arXiv subjects
Publications and source records attributed to Alexander Pushnitski.
We construct inverse spectral theory for finite rank Hankel operators acting on the Hardy space of the upper half-plane. A particular feature of our theory is that we completely characterise the set of spectral data. As an application of this theory, we prove the genericity of turbulent solutions of the cubic Szegő equation on the real line.
We consider the family of arithmetical matrices given explicitly by $$E(σ,τ)=\left\{\frac{n^σm^σ}{[n,m]^τ}\right\}_{n,m=1}^\infty,$$ where $[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters $σ$ and $τ$ satisfy $ρ:=τ-2σ>0$, $τ-σ>\frac12$ and $τ>0$. We prove that $E(σ,τ)$ is a compact self-adjoint positive definite operator on $\ell^2({\mathbb N})$, and the ordered sequence of eigenvalues of $E(σ,τ)$ obeys the asymptotic relation $$λ_n(E(σ,τ))=\frac{\varkappa(σ,τ)}{n^ρ}+o(n^{-ρ}), \quad n\to\infty,$$ with some $\varkappa(σ,τ)>0$. We give an application of this fact to the asymptotics of singular values of truncated multiplicative Toeplitz matrices with the symbol given by the Riemann zeta function on the vertical line with abscissa $σ<1/2$. We also point out a connection of the spectral analysis of $E(σ,τ)$ to the theory of generalised prime systems.
We consider infinite matrices obtained by restricting Hardy integral kernels to natural numbers. For a suitable class of Hardy kernels we describe the absolutely continuous spectrum, the essential spectrum and the asymptotic spectral density of these matrices.
We discuss an analogue of the First Szegő Limit Theorem for multiplicative Toeplitz operators and highlight the role of the multliplicative Følner condition in this topic.
We give a short proof of the main result of our previous paper [2]: every Schmidt subspace of a Hankel operator is the image of a model space by an isometric multiplier. This class of subspaces is closely related to nearly $S^*$-invariant subspaces, and our proof uses Hitt's theorem on the structure of such subspaces. We also give a formula for the action of a Hankel operator on its Schmidt subspace.
For a bounded Hankel matrix $Γ$, we describe the structure of the Schmidt subspaces of $Γ$, namely the eigenspaces of $Γ^* Γ$ corresponding to non zero eigenvalues. We prove that these subspaces are in correspondence with weighted model spaces in the Hardy space on the unit circle. Here we use the term "weighted model space" to describe the range of an isometric multiplier acting on a model space. Further, we obtain similar results for Hankel operators acting in the Hardy space on the real line. Finally, we give a streamlined proof of the Adamyan-Arov-Krein theorem using the language of weighted model spaces.
We consider the difference $f(H_1)-f(H_0)$, where $H_0=-Δ$ and $H_1=-Δ+V$ are the free and the perturbed Schrödinger operators in $L^2(\mathbb R^d)$, and $V$ is a real-valued short range potential. We give a sharp sufficient condition for this difference to belong to a given Schatten class $\mathbf S_p$, depending on the rate of decay of the potential and on the smoothness of $f$ (stated in terms of the membership in a Besov class). In particular, for $p>1$ we allow for some unbounded functions $f$.
We consider the difference $f(H_1)-f(H_0)$ for self-adjoint operators $H_0$ and $H_1$ acting in a Hilbert space. We establish a new class of estimates for the operator norm and the Schatten class norms of this difference. Our estimates utilise ideas of scattering theory and involve conditions on $H_0$ and $H_1$ in terms of the Kato smoothness. They allow for a much wider class of functions $f$ (including some unbounded ones) than previously available results do. As an important technical tool, we propose a new notion of Schatten class valued smoothness and develop a new framework for double operator integrals.
We consider a class of maps from integral Hankel operators to Hankel matrices, which we call restriction maps. In the simplest case, such a map is simply a restriction of the integral kernel onto integers. More generally, it is given by an averaging of the kernel with a sufficiently regular weight function. We study the boundedness of restriction maps with respect to the operator norm and the Schatten norms.
We characterize all bounded Hankel operators $Γ$ such that $Γ^*Γ$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.
We consider a class of compact Toeplitz operators on the Bergman space on the unit disk. The symbols of operators in our class are assumed to have a sufficiently regular power-like behaviour near the boundary of the disk. We compute the asymptotics of singular values of this class of Toeplitz operators. We use this result to obtain the asymptotics of singular values for a class of banded matrices.
A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite matrix with entries $\{a(jk)\}$ for $j,k\geq1$. Here the $(j,k)$'th term depends on the product $jk$. We study a self-adjoint Helson matrix for a particular sequence $a(j)=(\sqrt{j}\log j(\log\log j)^α))^{-1}$, $j\geq 3$, where $α>0$, and prove that it is compact and that its eigenvalues obey the asymptotics $λ_n\sim\varkappa(α)/n^α$ as $n\to\infty$, with an explicit constant $\varkappa(α)$. We also establish some intermediate results (of an independent interest) which give a connection between the spectral properties of a Helson matrix and those of its continuous analogue, which we call the integral Helson operator.
We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries $(\sqrt{mn}\log(mn))^{-1}$ for $m,n\geq2$. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with $[0,π]$. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on the tools of spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.
We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is a sequence of complex numbers. Helson matrices are considered as linear operators on $\ell^2(\mathbb{N})$. By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that $M(α)$ is non-negative if and only if $α$ is the moment sequence of a measure $μ$ on $\mathbb{R}^\infty$, assuming that $α$ does not grow too fast. We then characterize the non-negative bounded Helson matrices $M(α)$ as those where the corresponding moment measures $μ$ are Carleson measures for the Hardy space of countably many variables. Finally, we give a complete description of the Helson matrices of finite rank, in parallel with the classical Kronecker theorem on Hankel matrices.
Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in $L^2(\mathbb R_+)$. These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel $s^αt^α(s+t)^{-1-2α}$, where $α>-1/2$. Our analysis can be considered as an extension of J.Howland's 1992 paper which dealt with the unweighted case, corresponding to $α=0$.
We describe large classes of compact self-adjoint Hankel operators whose eigenvalues have power asymptotics and obtain explicit expressions for the coefficient in front of the leading term. The results are stated both in the discrete and continuous representations for Hankel operators. We also elucidate two key principles underpinning the proof of such asymptotic relations. We call them {\it the localization principle} and {\it the symmetry principle}. The localization principle says that disjoint components of the singular support of the symbol of a Hankel operator make independent contributions into the asymptotics of eigenvalues. The symmetry principle says that if the singular support of a symbol does not contain the points $1$ and $-1$ in the discrete case (or the points $0$ and $\infty$ in the continuous case), then the spectrum of the corresponding Hankel operator is asymptotically symmetric with respect to the reflection around zero.
We consider functions $ω$ on the unit circle $\mathbb T$ with a finite number of logarithmic singularities. We study the approximation of $ω$ by rational functions and find an asymptotic formula for the distance in the BMO-norm between $ω$ and the set of rational functions of degree $n$ as $n\to\infty$. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.
In the power scale, the asymptotic behavior of the singular values of a compact Hankel operator is determined by the behavior of the symbol in a neighborhood of its singular support. In this paper, we discuss the localization principle which says that the contributions of disjoint parts of the singular support of the symbol to the asymptotic behavior of the singular values are independent of each other. We apply this principle to Hankel integral operators and to infinite Hankel matrices. In both cases, we describe a wide class of Hankel operators with power-like asymptotics of singular values. The leading term of this asymptotics is found explicitly.