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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]

In the view of existing works on operators (namely, λ- Hankel and λ- Slant Hankel operators etc.) done by Avendano, Datt and Aggrawal, we take up an operator equation ... and solve it for λ, μ ∈ C. Its solution yields an important class of slant Hankel operators, (namely, (λ, μ)-slant Hankel operators). After that, we provide certain basic properties of (λ, μ)- slant Hankel operator on L²(T) of unit circle T. Finally, we come up with a results for (λ, μ)- slant Hankel operator, which is analogous to classical Kronecker theorem for Hankel matrix. [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]

The authors provide a correct proof of Theorem 1.2 and correct the statement of Theorem 2.6 in their paper, which appeared in Trans. Amer. Math. Soc. 375 (2022), 3733–3753. [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]

A class of operators is introduced (\(\mu\)-Hankel operators, \(\mu\) is a complex parameter), which generalizes the class of Hankel operators. Criteria for boundedness, compactness, nuclearity, and finite dimensionality are obtained for operators of this class, and for the case \(|\mu| = 1\) their description in the Hardy space is given. Integral representations of \(\mu\)-Hankel operators on the unit disk and on the Semi-Axis are also considered.

In this paper, we study the numerical ranges of (finite) Hankel matrices and Hankel operators (on an infinite-dimensional space). The main concern is which nonempty bounded convex set △ in the plane is the numerical range W (A) of a Hankel matrix or a Hankel operator A. In Section 1 below, we prove results for △ a line segment, an elliptic disc, or a polygonal region. For example, we show that if △ is a closed elliptic disc in the plane, then a necessary and sufficient condition for the existence of an n -by- n Hankel matrix A n with W (A n) equal to △ for all n ≥ 2 is that 0 is in △. In Section 2, we use the Megretskiĭ–Peller–Treil characterization of Hermitian Hankel operators to obtain an analogous condition for △ a (finite) line segment in the plane. [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]

We consider Hankel and Toeplitz operators on H 2 (T n) , the Hardy space of the n -torus T n. Given symbols φ and ψ in L ∞ (T n) with suitable properties, we obtain necessary and sufficient conditions for the Hankel operator H ψ , n and the Toeplitz operator T φ , n to commute. We then extend the study to the more general situation where no assumptions are imposed on φ , and provide new, non-trivial necessary conditions for the commutativity of H ψ , n and T φ , n. We also show that certain well known commutativity results between Hankel and Toeplitz operators in the one-variable case do not extend to the multivariable setting. [ABSTRACT FROM AUTHOR]

We give a complete characterization of Schatten class Hankel operators Hƒ acting on weighted Segal-Bargmann spaces F2(φ) using the notion of integral distance to analytic functions in Cn and Hörmander's ∂-theory. Using our characterization, for ƒ ∈ L∞and 1< p < ∞, we prove that Hƒ is in the Schatten class Sp if and only if Hƒ ∈ Sp, which was previously known only for the Hilbert-Schmidt class S2 of the standard Segal-Bargmann space F2(φ) with φ (z) = α |z|2. [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]

A class of operators is introduced (μ-Hankel operators, µ is a complex parameter), which generalizes the class of Hankel operators. Criteria for boundedness, compactness, nuclearity, and finite dimensionality are obtained for operators of this class, and for the case |μ| = 1 their description in the Hardy space is given. Integral representations of μ-Hankel operators on the unit disk and on the Semi-Axis are also considered. [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 consider Toepliz operators with integrable symbols acting on Bergman spaces Ap, 1

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]

We propose four new convolutions exhibiting convenient factorization properties associated with two finite interval integral transformations of Fourier‐type together with their norm inequalities. Moreover, we study the solvability of a class of integral equations of Wiener‐Hopf plus Hankel type (on finite intervals) with the help of the factorization identities of such convolutions. Fourier‐type series are used to produce the solution formula of such equations, and a Shannon‐type sampling formula is also obtained. [ABSTRACT FROM AUTHOR]

We show that Property (P) of ∂Ω , compactness of the ∂ ¯ -Neumann operators N 1 , and compactness of Hankel operator on a smooth bounded pseudoconvex complete Hartogs domain Ω = { (z , w 1 , w 2 , ... , w n) ∈ C n + 1 ∣ ∑ k = 1 n | w k | 2 < e − 2 ϕ (z) , z ∈ D } are equivalent, where D is a smooth bounded connected open set in C . [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]