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Let ρ be the defining function of a bounded strongly pseudoconvex domain D with smooth boundary in C n . In this paper, we study the essential norm of Hankel operators H f β which are considered as operators from weighted Bergman spaces A p (D , | ρ | α d V) to L q (D , | ρ | β d V) with 1 < p ≤ q < ∞ and - 1 < α , β < ∞ . For f ∈ L 1 (D , | ρ | β d V) , we obtain some quantities in terms of the symbol function f, which are comparable to the essential norm of the Hankel operator H f β . Furthermore, it is shown that the essential norm of H f β is equivalent to the distance norm from itself to compact Hankel operators. [ABSTRACT FROM AUTHOR]

We consider a class of unbounded nonhyperbolic complete Reinhardt domains D n , m , k μ , p , s := { (z , w 1 , ... , w m) ∈ C n × C k 1 × ... × C k m : ∥ w 1 ∥ 2 p 1 e − μ 1 ∥ z ∥ s + ... + ∥ w m ∥ 2 p m e − μ m ∥ z ∥ s < 1 } , where s, p1, ..., pm, μ1, ..., μm are positive real numbers and n, k1, ..., km are positive integers. We show that if a Hankel operator with anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space A 2 (D n , m , k μ , p , s) , then it must be zero. This gives an example of high dimensional unbounded complete Reinhardt domain that does not admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. [ABSTRACT FROM AUTHOR]

An extension of slant Hankel operator, namely, the k-th-order λ-slant Hankel operator on the Lebesgue space L 2 (T n) , where T is the unit circle and n ≥ 1, a natural number, is described in terms of the solution of a system of operator equations, which is subsequently expressed in terms of the product of a slant Hankel operator and a unitary operator. The study is further lifted in Calkin algebra in terms of essentially k-th-order λ-slant Hankel operators on L 2 (T n) . [ABSTRACT FROM AUTHOR]

We consider the weighted Bergman space A ψ 2 of all holomorphic functions on B n square integrable with respect to an exponential weight measure e - ψ d V on B n , where ψ (z) : = 1 1 - | z | 2 . We characterize boundedness (or compactness) of Toeplitz operators and Hankel operators on A ψ 2 . [ABSTRACT FROM AUTHOR]

The Bergman projection P α , induced by a standard radial weight, is bounded and onto from L ∞ to the Bloch space B . However, P α : L ∞ → B is not a projection. This fact can be emended via the boundedness of the operator P α : BMO 2 (Δ) → B , where BMO 2 (Δ) is the space of functions of bounded mean oscillation in the Bergman metric. We consider the Bergman projection P ω and the space BMO ω , p (Δ) of functions of bounded mean oscillation induced by 1 < p < ∞ and a radial weight ω ∈ M . Here M is a wide class of radial weights defined by means of moments of the weight, and it contains the standard and the exponential-type weights. We describe the weights such that P ω : BMO ω , p (Δ) → B is bounded. They coincide with the weights for which P ω : L ∞ → B is bounded and onto. This result seems to be new even for the standard radial weights when p ≠ 2 . [ABSTRACT FROM AUTHOR]

(μ; ν)-Hankel operators between separable Hilbert spaces were introduced and studied recently (A. Mirotin and E. Kuzmenkova, μ-Hankel operators on Hilbert spaces, Opuscula Math., 41 (2021), 881–899). This paper is devoted to generalization of (μ; ν)-Hankel operators to the case of (non-separable in general) 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 also considered. [ABSTRACT FROM AUTHOR]

Hankel operators have numerous realizations and form one of the most important classes of operators in the spaces of analytic functions. These operators can be defined as those having Hankel matrices (i.e. matrices whose entries depend only on the sum of the indices) with respect to some orthonormal basis for a separable Hilbert space. This paper continues the authors' research of 2021 which introduced a new class of operators in Hilbert spaces; i.e., -Hankel operators, with a complex parameter. These operators act in a separable Hilbert space and have matrices in some orthonormal basis of the space whose diagonals orthogonal to the main diagonal present geometric progressions with common ratio . Thus, the classical Hankel operators correspond to the case . The main result of the article is the normality criterion for -Hankel operators. By analogy with Hankel operators, the class of operators under consideration has concrete realizations in the form of integral operators which enables us to apply the abstract results, and thereby contribute to the theory of integral operators. We consider an example of these realizations in the Hardy space on the unit disk. Also, we give some criteria for the self-adjointness and normality of these operators. [ABSTRACT FROM AUTHOR]

In this paper the notion of slantification of a Hankel operator on the space H 2 (T n) , the Hardy space of n-torus, is introduced. Various properties including hyponormality, isometric behaviour, co-isometric behaviour and compactness of these operators are also studied. [ABSTRACT FROM AUTHOR]

The Berger–Coburn phenomenon of Hankel operators was recently reported by Hu and Virtanen for the Schatten classes C p , 1 < p < ∞ . But a careful reading of [14] finds that in the case 1 < p < 2 , there is a technical problem in their proof. In this paper we first fix this problem. We then establish the Berger–Coburn phenomenon for the Lorentz ideals C p + and C p - , 1 < p < ∞ . Last but not least, we show that there is no Berger–Coburn phenomenon for the trace class C 1 , for C 1 + , and for the Macaev ideal C ∞ - . [ABSTRACT FROM AUTHOR]

We prove equivalence of the classical Adamyan-Arov-Krein theorem (Adamyan, Arov and Krein in Mat Sb 86:34–75, 1971) and Mustafayev’s recent result in (Mustafayev in Integr Equ Oper Theory 92:1–21, 2020) about essential norm of Hankel and function of model operators, respectively. We also prove a quantitative version of Bessonov’s compactness theorem for truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

We introduce a k th-order slant Hankel operator on the Lebesgue space of n-torus, for k = (k 1 , ... , k n) , where each k t ≥ 1 , (1 ≤ t ≤ n) is an integer. Our main result is to obtain equivalent characterizations for a bounded operator on L 2 (T n) to be a k th-order slant Hankel operator. We also discuss various commutative, spectral and other properties of these operators. [ABSTRACT FROM AUTHOR]

There is a bounded Hankel operator on the Paley–Wiener space of a disc in R 2 which does not arise from a bounded symbol. [ABSTRACT FROM AUTHOR]

In this paper, we study the kernels of Hankel operators on the full Fock space. Firstly, we study the matrix and the symbol of S-Toeplitz operators and S-Hankel operators; Secondly, we show that there exists a strong cyclic vector for S ∗ by Champernowne sequence; as an application, we show that the invariant subspaces of S and the kernels of Hankel operators are one to one. [ABSTRACT FROM AUTHOR]

In this survey paper, we consider the problem of which nonempty bounded convex subset of the complex plane is the numerical range of some bounded linear operator on a complex separable Hilbert space. We start in Section 1 with general operators and move subsequently to operators in certain special classes such as normal and hyponormal operators in Section 2, Toeplitz operators in Section 3, Hankel operators in Section 4, compact operators in Section 5, nilpotent operators and roots of identity in Section 6, Sn-matrices and companion matrices in Section 7, and, finally, nonnegative matrices in Section 8. The known main results will be briefly sketched, which are interspersed with relevant unsolved problems. [ABSTRACT FROM AUTHOR]

On the setting of the Siegel upper half-space we study the spaces of bounded and vanishing mean oscillations which are defined in terms of the Berezin transform, we use them to characterize bounded and compact Hankel operators on Bergman space. When the symbols are specialized to be holomorphic, we study its corresponding Hankel operators in terms of Bloch space of the Siegel upper half-space. [ABSTRACT FROM AUTHOR]

We completely describe the boundedness and compactness of Hankel operators with general symbols acting on Bergman spaces with exponential type weights. [ABSTRACT FROM AUTHOR]

We study algebraic properties of two Toeplitz operators on the weighted Bergman space on the unit disk with harmonic symbols. In particular the product property and commutative property are discussed. Further we apply our results to solve a compactness problem of the product of two Hankel operators on the weighted Bergman space on the unit bidisk. [ABSTRACT FROM AUTHOR]

We study big Hankel operators H f ν : A ω p → L ν q generated by radial Bekollé–Bonami weights ν , when 1 < p ≤ q < ∞ . Here the radial weight ω is assumed to satisfy a two-sided doubling condition, and A ω p denotes the corresponding weighted Bergman space. A characterization for simultaneous boundedness of H f ν and H f ¯ ν is provided in terms of a general weighted mean oscillation. Compared to the case of standard weights that was recently obtained by Pau et al. (Indiana Univ Math J 65(5):1639–1673, 2016), the respective spaces depend on the weights ω and ν in an essentially stronger sense. This makes our analysis deviate from the blueprint of this more classical setting. As a consequence of our main result, we also study the case of anti-analytic symbols. [ABSTRACT FROM AUTHOR]

We consider the Hankel operator with symbols of forms and study the relationship between the compactness of this operator and the compactness of ∂̄-Neumann operator. [ABSTRACT FROM AUTHOR]

In this paper, we characterize bounded Toeplitz product T f T g and Hankel product H f ∗ H g on Fock space F α 2 for two polynomials f and g in z and z ¯ . As a consequence, we obtain when Toeplitz operator T f or Hankel operator H f with the polynomial symbol f in z and z ¯ is bounded. [ABSTRACT FROM AUTHOR]

Motivated by results in uniform algebras, a distance localization formula in C*-algebras is established in the framework of the Allan–Douglas localization principle, and is used to derive a locality result for products of Hankel operators as compact perturbations of Hankel operators. Using localization and certain QC functions, it is proved that the essential spectrum of the commutator of a Toeplitz and a Hankel operator is antipodal symmetric under a mild condition on the Hankel symbol function. Under the same condition the essential spectrum of a Hankel operator also exhibits this symmetry. Conjugates of interpolating Blaschke products and characteristic functions are constructed that satisfy the condition, while examples show the condition is only sufficient. [ABSTRACT FROM AUTHOR]

Kolmogorov n-widths and Hankel singular values are two commonly used concepts in model reduction. Here, we show that for the special case of linear time-invariant (LTI) dynamical systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov n-width of an LTI system equals its (n + 1)st Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov n-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov n-width. [ABSTRACT FROM AUTHOR]

Given some regular weight ω on the unit disk D , let L ω p be the space of all Lebesgue measurable functions on D for which ‖ f ‖ L ω p = ∫ D | f (z) | p ω (z) d A (z) 1 p < ∞ , and let A ω p be the weighted Bergman space of all holomorphic functions f ∈ L ω p . For all possible 1 < p , q < ∞ , we characterize these symbols f for which the induced Hankel operators H f are bounded (or compact) from A ω p to L ω q . While doing that we develop an approach to obtain some L ω p estimates on the canonical solution to ∂ ¯ -equation. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the concept of asymptotic Hankel operators on H 2 (D) to the Hardy space H 2 (D n) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce ith-partial Hankel operators on H 2 (D n) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on H 2 (D n) . It is also shown that a Toeplitz operator with symbol ϕ is asymptotic Hankel if and only if ϕ is holomorphic function in L ∞ (T n) . [ABSTRACT FROM AUTHOR]

We survey the main properties of the cubic Szegő equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of high Sobolev norms and the effects of weak damping. [ABSTRACT FROM AUTHOR]

Based on the results in A. Feintuch (1989), this work sheds light upon some interesting properties of strongly asymptotically Toeplitz and Hankel operators, and relations between these two classes of operators. Indeed, among other things, two main results here are (a) vanishing Toeplitz and Hankel operators forms an ideal, and (b) finding the distance of a strongly asymptotically Toeplitz operator from the set of vanishing Toeplitz operators. [ABSTRACT FROM AUTHOR]

The zero sets of the Bergman space A ω p induced by either a radial weight ω admitting a certain doubling property or a non-radial Bekollé-Bonami type weight are characterized in the spirit of Luecking's results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz's factorization result from 1977 for functions in A ω p . The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling measures for the weighted Bergman space A ω p induced by a doubling weight are also studied. Several open problems related to the scheme of the paper are posed. [ABSTRACT FROM AUTHOR]

The trace-class property of Hankel operators (and their derivatives with respect to the parameter) with strongly oscillating symbol is studied. The approach used is based on Peller’s criterion for the trace-class property of Hankel operators and on the precise analysis of the arising triple integral using the saddle-point method. Apparently, the obtained results are optimal. They are used to study the Cauchy problem for the Korteweg-de Vries equation. Namely, a connection between the smoothness of the solution and the rate of decrease of the initial data at positive infinity is established. [ABSTRACT FROM AUTHOR]

We prove a Weiss conjecture on ß-admissibility of observation operators for discrete and continuous γ-hypercontractive semigroups of operators, by representing them in terms of shifts on weighted Bergman spaces and using a reproducing kernel thesis for Hankel operators. Particular attention is paid to the case γ = 2, which corresponds to the unweighted Bergman shift. [ABSTRACT FROM AUTHOR]

Consider Hankel operators Hf on the weighted Bergman space La2(B,dvα). In this paper we characterize the membership of Hf∗Hfs/2=|Hf|s in the norm ideal CΦ, where 0 and the symmetric gauge function Φ is allowed to be arbitrary. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded convex Reinhardt domain in C2 and ϕ∈C(Ω¯). We show that the Hankel operator Hϕ is compact if and only if ϕ is holomorphic along every non-trivial analytic disc in the boundary of Ω. [ABSTRACT FROM AUTHOR]

We here study finite impulse response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class $$\mathcal {U}$$ ). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in $$\mathcal {U}$$ , as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs. Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in $$\mathcal {U}$$ , so is the whole family. A key role is played by Hankel matrices. [ABSTRACT FROM AUTHOR]

We establish concrete rank formulas for a class of finite rank truncated Toeplitz operators on the model spaces $$K^2_u:= H^2 \ominus uH^2$$ in terms of their symbol functions. We also establish some rank inequalities for truncated Toeplitz operators on finite dimensional model spaces. [ABSTRACT FROM AUTHOR]

On complete pseudoconvex Reinhardt domains in ℂ, we show that there is no nonzero Hankel operator with anti-holomorphic symbol that is Hilbert-Schmidt. In the proof, we explicitly use the pseudoconvexity property of the domain. We also present two examples of unbounded non-pseudoconvex domains in ℂ that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. In the first example the Bergman space is finite dimensional. However, in the second example the Bergman space is infinite dimensional and the Hankel operator $${H_{{{\bar z}_1}{{\bar z}_2}}}$$ is Hilbert-Schmidt. [ABSTRACT FROM AUTHOR]

The classes of right and left γ-generating matrices were introduced by D. Z. Arov in the 1980s. These matrices play an important role in the description of solutions of the indeterminate Nehari problem. In the present paper, the classes of the so-called generalized right and left γ-generating matrices, being resolvent matrices of the Nehari-Takagi problem, are introduced. For matrices of these classes, some factorization theorems are proved, and the connection between the class of generalized γ-generating matrices and the class of generalized j -inner matrix valued functions is found. Subclasses of singular, regular, and strongly regular generalized -generating matrices are introduced and studied. [ABSTRACT FROM AUTHOR]

Let M be a von Neumann algebra with a faithful normal semifinite trace τ. The noncommutative Hardy space Hp(M) associates with A, which is a subdiagonal algebra of M. We define the Hankel operator Ht on Hp(M), and we obtain that the norm ∥Ht∥ is equal to d(t;A) and is also the equivalent of the BMO(Msa) norm of t for every t∈M, where Msa are the self-adjoint operators in M. [ABSTRACT FROM AUTHOR]

In this paper, we show that, on the generalized Fock space $${F^p(\varphi)}$$ with $${1 < p < \infty}$$ , the essential norm of a noncompact Toeplitz operator $${T_\mu}$$ with $${|\mu|}$$ being a Fock-Carleson measure equals its distance to the set of compact Toeplitz operators. Moreover, the distance is realized by infinitely many compact Toeplitz operators. Our approach is also available on the Bergman space setting. [ABSTRACT FROM AUTHOR]

Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators in both frameworks of standard and variable exponent Lebesgue spaces are considered in this paper. The main aim is to describe certain dependencies between the Fredholm property of some Wiener-Hopf operators acting between variable exponent Lebesgue spaces and the invertibility of Wiener-Hopf plus and minus Hankel operators on all the standard Lebesgue spaces. Different types of Fourier symbols will be used but special focus will be considered on the Wiener subclass of almost periodic matrix functions. In the first part of the paper we will give a survey of investigations on related results. This will be useful at the end of the paper to derive the above mentioned dependencies between the operators under study. [ABSTRACT FROM AUTHOR]

In this paper we consider Hankel operators on a family of Fock-type spaces and characterize their boundedness and compactness in terms of a certain notion of bounded and vanishing mean oscillation. This extends the main results of Seip and Youssfi (J Geom Anal 23:170-201, ) to symbol functions that are not necessarily anti-holomorphic. We also give geometric descriptions for the spaces BMO and VMO which were defined in terms of the Berezin transform. [ABSTRACT FROM AUTHOR]

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The distance from the nonconstant function φ in L∞(T) to the set Fconst of all constant functions is estimated in terms of Hankel operators on the Hardy space H2(D) over the unit disk D = {z ∊ C : : ∣z∣ < 1}. We give a sufficient condition ensuring the equality dist(φ,Fconst) = ∥'k∥L∞. Some other dist-formulas are also discussed. [ABSTRACT FROM AUTHOR]

This paper deals with the inverse problem for the class of orthogonal functions that for the scalar case was introduced by Ellis and Gohberg (J Funct Anal 109:155-198, ). The problem is reduced to a linear equation with a special right hand side. This reduction allows one to solve the inverse problem for square matrix functions under conditions that are natural generalizations of those appearing in the scalar case. These conditions lead to a unique solution. Special attention is paid to the polynomial case. A number of partial results are obtained for the non-square case. Various examples are given to illustrate the main results and some open problems are presented. [ABSTRACT FROM AUTHOR]

Singular spectrum analysis has recently become an attractive tool in a broad range of applications. Its main mechanism of alternating between rank reduction and Hankel projection to produce an approximation to a particular component of the original time series, however, deserves further mathematical justification. One paramount question to ask is how good an approximation that such a straightforward apparatus can provide when comparing to the absolute optimal solution. This paper reexamines this issue by exploiting a natural parametrization of a general Hankel matrix via its Vandermonde factorization. Such a formulation makes it possible to recast the notion of singular spectrum analysis as a semi-linear least squares problem over a compact feasible set, whence global optimization techniques can be employed to find the absolute best approximation. This framework might not be immediately suitable for practical application because global optimization is expectedly more expensive, but it does provide a theoretical baseline for comparison. As such, our empirical results indicate that the simpler SSA algorithm usually is amazingly sufficient as a handy tool for constructing exploratory model. The more complicated global methods could be used as an alternative of rigorous affirmative procedure for verifying or assessing the quality of approximation. [ABSTRACT FROM AUTHOR]

Let $${\mathcal{R}}$$ be an arbitrary bounded complete Reinhardt domain in $${\mathbb{C}^n}$$ . We show that for $${n \geq 2}$$ , if a Hankel operator with an anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $${A^2(\mathcal{R})}$$ , then it must equal zero. This fact has previously been proved only for strongly pseudoconvex domains and for a certain class of pseudoconvex domains. [ABSTRACT FROM AUTHOR]

Based on the different kinds of auxiliary operators and corresponding operator relations, we will present conditions which characterize the invertibility of matrix Wiener–Hopf plus Hankel operators having different Fourier symbols in the class of almost periodic elements. To reach such invertibility characterization, we introduce a new kind of factorization for AP matrix functions. Additionally, under certain conditions, we will obtain the one-sided and two-sided inverses of the matrix Wiener–Hopf plus Hankel operators in study. [ABSTRACT FROM AUTHOR]

In this paper we perform an explicit diagonalization of Hankel integral operators K, K, K,... 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 (Proc Am Math Soc 136:2067-2071, ). [ABSTRACT FROM AUTHOR]

An explicit state-space approach is presented for solving the super-optimal Nehari-extension problem. The approach is based on the all-pass dilation technique developed in (Jaimoukha and Limebeer in SIAM J Control Optim 31(5):1115-1134, ) which offers considerable advantages compared to traditional methods relying on a diagonalisation procedure via a Schmidt pair of the Hankel operator associated with the problem. As a result, all derivations presented in this work rely only on simple linear-algebraic arguments. Further, when the simple structure of the one-block problem is taken into account, this approach leads to a detailed and complete state-space analysis which clearly illustrates the structure of the optimal solution and allows for the removal of all technical assumptions (minimality, multiplicity of largest Hankel singular value, positive-definiteness of the solutions of certain Riccati equations) made in previous work (Halikias et al. in SIAM J Control Optim 31(4):960-982, ; Limebeer et al. in Int J Control 50(6):2431-2466, ). The advantages of the approach are illustrated with a numerical example. Finally, the paper presents a short survey of super-optimization, the various techniques developed for its solution and some of its applications in the area of modern robust control. [ABSTRACT FROM AUTHOR]