Quasi-diagonalization of Hankel operators
We show that all Hankel operators H realized as integral operators with kernels h(t + s) in L 2(R+) can be quasi-diagonalized as H = L*ΣL. Here L is the Laplace transform, Σ is the operator of multiplication by a function (distribution) σ(λ), λ ∈ R. We find a scale of spaces of test functions on which L acts as an isomorphism. Then L* is an isomorphism of the corresponding spaces of distributions. We show that h = L*σ, which yields a one-to-one correspondence between kernels h(t) and sigma-functions σ(λ) of Hankel operators. The sigma-function of a self-adjoint Hankel operator H contains substantial information about its spectral properties. Thus we show that the operators H and Σ have the same number of positive and negative eigenvalues. In particular, we find necessary and sufficient conditions for sign-definiteness of Hankel operators. These results are illustrated with examples of quasi-Carleman operators generalizing the classical Carleman operator with kernel h(t) = t −1 in various directions. The concept of the sigmafunction leads directly to a criterion (equivalent, of course, to the classical Nehari theorem) for boundedness of Hankel operators. Our construction also shows that every Hankel operator is unitarily equivalent by the Mellin transform to a pseudodifferential operator with amplitude which is a product of functions of one variable (x ∈ R and its dual variable).
Unable to display preview. Download preview PDF.
- N. K. Nikolski, Operators, Functions, and Systems: an Easy Reading, Vol. 1: Hardy, Hankel, and Toeplitz, Amer. Math. Soc., Providence, RI, 2002.Google Scholar
- D. R. Yafaev, The discrete spectrum in the singular Friedrichs model, in Differential Operators and Spectral Theory, American Mathematical Society, Providence, RI, 1999, pp. 255–274.Google Scholar