If $E$ is an operator space, the non-commutative vector valued $L^p$ spaces $S^p[E]$ have been defined by Pisier for any $1 \leq p \leq \infty$. In this paper a necessary and sufficient condition for a Hankel matrix of the form $(a_{i+j})_{0 \le i,j}$ with $a_k \in E$ to be bounded in $S^p[E]$ is established. This extends previous results of Peller where $E=\C$ or $E=S^p$. The main theorem states that if $1 \leq p < \infty$, $(a_{i+j})_{0 \le i,j}$ is bounded in $S^p[E]$ if and only if there is an analytic function $ϕ$ in the vector valued Besov Space $B_p^{1/p}(E)$ such that $a_n = \hat ϕ(n)$ for all $n \in \N$. In particular this condition only depends on the Banach space structure of $E$. We also show that the norm of the isomorphism $ϕ\mapsto (\hat ϕ(i+j))_{i,j}$ grows as $\sqrt p$ as $p \to \infty$, and compute the norm of the natural projection onto the space of Hankel matrices.
We study small Hankel operators $h_b$ with operator-valued holomorphic symbol $b$ on a class of vector-valued Fock type spaces. We show that the boundedness / compactness of $h_b$ is equivalent to the membership of $b$ to a specific growth space, which is described via a Littlewood-Paley type condition and a Bergman type projection, and estimate the norm of $h_b$. We also establish some properties of duality and density for these Fock spaces.
For a matrix-valued function $Φ\in L^2_{M_{n\times m}}$, it is well-known that the kernel of a block Hankel operator $H_Φ$ is an invariant subspace for the shift operator. Thus, if the kernel is nontrivial, then $\ker H_Φ= ΘH^2_{\mathbb C^r}$ for a natural number $r$ and an $m\times r$ matrix inner function $Θ$ by Beurling-Lax-Halmos Theorem. It will be shown that the size of the matrix inner function $Θ$ associated with the kernel of a block Hankel operator $H_Φ$ is closely related with a certain independency of the columns of $Φ$, which is defined in this paper. As an important application o…
We show that some matrices are Schur multipliers and this is applied to obtain classes of operator-valued Foguel-Hankel operators similar to contractions. This provides partial answers to a problem of K. Davidson and the second author concerning CAR-valued Foguel-Hankel operators.
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.
We show that an infinite Toeplitz+Hankel matrix $T(φ) + H(ψ)$ generates a bounded (compact) operator on $\ell^p(\mathbb{N}_0)$ with $1\leq p\leq \infty$ if and only if both $T(φ)$ and $H(ψ)$ are bounded (compact). We also give analogous characterizations for Toeplitz+Hankel operators acting on the reflexive Hardy spaces. In both cases, we provide an intrinsic characterization of bounded operators of Toeplitz+Hankel form similar to the Brown-Halmos theorem. In addition, we establish estimates for the norm and the essential norm of such operators.
We compute the asymptotics of the determinants of certain $n\times n$ Toeplitz + Hankel matrices $T_n(a)+H_n(b)$ as $n\to\infty$ with symbols of Fisher-Hartwig type. More specifically we consider the case where $a$ has zeros and poles and where $b$ is related to $a$ in specific ways. Previous results of Deift, Its and Krasovsky dealt with the case where $a$ is even. We are generalizing this in a mild way to certain non-even symbols.
In this paper we establish several relations between the determinants of the following structured matrices: Hankel matrices, symmetric Toeplitz + Hankel matrices and Toeplitz matrices. Using known results for the asymptotic behavior of Toeplitz determinants, these identities are used in order to obtain Fisher-Hartwig type results on the asymptotics of certain skewsymmetric Toeplitz determinants and certain Hankel determinants.
Paley's theorem about lacunary coefficients of functions in the classical space $H^1$ on the unit circle is equivalent to the statement that certain Hankel matrices define bounded operators on $\ell^2$ of the nonnegative integers. Since that statement reduces easily to the case where the entries in the matrix are all nonnegative, it must be provable by the Schur test. We give such proofs with interesting patterns in the vectors used in the test, and we recover the best constant in the main case. We use related ideas to reprove the characterization of Paley multipliers from $H^1$ to $H^2$.
In this paper, we study the boundedness and the compactness of the little Hankel operators $h_b$ with operator-valued symbols $b$ between different weighted vector-valued Bergman spaces on the open unit ball $\mathbb{B}_n$ in $\mathbb{C}^n.$ More precisely, given two complex Banach spaces $X,Y,$ and $0 < p,q \leq 1,$ we characterize those operator-valued symbols $b: \mathbb{B}_{n}\rightarrow \mathcal{L}(\overline{X},Y)$ for which the little Hankel operator $h_{b}: A^p_α(\mathbb{B}_{n},X) \longrightarrow A^q_α(\mathbb{B}_{n},Y),$ is a bounded operator. Also, given two reflexive complex Banach s…
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)…
We establish asymptotic formulas for the determinants of finite Toeplitz + Hankel matrices of size N, as N goes to infinity for singular generating functions defined on the unit circle in the special case where the generating function is even, i.e., where the Toeplitz + Hankel matrices are symmetric.
The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.
The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of the form $ (a_{i-j} \pm a_{i+j+1-k})_{i,j=0... N-1} $ with $k$ is fixed. We will show that this example as well as some general classes of operators have expansions that are similar to those that appear in the Strong Szegö Limit Theorem. We also obtain exact identitities for some of the determinants that are analogous to the one derived independently by Ger…
The purpose of this paper is to compute asymptotically Hankel determinants for weights that are supported in a semi-infinite interval.The main idea is to reduce the problem to determinants of other operators whose determinant asymptotics are well known.
Banach Spaces Generated by Finite-Valued Functions: Superreflexive Rigidity, Hankel Operators, and Universality
Konstantin Preslavsky University of Shumen（舒门康斯坦丁·普拉斯夫斯基大学）；University of Plovdiv Paisii Hilendarski（普罗夫迪夫帕伊西·希伦达斯基大学）；Technical University of Varna（瓦尔纳技术大学）
In a 1990 paper Helton and Young showed that under certain conditions the optimal solution of the Nehari problem corresponding to a finite rank Hankel operator with scalar entries can be efficiently approximated by certain functions defined in terms of finite dimensional restrictions of the Hankel operator. In this paper it is shown that these approximants appear as optimal solutions to restricted Nehari problems. The latter problems can be solved using relaxed commutant lifting theory. This observation is used to extent the Helton and Young approximation result to a matrix-valued setting. As …
We establish an asymptotic formula for determinants of truncated Wiener-Hopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.
Necessary and sufficient conditions are given for the solvability of the operator valued two-variable autoregressive filter problem. In addition, in the two variable suboptimal Nehari problem sufficient conditions are given for when a strictly contractive little Hankel has a strictly contractive symbol.
We give a characterization of compact and Fredholm operators on polyanalytic Fock spaces in terms of limit operators. As an application we obtain a generalization of the Bauer-Isralowitz theorem using a matrix valued Berezin type transform. We then apply this theorem to Toeplitz and Hankel operators to obtain necessary and sufficient conditions for compactness. As it turns out, whether or not a Toeplitz or Hankel operator is compact does not depend on the polyanalytic order. For Hankel operators this even holds on the true polyanalytic Fock spaces.
In this paper we show that the theory of Hankel operators in the torus $\T^d$, for $d > 1$, presents striking differences with that on the circle $\T$, starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbols $L^\infty (\T)$ by BMOr$(\T^d)$, a subspace of product BMO, and the singular numbers of Hankel operators by so-called sigma numbers. This leads to versions of the Nehari--AAK and Kronecker theorems, and provides conditions for the existence of solutions of product Pick problems through finite Pick-type matrice…
We establish a connection between compactness of Hankel operators and geometry of the underlying domain through compactness multipliers for the $\overline{\partial}$-Neumann operator. In particular, we prove that any compactness multiplier induces a compact Hankel operator. We also generalize the notion of compactness multipliers to vector fields and matrices and then we use this generalization to generate compact Hankel operators.
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…
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.
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.…
Let $Φ$ be a continuous $n\times n$ matrix-valued function on the unit circle $\T$ such that the $(k-1)$th singular value of the Hankel operator with symbol $Φ$ is greater than the $k$th singular value. In this case, it is well-known that $Φ$ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty}_{(k)}$; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb{D}$ (i.e. the McMillan degree of $Q$ in $\mathbb{D}$ is at most $k$) and $Q$ minimizes the essential suprema of singular values $s_{j}((Φ-Q)(ζ))$, $j\geq0$, with respect to the lexicographic ordering. For each $j\geq 0$…
