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In this paper, we characterize the boundedness and compactness of the block dual Toeplitz operators acting on the orthogonal complement of the Fock spaces, and the compactness of the finite sum of two dual Toeplitz operators products. The commutator and semi-commutator induced by the block dual Toeplitz operators are considered. [ABSTRACT FROM AUTHOR]

In this paper, the authors completely characterize when two dual truncated Toeplitz operators are essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact. Their main idea is to study dual truncated Toeplitz operators via Hankel operators, Toeplitz operators and function algebras. [ABSTRACT FROM AUTHOR]

In this paper, we describe the minimal reducing subspaces of Toeplitz operators induced by non-analytic monomials on the weighted Bergman spaces and Dirichlet spaces over the unit ball B 2 . It is proved that each minimal reducing subspace M is finite dimensional, and if dim M ≥ 3, then M is induced by a monomial. Furthermore, the structure of commutant algebra ν (T w ¯ N z N ) := { M w N ∗ M z N , M z N ∗ M w N } ′ is determined by N and the two dimensional minimal reducing subspaces of T w ¯ N z N . We also give some interesting examples. [ABSTRACT FROM AUTHOR]

In this paper, we first completely characterize the complex symmetric Toeplitz operators T φ on the Hardy spaces H 2 (D) with conjugations C p i , j and C n . Next, we give a method to determine the coefficients of φ (z) when T φ is complex symmetric on H 2 (D) with the conjugation C σ , which partially solves a problem raised by [2]. Finally, we consider the complex symmetric Toeplitz operators T φ on the weighted Bergman spaces A 2 (B n) and the pluriharmonic Bergman spaces b 2 (B n) with conjugations C V , where V is a symmetric permutation matrix. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the various properties of the Toeplitz operators acting on the Dirichlet spaces. First, we consider the matrix representation of Toeplitz operators with harmonic and monomial symbols. Second, we establish the expansivity and contractivity of the Toeplitz operators T φ with monomial symbols φ . Third, we give a necessary and sufficient conditions for the normality and hyponormality of the Toeplitz operators T φ with such symbols on the Dirichlet spaces. [ABSTRACT FROM AUTHOR]

This paper is dedicated to extending the focusing/defocusing Calogero–Moser–Sutherland cubic derivative Schrödinger equations (CMSdNLS) i ∂ t u + ∂ x 2 u = ± u D + | D | | u | 2 , D = - i ∂ x , x ∈ R or x ∈ T : = R / 2 π Z , which were initially introduced in Matsuno (Phys Lett A 278(1–2):53–58, 2000; Inverse Probl 18:1101–1125, 2002; J Phys Soc Jpn 71(6):1415–1418, 2002; Inverse Prob 20(2):437–445, 2004), Abanov et al. (J Phys A 42(13): 135201, 2009), Gérard and Lenzmann (The Calogero–Moser derivative nonlinear Schrödinger equation, Communications on Pure and Applied Mathematics. ) and Badreddine (On the global well-posedness of the Calogero–Sutherland derivative nonlinear Schrödinger equation, Pure and Applied Analysis. ; Traveling waves and finite gap potentials for the Calogero–Sutherland derivative nonlinear Schrödinger equation, Annales de l’Institut Henri Poincaré, Analyse Non Linéaire. ), to a system of two matrix-valued variables, leading to the following intertwined system, i ∂ t U + ∂ x 2 U = - 1 2 U D + | D | V ∗ U - 1 2 V D + | D | U ∗ U , i ∂ t V + ∂ x 2 V = - 1 2 V D + | D | U ∗ V - 1 2 U D + | D | V ∗ V. This system enjoys a Lax pair structure, enabling the establishment of an explicit formula for general solutions on both the 1-dimensional torus and the real line. Consequently, this system can be regarded as an integrable perturbation and extension of both the linear Schrödinger equation and the CMSdNLS equations. [ABSTRACT FROM AUTHOR]

A truncated Hankel operator $ B_{f}^{\theta }: K_{\theta }\rightarrow K_{\theta } $ B f θ : K θ → K θ is a compression of a Hankel operator $ H_{f}:H^{2}\rightarrow H^{2} $ H f : H 2 → H 2 to a model space $ K_{\theta } $ K θ . In this paper we firstly characterize when a bounded truncated Hankel operator is compact. Then we study the product of a truncated Hankel operator and a truncated Toeplitz operator, and prove, for some special symbols, several necessary and sufficient conditions for the product to be zero or compact respectively. [ABSTRACT FROM AUTHOR]

We discuss de Branges–Rovnyak spaces ℋ ⁢ (b) generated by nonextreme and rational functions b and local Dirichlet spaces of order m introduced in [S. Luo, C. Gu and S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, Adv. Math. 385 2021, Paper No. 107748]. In that paper, the authors characterized nonextreme b for which the operator Y = S | ℋ ⁢ (b) , the restriction of the shift operator S on H 2 to ℋ ⁢ (b) , is a strict 2 ⁢ m -isometry and proved that such spaces ℋ ⁢ (b) are equal to local Dirichlet spaces of order m. Here we give a characterization of local Dirichlet spaces of order m in terms of the m-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for b in the case when ℋ ⁢ (b) coincides with local Dirichlet space of order m with equality of norms. Finally, we prove a property of wandering vectors of Y analogous to the property of wandering vectors of the restriction of S to harmonically weighted Dirichlet spaces obtained in [D. Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 1998, 2, 186–213]. [ABSTRACT FROM AUTHOR]

In this paper, we give a universal description of the boundedness and compactness of Toeplitz operator T μ ω between Bergman spaces A η p and A ν q when μ is a positive Borel measure, 1 < p,q < ∞ and ω, η, ν are regular weights. By using Khinchin's inequality and Kahane's inequality, we get a new characterization of the Carleson measure for Bergman spaces induced by regular weights. [ABSTRACT FROM AUTHOR]

In this paper, we consider several questions about the eigenvalues, the numerical ranges, and the invariant subspaces of the Toeplitz operator on the Bergman space over the bidisk and we obtain the corresponding results. [ABSTRACT FROM AUTHOR]

In this paper, we study several properties of an orthonormal basis { N n (z) } for the Newton space N 2 (P) . In particular, we investigate the product of N m and N m and the orthogonal projection P of N n ‾ N m that maps from L 2 (P) onto N 2 (P) . Moreover, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space N 2 (P) . [ABSTRACT FROM AUTHOR]

In this paper, we construct a symbol calculus yielding short exact sequences for the dual Toeplitz algebra generated by all bounded dual Toeplitz operators on the Hardy space associated with the polydisk D n in the unitary space C n , that have been introduced and well studied in our earlier paper (Benaissa and Guediri in Taiwan J Math 19: 31–49, 2015), as well as for the C*-subalgebra generated by dual Toeplitz operators with symbols continuous on the associated hypertorus T n . [ABSTRACT FROM AUTHOR]

In this paper, we study Toeplitz and Hankel operators on exponential weighted Bergman spaces. For 0 < p < ∞ , we obtain sufficient and necessary conditions for Toeplitz and Hankel operators to belong to Schatten-p class by the averaging functions of symbols. For a continuous increasing convex function h, the Schatten-h class Toeplitz and Hankel operators are also characterized. [ABSTRACT FROM AUTHOR]

In the setting of Fock–Sobolev spaces of positive orders over the complex plane, Choe and Yang showed that if the one of the symbols of two commuting Toeplitz operators with bounded symbols is non-trivially radial, then the other must also be radial. In this paper, we extend this result to the Fock–Sobolev space of negative order using the Fock-type space with a confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the contractive real symmetric matrix completion problems motivated in part by studies on sparse (or dense) matrices for weighted sparse recovery problems and rating matrices with rating density in recommender systems. We completely characterize symmetric patterns P with the property (C) that every partially contractive real symmetric matrix with pattern P has a contractive real symmetric completion using graphs. [ABSTRACT FROM AUTHOR]

In this paper, we establish the convergence of several Galerkin–Petrov methods, including the finite section method, the polynomial collocation method and the analytic element collocation method for Toeplitz operators on Dirichlet type spaces. In particular, we show that such methods converge if the basis functions and test functions own certain circular symmetry. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space. First we show that the dual Toeplitz operator with the bounded symbol is hyponormal if and only if it is normal. Then we obtain a necessary and sufficient condition for the dual Toeplitz operator Sφ with the symbol φ (z) = a z n 1 − m 1 z + b z n 2 − m 2 z (n1, n2, m1, m 2 ∈ N and a, b ∈ ℂ) to be hyponormal. Finally, we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank. [ABSTRACT FROM AUTHOR]

We introduce an extended class of cross-Toeplitz operators which act between Fock–Segal–Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-variant transforms, twisted convolutions, symplectic Fourier transform, etc. We discuss connections of (cross-)Toeplitz operators with pseudo-differential operators, localisation operators in time–frequency analysis, and characterisation of kernels in terms of ladder operators. The paper is written in a detailed and reasonably self-contained manner to be suitable as an introduction into group-theoretical methods in phase space and time–frequency operator theory. [ABSTRACT FROM AUTHOR]

In this paper, we characterize those positive Borel measurable symbols μ on C n that induce the Toeplitz operators T μ α to be bounded or compact between different Fock–Sobolev–type spaces F α p and F α ∞ with 0 < p ≤ ∞ . [ABSTRACT FROM AUTHOR]

In this paper we describe the Beurling degree of inner matrix functions in terms of their entries. As a corollary, we recapture the well-known formula for the spectral multiplicity of the model operator. [ABSTRACT FROM AUTHOR]

In this paper, we consider the hyponormality of Toeplitz operators acting on the weighted Bergman space A α 2 (D). We establish necessary or sufficient conditions for the hyponormality of Toeplitz operators with non-harmonic symbols φ (z) that are bivariate polynomials in z and z ¯ (i.e., φ (z) = a z m z ¯ n + b z s z ¯ t ). In particular, we present some characteristics according to the coefficients and degree of φ (z). [ABSTRACT FROM AUTHOR]

Using some concepts and tools from Functional Analysis, we express explicit forms of 'eigenvectors' of two Toeplitz operators: An operator of multiplicity one, which appears e.g. in Autoregressive Process, and a banded operator of high multiplicity. Then we investigate the approximate eigenvectors of their associated Toeplitz matrices of large sizes, and after some numerical experiments, the paper concludes with an application in the principal component analysis method for dimension reduction. [ABSTRACT FROM AUTHOR]

In this paper, we study Hankel operators on the doubling Fock spaces for all possible 1 ≤ p , q < ∞ . We characterize those symbols f for which the Hankel operators H f and H f ¯ are simultaneously bounded or compact from doubling Fock space F φ p to Lebesgue space L φ q , where φ is a nonzero subharmonic function such that Δ φ d A is a doubling measure. [ABSTRACT FROM AUTHOR]

In this paper, N ψ -type quotient modules H 2 (T n) / K of the Hardy module on polydisc are defined, where K is the submodule generated by { z 1 - ψ (z k) , 2 ≤ k ≤ n } for a finite Blaschke product. Alternative characterizations are given and an orthonormal basis is constructed. Then we show that the self-commutators and cross-commutators are in trace class, self-commutators are Hilbert–Schmidt. Moreover, the traces and the Hilbert–Schmidt norms are given, respectively. [ABSTRACT FROM AUTHOR]

In this paper, using some properties about Toeplitz kernels, we present some results about finite-rank properties of the commutator [ A f , A g ] . Firstly, we show that [ A B n , A v ∗ ] must have a finite rank on the model space K u 2 , where B n is a finite Blaschke product and v is an inner function. Next, we present that when ker T u ¯ B n is an invariant subspace of T ϕ ∗ , then [ A B n , A ϕ ∗ ] has a finite rank on K u 2 for ϕ ∈ H ∞ . Finally, we prove that [ A B n , A ϕ ∗ ] must have a finite rank on K u 2 when u = B n u 1 for an inner function u 1 . [ABSTRACT FROM AUTHOR]

A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact. [ABSTRACT FROM AUTHOR]

The paper is devoted to weighted estimates for operators of fractional integration of variable order of Bergman type in generalized variable Hölder spaces of holomorphic functions on the unit disc D . Due to the choice of the weight, we can include in consideration the case when the real part of the complex power of the operator degenerates. We prove the estimates of Zygmund type for the modulus of continuity, and then we obtain the corresponding weighted boundedness result. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate the boundedness of Toeplitz product TfTg and Hankel product H*fHg on Fock-Sobolev space for f , g ∈ P . As a result, the boundedness of Toeplitz operator Tf and Hankel operator Hf with f ∈ P is characterized. [ABSTRACT FROM AUTHOR]

In the paper we develop the Gelfand theory for a new class of commutative Banach algebras generated by Toeplitz operators both in the Bergman and the Hardy space cases. We introduce the so-called generalized pseudo-homogeneous symbols, the functions of the form c (θ , t 1 t ¯ 2) , where (t 1 , t 2) ∈ T × T , thus t 1 t ¯ 2 ∈ T , and c ∈ L ∞ ([ 0 , π 2 ] × T). This class of symbols is a result of step-by-step generalization of previously considered (and studied) symbols. For the commutative algebras under study we describe the compact sets of their maximal ideals, Gelfand transform, and give a criterion on whether our algebra is semi-simple or has a radical. In the last case, we give a complete characterization of the radical. [ABSTRACT FROM AUTHOR]

In this paper, we consider compressions of kth-order slant Toeplitz operators to the backward shift-invariant subspaces of the classical Hardy space H2. In particular, we characterize these operators using compressed shifts and finite-rank operators of special kind. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the frequent hypercyclicity of the Toeplitz operator T Φ and their tensor products on the Hardy space with the symbol of the form Φ (z) = p (1 z) + φ (z) , where p is a polynomial and φ ∈ H ∞ . We also give some sufficient conditions for T Φ 1 ⊗ T Φ 2 to be hypercyclic and construct an example such that neither T Φ 1 nor T Φ 2 is hypercyclic, but T Φ 1 ⊗ T Φ 2 is frequently hypercyclic. Moreover, we also characterize the mixing property and chaoticity of T Φ and T Φ 1 ⊗ T Φ 2 . [ABSTRACT FROM AUTHOR]

In this paper, we characterize the compactness and Fredholmness of Toeplitz operators and Toeplitz products on Bergman-Sobolev spaces over the unit polydisk. We also calculate the essential norm of finite sums of finite Toeplitz products on these spaces. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the kernels and finite rank properties of the Hankel operators and small Hankel operators with pluriharmonic symbols on Hardy-Sobolev spaces H β 2 and pluriharmonic Hardy-Sobolev spaces H β , p h 2 . For β = 1 2 , the result was given by Y. J. Lee. [ABSTRACT FROM AUTHOR]