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..
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 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}_{+}) $.
In a paper from 2016 D. R. Yafaev considers Hankel operators associated with Hamburger moment sequences q_n and claims that the corresponding Hankel form is closable if and only if the moment sequence tends to 0. The claim is not correct, since we prove closability for any indeterminate moment sequence but also for certain determinate moment sequences corresponding to measures with finite index of determinacy. It is also established that Yafaev's result holds if the moments satisfy \root{2n}\of{q_{2n}}=o(n).
In a paper from 2016 D. R. Yafaev initiated a study of closable Hankel forms associated with the moments $(m_n)$ of a positive measure with infinite support on the real line. If $m_n=o(1)$ Yafaev characterized the closure of the form based on earlier work on quasi-Carleman operators. We give a new proof of the description of the closure based entirely on moment considerations. The main purpose of the present paper is a description of the self-adjoint Hankel operators associated with closed Hankel forms in the Hilbert space of square summable sequences. We do this not only in the case $m_n=o(1)…
In this paper we give solutions to Hamburger moment problems with missing entries. The problem of completing partial positive sequences is considered. The main result is a characterization of positive definite completable patterns, namely patterns that guarantee the existence of Hamburger moment completion of a partial positive definite sequence. Moreover, several patterns which are not positive definite completable are given.
We consider operators of the type $D^α:H^2(\mathcal{H})\to H^2(\mathcal{H})$, where $D^α$ denotes a fractional differentiation operator, and $Γ_ϕ$ is a Hankel operator. For $α>0$, we characterize boundedness in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries. The first is that our main result does not extend to $α=0$, i.e. Nehari-Page BMOA is not characterized by the natural anti-analytic Carleson embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedne…
It is shown that a positive (bounded linear) operator on a Hilbert space with trivial kernel is unitarily equivalent to a Hankel operator that satisfies double positivity condition if and only if it is non-invertible and has simple spectrum (that is, if this operator admits a cyclic vector). More generally, for an arbitrary positive (bounded linear) operator A on a Hilbert space H with trivial kernel the collection V(A) of all linear isometries V from H into H such that AV is positive as well is investigated. In particular, operators A such that V(A) contains a pure isometry with a given defic…
A Pick function of $d$ variables is a holomorphic map from $Π^d$ to $Π$, where $Π$ is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series $\sum_{n=1}^\infty ρ_n z^{-n}$ with real numbers $ρ_n$ that gives an asymptotic expansion on non-tangential approach regions to infinity. H. Hamburger in 1921 characterized which sequences $\{ρ_n\} $ can occur. We give an extension of Hamburger's results to Pick functions of two variables.
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 H…
In this paper we prove the global in time solvability of the continuous growth--fragmentation--coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth--fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic.
Truncated moment problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established, covering both the even and odd problems, including complete parametrizations of solutions. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new effects which reveals in the indefinite case is that the degenerated moment problem may have infinitely many solutions. However, with a careful application of an indefinite analogue of a step-by-step Schur algorithm a complete description of the set…
Fundamental properties of unbounded composition operators in $L^2$-spaces are studied. Characterizations of normal and quasinormal composition operators are provided. Formally normal composition operators are shown to be normal. Composition operators generating Stieltjes moment sequences are completely characterized. The unbounded counterparts of the celebrated Lambert's characterizations of subnormality of bounded composition operators are shown to be false. Various illustrative examples are supplied.
The power moments of a positive measure on the real line or the circle are characterized by the non-negativity of an infinite matrix, Hankel, respectively Toeplitz, attached to the data. Except some fortunate configurations, in higher dimensions there are no non-negativity criteria for the power moments of a measure to be supported by a prescribed closed set. We combine two well studied fortunate situations, specifically a class of curves in two dimensions classified by Scheiderer and Plaumann, and compact, basic semi-algebraic sets, with the aim at enlarging the realm of geometric shapes on w…
A new method of verifying the subnormality of unbounded Hilbert space operators based on an approximation technique is proposed. Diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established. An approach to this issue via consistent systems of probability measures is invented. The role played by determinate Stieltjes moment sequences is elucidated. Lambert's characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi-analytic vectors, which is a new p…
A new method of verifying the subnormality of unbounded Hilbert space operators based on an approximation technique is proposed. Diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established. An approach to this issue via consistent systems of probability measures is invented. The role played by determinate Stieltjes moment sequences is elucidated. Lambert's characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi-analytic vectors, which is a new p…
We characterize the existence of the $L^1$ solutions of the truncated moments problem in several real variables on unbounded supports by the existence of the maximum of certain concave Lagrangian functions. A natural regularity assumption on the support is required.
In this article we solve four special cases of the truncated Hamburger moment problem (THMP) of degree $2k$ with one or two missing moments in the sequence. As corollaries we obtain, by using appropriate substitutions, the solutions to bivariate truncated moment problems of degree $2k$ for special curves. Namely, for the curves $y=x^3$ (first solved by Fialkow), $y^2=x^3$, $y=x^4$ where a certain moment of degree $2k+1$ is known and $y^3=x^4$ with a certain moment given. The main technique is the completion of the partial positive semidefinite matrix (ppsd) such that the conditions of Curto an…
We prove a solvability theorem for the Stieltjes moment problem on $R^d$ which is based on the multivariate Stieltjes condition $\sum_{n=1}^\infty L(x_j^n)^{-1/(2n)}=+\infty$, $j=1,\dots,d.$ This result is applied to derive a new solvability theorem for the moment problem on unbounded semi-algebraic subsets of $R^d$.
We will consider the indefinite truncated multidimensional moment problem. Necessary and sufficient conditions for a given truncated multisequence to have a signed representing measure $μ$ with ${\rm card}\,{\rm supp}\, μ$ as small as possible are given by the existence of a rank preserving extension of a multivariate Hankel matrix (built from the given truncated multisequence) such that the corresponding associated polynomial ideal is real radical. This result is a special case of a more general characterisation of truncated multisequences with a minimal complex representing measure whose sup…
We characterize the existence of the Lebesgue integrable solutions of the truncated problem of moments in several variables on unbounded supports by the existence of some maximum entropy -- type representing densities and discuss a few topics on their approximation in a particular case, of two variables and 4th order moments.
A criterion for subnormality of unbounded composition operators in L2-spaces, written in terms of measurable families of probability measures satisfying the so-called consistency condition, is established. It becomes a new characterization of subnormality in the case of bounded composition operators. Pseudo-moments of a measurable family of probability measures that satisfies the consistency condition are proved to be given by the Radon-Nikodym derivatives which appear in Lambert's characterization of bounded composition operators. A criterion for subnormality of composition operators induced …
The strong truncated Hamburger moment problem (STHMP) of degree $(-2k_1,2k_2)$ asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on $\mathbb{R}\setminus \{0\}$, such that $β_i=\int x^idμ\; (-2k_1\leq i\leq 2k_2)$. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we solve the STHMP. Then, using the equivalence with the STHMP of degree $(-2k,2k)$, we obtain the solution of the 2-dimensional truncated moment problem (TMP) of degree $2k$ with variety $xy=1$, first solved by Curto and Fialkow. …
