Unbounded Hankel operators and moment problems

We find simple conditions for a non-negative Hankel quadratic form to be closable. Under some mild a priori assumption on the associated moments these sufficient conditions turn out to be also necessary. We also describe the domain of the corresponding closed form. This allows us to define unbounded non-negative Hankel operators under minimal assumptions on their matrix elements. The results obtained supplement the classical Widom condition for a Hankel operator to be bounded..Comment: An a priori condition on moments has been omitted in the previous versio

Diagonalizations of two classes of unbounded Hankel operators

We show that every Hankel operator $H$ is unitarily equivalent to a pseudo-differential operator $A$ of a special structure acting in the space $L^2 ({\Bbb R})$. As an example, we consider integral operators $H$ in the space $L^2 ({\Bbb R}_{+})$ with kernels $P (\ln (t+s)) (t+s)^{-1}$ where $P(x)$ is an arbitrary real polynomial of degree $K$. In this case, $A$ is a differential operator of the same order $K$. This allows us to study spectral properties of Hankel operators $H$ with such kernels. In particular, we show that the essential spectrum of $H$ coincides with the whole axis for $K$ odd, and it coincides with the positive half-axis for $K$ even. In the latter case we additionally find necessary and sufficient conditions for the positivity of $H$. We also consider Hankel operators whose kernels have a strong singularity at some positive point. We show that spectra of such operators consist of the zero eigenvalue of infinite multiplicity and eigenvalues accumulating to $+\infty$ and $-\infty$. We find the asymptotics of these eigenvalues

Trace-class approach in scattering problems for perturbations of media

We consider the operators $H_0=M_0^{-1}(x) P(D)$ and $H =M^{-1} (x) P(D)$ where $M_0 (x)$ and $M (x)$ are positively definite bounded matrix-valued functions and $P(D)$ is an elliptic differential operator. Our main result is that the wave operators for the pair $H_0$, $H$ exist and are complete if the difference $M(x)-M_0(x)=O(|x|^{- rho})$, $rho>d$, as $|x| to infty$. Our point is that no special assumptions on $M_0(x)$ are required. Similar results are obtained in scattering theory for the wave equation.Comment: 11 page

Hankel and Toeplitz operators: continuous and discrete representations

We find a relation guaranteeing that Hankel operators realized in the space of sequences $\ell^2 ({\Bbb Z}_{+})$ and in the space of functions $L^2 ({\Bbb R}_{+})$ are unitarily equivalent. This allows us to obtain exhaustive spectral results for two classes of unbounded Hankel operators in the space $\ell^2 ({\Bbb Z}_{+})$ generalizing in different directions the classical Hilbert matrix. We also discuss a link between representations of Toeplitz operators in the spaces $\ell^2 ({\Bbb Z}_{+})$ and $L^2 ({\Bbb R}_{+})$.Comment: Compared to he previous version, Appendix is written in a more detailed wa

A trace formula for the Dirac operator

Our goal is to extend the theory of the spectral shift function to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, we consider a couple of Dirac operators.Comment: 10 page

On semibounded Toeplitz operators

We show that a semibounded Toeplitz quadratic form is closable in the space $\ell^2({\Bbb Z}_{+})$ if and only if its matrix elemens are Fourier coefficients of an absolutely continuous measure. We also describe the domain of the corresponding closed form. This allows us to define semibounded Toeplitz operators under minimal assumptions on their matrix elements.Comment: This is a slightly revised version of the article, arXiv:1603.06229v1, with the same tittle. Some misprints has been removed and some arguments has been made more clear. The results are unchaged. To appear in J. Operator theor