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Quasi-diagonalization of Hankel operators. (English) Zbl 06822165

Summary: We show that all Hankel operators \(H\) realized as integral operators with kernels \(h(t+s)\) in \(L^2(\mathbb R_+)\) can be quasi-diagonalized as \(H=L^*\Sigma\mathrm L\). Here \(\mathrm L\) is the Laplace transform, \(\Sigma\) is the operator of multiplication by a function (distribution) \(\sigma(\lambda)\), \(\lambda\in\mathbb R\). We find a scale of spaces of test functions on which \(\mathrm L\) acts as an isomorphism. Then \(\mathrm L^*\) is an isomorphism of the corresponding spaces of distributions. We show that \(h=\mathrm L^* \sigma\), which yields a one-to-one correspondence between kernels \(h(t)\) and sigma-functions \(\sigma(\lambda)\) 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 \(\Sigma\) 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 (\(x\in\mathbb R\) and its dual variable).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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