Quasi-diagonalization of Hankel operators

• Published in: Journal d'analyse mathématique (Jerusalem) 2017-10, Vol.133 (1), p.133-182
• Main Author: Yafaev, D. R.
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Analysis; Analysis of PDEs; Dynamical Systems and Ergodic Theory; Eigenvalues; Equivalence; Functional Analysis; Isomorphism; Kernels; Laplace transforms; Mathematics; Mathematics and Statistics; Mellin transforms; Operators (mathematics); Partial Differential Equations
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-017-0030-7

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).