In particular, for Hankel operators of finite rank, we find an explicit formula for the total multiplicity of their negative and positive spectra.