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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 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]

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]

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]

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]

We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R 3 subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L 2 ln (L) , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region L Λ ⊂ R 3 as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln (L) factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂ Λ , is not solely on its area but on the "surface perpendicular to the direction of the magnetic field". We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L 2 of the relevant trace for piecewise C 1 , α smooth surfaces ∂ Λ . [ABSTRACT FROM AUTHOR]

We consider the weighted Bergman space A α 2 (픻) of analytic functions f on the unit disc 픻 = {z ∈ ℂ :|z| < 1} such that f A α 2 2 := ∫ D f z 2 d A α z < + ∞ , where d Aα(z) = (α+1) (1 −|z|2)α d A(z) and d A z = d x d y π (the normalized Lebesgue area measure). We investigate the generalized Riccati operator equation X A 1 X + A 2 X A 3 + A 4 Y A 5 + A 6 = 0 with Ai ∈ 퓑( A α 2 (픻)), i = 1, 6. Also we study the solvability of the operator equation X 1 T φ 1 + X 2 T φ 2 + ⋯ + X n T φ n = I A α 2 , in the set of Toeplitz operators on the weighted Bergman space A α 2 (픻). Moreover, we characterize compactness of the operator Tφ Tψ − Tη in terms of Berezin transforms. [ABSTRACT FROM AUTHOR]

We investigate the p-Carleson measure for the Bloch space B and introduce a holomorphic function space W B p , α associated with this measure. An integral operator which preserves the p-Carleson measure for B is established. As applications, we give a generalized Jones’ formula for W B p , α , characterize the bounded small Hankel operator h s , f from B to the Bergman space A α p , and give an atomic decomposition of W B p , α . [ABSTRACT FROM AUTHOR]

Let μ be a positive Borel measure on the complex plane ℂn and let j = (j1, ..., jn) with ji ∈ ℕ. We study the generalized Toeplitz operators T μ (j) on the Fock space F α 2 . We prove that T μ (j) is bounded (or compact) on F α 2 if and only if μ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for T μ (j) to be in the Schatten p-class for 1 ⩽ p < ∞. [ABSTRACT FROM AUTHOR]