Search

In this paper, we consider a class of logarithmically convex domains in C n , called elementary Reinhardt domains, which can be regarded as a natural generalization of Hartogs triangles. The purpose of this paper is twofold. On one hand, we will compute the explicit forms of the Bergman kernel of weighted Hilbert space with radial symbols. On the other hand, by using the expressions of the weighted Bergman kernel, we will show the regularity of the Berezin transform on the elementary Reinhardt domains. [ABSTRACT FROM AUTHOR]

In a previous paper (Barrera-Castelán et al. in Bol Soc Mat Mex 27:43, 2021. ), using disk polynomials as an orthonormal basis in the n-analytic weighted Bergman space, we showed that for every bounded radial generating symbol a, the associated Toeplitz operator, acting in this space, can be identified with a matrix sequence γ (a) , where the entries of the matrices are certain integrals involving a and Jacobi polynomials. In this paper, we suppose that the generating symbols a have finite limits on the boundary and prove that the C*-algebra generated by the corresponding matrix sequences γ (a) is the C*-algebra of all matrix sequences having scalar limits at infinity. We use Kaplansky’s noncommutative analog of the Stone–Weierstrass theorem and some ideas from several papers by Loaiza, Lozano, Ramírez-Ortega, Ramírez-Mora, and Sánchez-Nungaray. We also prove that for n ≥ 2 , the closure of the set of matrix sequences γ (a) is not equal to the generated C*-algebra. [ABSTRACT FROM AUTHOR]

In this paper, commutativity of the compressions of kth order slant Toeplitz operators is discussed. We show that the commutativity and essential commutativity of two compressions of kth order slant Toeplitz operators on H 2 are same. We also establish some results on the compactness of the compressions of kth order slant Toeplitz operators. In the last section, we discuss various similarities and differences between compactness and non-compactness of these operators by simply looking at their graphs. [ABSTRACT FROM AUTHOR]

If U is a unitary operator on a separable complex Hilbert space H , an application of the spectral theorem says there is a conjugation C on H (an antilinear, involutive, isometry on H ) for which C U C = U ∗. In this paper, we fix a unitary operator U and describe all of the conjugations C which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for U if and only if it is invariant for any conjugation C for which C U C = U ∗ . [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 semicommutators of truncated Toeplitz operators on some model space are discussed. We investigate when the product or the semicommutator of two truncated Toeplitz operators is zero, respectively. We also obtain a necessary and sufficient condition for the semicommutator of two truncated Toeplitz operators to be compact. [ABSTRACT FROM AUTHOR]

Inspired by Xiao’s work about the Hankel measures for the weighted Bergman spaces, in this paper, if β > 0 and the measure μ is a complex Borel measure on the unit disk D , we define the Hankel type operator K μ , β by K μ , β : f ⟼ ∫ D (1 - w z) - (β) f (w) d μ (w). The operator itself has been widely studied when μ is a positive Borel measure supported on the interval [0, 1). We study the boundedness of K μ , 1 acting on Hardy spaces and the boundedness of K μ , α , α > 1 acting on weighted Bergman spaces. Then we raise and answer some questions about the boundedness of those operators. Also, we find some special measures μ ′ s such that s-Hankel measure is equal to s-Carleson measure. [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]

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]

Let μ be a finite positive Borel measure on [0, 1) and let H (D) be the space of all analytic function in the unit disc D . The Cesàro-like operator C μ is defined in H (D) as follows: If f ∈ H (D) , f (z) = ∑ n = 0 ∞ a n z n (z ∈ D) , then C μ (f) (z) = ∑ n = 0 ∞ μ n ∑ k = 0 n a k z n , z ∈ D , where for n ≥ 0 , μ n denotes the n-th moment of the measure μ , that is, μ n = ∫ 0 1 t n d μ (t) . For s > 1 , let X be a Banach subspace of H (D) lying between the mean Lipschtz space Λ 1 s s and the Bloch space B . In this paper we characterize the measures μ as above for which C μ is bounded (compact) from X into any of the Hardy spaces H p (1 ≤ p ≤ ∞) . [ABSTRACT FROM AUTHOR]

In this paper, we discuss block Toeplitz operators T Φ whose symbols are in the class of normal functions Φ (z) = G ∗ (z) + F (z) , where F (z) = A m z m + A N z N and G (z) = A - m z m + A - N z N (A i ∈ M 2) , on the weighted Bergman space. Additionally, we present a necessary and sufficient condition for the hyponormality of the block Toeplitz operators T Φ . [ABSTRACT FROM AUTHOR]

The main continuous time Leech problems considered in this paper are based on stable rational finite dimensional operator-valued functions G and K. Here stable means that G and K do not have poles in the closed right half plane including infinity, and the Leech problem is to find a stable rational operator solution X such that G (s) X (s) = K (s) (s ∈ C +) and sup { ‖ X (s) ‖ : ℜ s ≥ 0 } < 1. In the paper the solution of the Leech problem is given in the form of a state space realization. In this realization the finite dimensional operators involved are expressed in the operators of state space realizations of the functions G and K. The formulas are inspired by and based on ideas originating from commutant lifting techniques. However, the proof mainly uses the state space representations of the rational finite dimensional operator-valued functions involved. The solutions to the discrete time Leech problem on the unit circle are easier to develop and have been solved earlier; see, for example, Frazho et al. (Indagationes Math 25:250–274 2014). [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]

The paper is devoted to studying the Fredholmness of operators from the Banach algebra B of singular integral operators with complex conjugation and slowly oscillating coefficients on the Lebesgue spaces L p (Γ) with p ∈ (1 , ∞) over star-like curves Γ with cusps. Applying reductions to a Banach algebra of Wiener–Hopf operators with slowly oscillating coefficients on weighted Lebesgue spaces over the half-line R + and a Banach algebra of convolution type operators with piecewise slowly oscillating coefficients on weighted Lebesgue spaces over the real line R , we construct a Fredholm symbol calculus for the Banach algebra B and establish a Fredholm criterion for the operators in B . [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]

(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]

In this paper, we consider Hankel operators on domains with bounded intrinsic geometry. For these domains we characterize the L 2 -symbols where the associated Hankel operator is compact (respectively bounded) on the space of square integrable holomorphic functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the weighted composition operators W f , φ on the Hardy space H 2 that are (T u , λ) -Toeplitz, where u is a nonconstant inner function and λ is a complex number with | λ | ≤ 1 ; that is, T u ∗ W f , φ T u = λ W f , φ . In particular, we prove that for an analytic self-map φ of D , not identically zero on D , and f ∈ H ∞ \ { 0 } , if λ ∈ D ¯ \ { 0 } such that 1 λ (u ∘ φ) is a self-map of D , then W f , φ is (T u , λ) -Toeplitz precisely when C φ u = λ u . We also give several applications of this result. [ABSTRACT FROM AUTHOR]

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space H 2 . The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly S ∗ -invariant subspace of H 2 , is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators. [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]

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]

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]

Let Ω + be either the open unit disc or the open upper half plane or the open right half plane. In this paper, we compute the norm of the basic operator A α = Π Θ T b α | H (Θ) in the vector valued model space H (Θ) = H 2 m ⊖ Θ H 2 m associated with an m × m matrix valued inner function Θ in Ω + and show that the norm is attained. Here Π Θ denotes the orthogonal projection from the Lebesgue space L 2 m onto H (Θ) and T b α is the operator of multiplication by the elementary Blaschke factor b α of degree one with a zero at a point α ∈ Ω + . We show that if A α is strictly contractive, then its norm may be expressed in terms of the singular values of Θ (α) . We then extend this evaluation to the more general setting of vector valued de Branges spaces. [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, we establish an equivariant version of Dai–Zhang's Toeplitz index theorem for compact odd-dimensional spin manifolds with even-dimensional boundary. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study several new Hilbert-type operators acting between the weighted Fock spaces. We provide some sufficient and necessary conditions for the boundedness and compactness of certain Hilbert-type operators from one weighted Fock space to another. [ABSTRACT FROM AUTHOR]