For $\a,\b>0$ and for a locally integrable function (or, more generally, a distribution) $\f$ on $(0,\be)$, we study integral ooperators ${\frak G}^{\a,\b}_\f$ on $L^2(\R_+)$ defined by $\big({\frak G}^{\a,\b}_\f f\big)(x)=\int_{\R_+}\f\big(x^\a+y^\b\big)f(y)dy$. We describe the bounded and compact operators ${\frak G}^{\a,\b}_\f$ and operators ${\frak G}^{\a,\b}_\f$ of Schatten--von Neumann class $\bS_p$. We also study continuity properties of the averaging projection $\Q_{\a,\b}$ onto the operators of the form ${\frak G}^{\a,\b}_\f$. In particular, we show that if $\a\le\b$ and $\b>1$, then ${\frak G}^{\a,\b}_\f$ is bounded on $\bS_p$ if and only if $2\b(\b+1)^{-1}<p<2\b(\b-1)^{-1}$.
In this paper we perform an explicit diagonalization of Hankel integral operators $ K^{(0)}, K^{(1)}, K^{(2)}, ... $ It turns out that each of these operators has a simple purely absolutely continuous spectrum filling in the interval $ [-1,1] $. This generalizes a result of Kostrykin and Makarov (2008).
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.
It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener-Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More general, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which are Fredholm can be expressed in terms of the partial indices of an associated Wiener-Hopf factorization problem. In this paper we establish corresponding results for Toeplitz + Hankel operators and singular integral operators with flip under the assumption th…
$(μ;ν)$-Hankel operators between separable Hilbert spaces were introduced and studied recently (\textit{$μ$-Hankel operators on Hilbert spaces}, Opuscula Math., \textbf{41} (2021), 881--899). This paper, is devoted to generalization of $(μ;ν)$-Hankel operators to the (non-separable in general) case of Hardy spaces over compact and connected Abelian groups. In this setting bounded $(μ;ν)$-Hankel operators are fully described under some natural conditions. Examples of integral operators are considered.
We deal with a class of one-parameter family of integral transforms of Bargmann type arising as dual transforms of fractional Hankel transform. Their ranges are identified to be special subspaces of the weighted hyperholomorphic left Hilbert spaces, generalizing the slice Bergman space of the second kind. Their reproducing kernel is given by closed expression involving the $\star$-regularization of Gauss hypergeometric function. We also discuss their basic properties such as their boundedness and we determinate their singular values. Moreover, we describe their compactness and membership in $p…
The paper describes various approaches to the invertibility of Toeplitz plus Hankel operators in Hardy and $l^p$-spaces, integral and difference Wiener-Hopf plus Hankel operators and generalized Toeplitz plus Hankel operators. Special attention is paid to a newly developed method, which allows to establish necessary, sufficient and also necessary and sufficient conditions of invertibility, one-sided and generalized invertibility for wide classes of operators and derive efficient formulas for the corresponding inverses. The work also contains a number of problems whose solution would be of inte…
A characterisation is given of bounded embeddings from weighted $L^2$ spaces on bounded intervals into $L^2$ spaces on the half-plane, induced by isomorphisms given by the Laplace transform onto weighted Hardy and Bergman spaces (Zen spaces). As an application necessary and sufficient conditions are given for the boundedness of truncated Hankel and Toeplitz integral operators, including the weighted case.
In this note, we find sufficient conditions for an operator with kernel of the form $A(x)B(y)-A(x)B(y)/(x-y)$ (which we call a Tracy-Widom type operator) to be the square of a Hankel operator. We consider two contexts: infinite matrices on $\ell^2$, and integral operators on the Hardy space $H^2(\mathbb{T})$. The results can be applied to the discrete Bessel kernel, which is significant in random matrix theory.
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.…
We consider diffusive systems, regarded as input/output systems with a kernel given as the Fourier--Borel transform of a measure in the left half-plane. Associated with these are a family of weighted Hankel integral operators, and we provide conditions for them to be bounded, Hilbert--Schmidt or nuclear, thereby generalizing results of Widom, Howland and others.
We find a concrete integral formula for the class of generalized Toeplitz operators $T_a$ in Bergman spaces $A^p$, $1<p<\infty$, studied in an earlier work by the authors. The result is extended to little Hankel operators. We give an example of an $L^2$-symbol $a$ such that $T_{|a|} $ fails to be bounded in $A^2$, although $T_a : A^2 \to A^2$ is seen to be bounded by using the generalized definition. We also confirm that the generalized definition coincides with the classical one whenever the latter makes sense.
We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows. Let $Ξ= (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2Ξ$, consider the Hankel operator $$Γ_f (g)(x)=\int_Ξ f(x+y) g(y) \, dy, \quad x \inΞ.$$ Then $Γ_f$ extends to a bounded operator on $L^2(Ξ)$ if and only if there is a bounded function $b$ on $\mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2…
For bounded Lebesgue measurable functions $f,g,ϕ$ and $ψ$ on the unit circle, $P_{+}fP_{+}+P_{-}gP_{+} +P_{+}ϕP_{-}+P_{-}ψP_{-}$ is called a generalized singular integral operator (GSIO) on $L^{2}(\mathbb{T})$, where $P_{+}$ is the Riesz projection, $P_{-}=I-P_{+}.$ In this paper, we relate GSIOs to a number of operators, including Cauchy singular integral operator, (dual) truncated Toeplitz operator, Foguel-Hankel operator, multiplication operator, Toeplitz plus Hankel operator etc. We establish the short exact sequences associated of the $C^{*}-$algebras generated by GSIOs with bounded or qu…
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 $K$ o…
We show that all Hankel operators $H$ realized as integral operators with kernels $h(t+s)$ in $L^2 ({\Bbb R}_{+}) $ can be quasi-diagonalized as $H= {\sf L}^* Σ{\sf L} $. Here ${\sf L}$ is the Laplace transform, $Σ$ is the operator of multiplication by a function (distribution) $σ(λ)$, $λ\in {\Bbb R}$. We find a scale of spaces of test functions where ${\sf L} $ acts as an isomorphism. Then ${\sf L}^*$ is an isomorphism of the corresponding spaces of distributions. We show that $h= {\sf L}^* σ$ which yields a one-to-one correspondence between kernels $h(t)$ and sigma-functions $σ(λ)$ of Hankel…
For a fixed reals $c>0$, $a>0$ and $α>-\frac{1}{2}$, the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions as some authors call it, are the eigenfunctions of the finite Hankel transform operator, denoted by $\mathcal{H}_c^α$, which is the integral operator defined on $L^2(0,1)$ with kernel $H_c^α(x,y)=\sqrt{cxy}J_α(cxy)$. Also, they are the eigenfunctions of the positive, self-adjoint compact integral operator $\mathcal{Q}_c^α=c\mathcal{H}_c^α\mathcal{H}_c^α.$ The CPSWFs play a central role in many applications such as the analysis of 2d-radial signals. Moreover, a re…
The Carleman operator is defined as integral operator with kernel $(t+s)^{-1}$ in the space $L^2 ({\Bbb R}_{+}) $. This is the simplest example of a Hankel operator which can be explicitly diagonalized. Here we study a class of self-adjoint Hankel operators (we call them quasi-Carleman operators) generalizing the Carleman operator in various directions. We find explicit formulas for the total number of negative eigenvalues of quasi-Carleman operators and, in particular, necessary and sufficient conditions for their positivity. Our approach relies on the concepts of the sigma-function and of th…
We consider a class of Hankel operators $H$ realized in the space $L^2 ({\Bbb R}_{+}) $ as integral operators with kernels $h(t+s)$ where $h(t)=P (\ln t) t ^{-1}$ and $P(X)= X^n+p_{n-1} X^{n-1}+\cdots$ is an arbitrary real polynomial of degree $n$. This class contains the classical Carleman operator when $n =0$. We show that a Hankel operator $H$ in this class can be reduced by an {\it explicit} unitary transformation (essentially by the Mellin transform) to a differential operator $A = v Q(D) v$ in the space $L^2 ({\Bbb R}) $. Here $Q(X)= X^n+ q_{n-1} X^{n-1}+\cdots$ is a polynomial determine…
We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order $0$ with geometric origin and $f$ a multiplication operator by a function. When $f$ is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudo-differential calculus is available, variations of Connes' residue trace theorem and related integral fo…
