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Let H^2(S) be the Hardy space on the unit sphere in \mathbf {C}^n. We show that there are Toeplitz operators T_f and T_g on H^2(S) such that the product T_fT_g is not compact and yet \|T_fT_gk_z\| tends to 0 as |z| \rightarrow 1. Consequently, the Berezin transform \langle T_fT_gk_z,k_z\rangle tends to 0 as |z| \rightarrow 1. [ABSTRACT FROM AUTHOR]

A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set Θ ⊆ C such that the evaluation functionals φ τ (f) = f (τ) , τ ∈ Θ , are continuous on ℋ. The Berezin number of an operator X is defined by ber (X) = sup τ ∈ Θ | X ˜ (τ) | = sup τ ∈ Θ | 〈 X k ˆ τ , k ˆ τ 〉 | , where the operator X acts on the reproducing kernel Hilbert space H = H (Θ) over some (non-empty) set Θ . In this paper, we introduce a new family involving means ∥ ⋅ ∥ σ t between the Berezin radius and the Berezin norm. Among other results, it is shown that if X ∈ L (H) and f , g are two non-negative continuous functions defined on [ 0 , ∞) such that f (t) g (t) = t , (t ⩾ 0) , then ∥ X ∥ σ 2 ⩽ ber (1 4 (f 4 (| X |) + g 4 (| X ∗ |)) + 1 2 | X | 2) and ∥ X ∥ σ 2 ⩽ 1 2 ber (f 4 (| X |) + g 2 (| X | 2)) ber (f 2 (| X | 2) + g 4 (| X ∗ |)) , where σ is a mean dominated by the arithmetic mean ∇ . [ABSTRACT FROM AUTHOR]

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 this note, we study the regularity of the Berezin transform on the generalized Hartogs triangles. By introducing a rotation invariant weight function, we show the unboundedness of the Berezin transform of weighted Hilbert spaces defined on the generalized Hartogs triangles. [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]

In this note, we introduce Bergman spaces on the complex upper-right quarter-plane (the first quadrant), and establish some of their fundamental properties. Moreover, we study the associated Berezin transform, and we deduce characterizations of bounded as well as compact Toeplitz operators with symbols that are either harmonic or continuous. [ABSTRACT FROM AUTHOR]

In this work, we provide a complete characterization of the boundedness of two classes of multiparameter Forelli–Rudin type operators from one mixed-norm Lebesgue space L p → to another space L q → , when 1 ≤ p → ≤ q → < ∞ , equipped with possibly different weights. Using these characterizations, we establish the necessary and sufficient conditions for both L p → - L q → boundedness of the weighted multiparameter Berezin transform and L p → - A q → boundedness of the weighted multiparameter Bergman projection, where A q → denotes the mixed-norm Bergman space. Our approach presents several novelties. Firstly, we conduct refined integral estimates of holomorphic functions on the unit ball in C n . Secondly, we adapt the classical Schur’s test to different weighted mixed-norm Lebesgue spaces. These improvements are crucial in our proofs and allow us to establish the desired characterization and sharp conditions. [ABSTRACT FROM AUTHOR]

İşlevsel Hilbert uzayları, istatistik, yaklaşım teorisi, grup temsili teorisi, vb. dahil olmak üzere birçok alanda ortaya çıkar. İşlevsel Hilbert uzay sayesinde tanımlanan Berezin dönüşümü ise, düzgün fonksiyonları analitik fonksiyonların Hilbert uzayları üzerindeki operatörlerle ilişkilerini inceler. Berezin yarıçapını ve Berezin normunu karakterize etmek için bazı çalışmalarda birçok eşitsizlik ve bunların özellikleri vardır. Bu çalışmada fonksiyonel bir Hilbert uzayı üzerinde tanımlanan sınırlı lineer operatörlerin Berezin normu ve Berezin sayısı için yeni eşitsizlikler sunulmuştur. Bu makalenin benzersizliği veya yeniliği, iki operatör için yeni Berezin sayıları tahminlerinden oluşmaktadır. Bu tahminler, diğer benzer makaleler tarafından elde edilen Berezin sayılarının üst sınırlarını iyileştirmiştir. Daha sonra El-Haddad and Kittaneh ([10]) tarafından verilen eşitsizlik genelleştirilmiş ve iyileştirilmiştir. Bu çalışmada fikir ve sunulan metodolojiler, bu alanda gelecekteki araştırmalar için bir başlangıç noktası olarak hizmet edebilir.

In this paper, we investigate the relationship between sampling measure \mu, Berezin transform \tilde {\mu } and r-averaging transform \widehat {\mu }_r on Bergman spaces. Compared with some results of Luecking [Amer. J. Math. 107 (1985), pp. 85–111], our results provide an equivalent description of sampling measures, which reveals the reason why 1/\hat {\mu }_r\in L^\infty or 1/\tilde {\mu }\in L^\infty does not make sure that \mu is a sampling measure on Bergman spaces. [ABSTRACT FROM AUTHOR]

We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the non-negative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space. [ABSTRACT FROM AUTHOR]

The Berezin range of a bounded operator T acting on a reproducing kernel Hilbert space H is the set Ber (T) : = { ⟨ T k ^ x , k ^ x ⟩ H : x ∈ X } , where k ^ x is the normalized reproducing kernel for H at x ∈ X . In general, the Berezin range of an operator is not convex. In this paper, we discuss the convexity of range of the Berezin transforms. We characterize the convexity of the Berezin range for a class of composition operators acting on the Hardy space and the Bergman space of the unit disk. Also for so-called superquadratic functions, we prove the Berezin set mapping theorem for positive self-adjoint operators A on the reproducing kernel Hilbert space H (Ω) , namely we prove that f (Ber (Φ (A))) = Ber (Φ (f (A))) , where Φ : B H Ω → B K (Q) is a normalized positive linear map. [ABSTRACT FROM AUTHOR]

We completely characterize the sampling measures μ for a family of Fock space F ϕ p (0 < p < ∞) induced by a non-radial doubling weight by refining μ ^ r and related dominating sets. Using this characterization, we illustrate the reason why μ ^ r , 1 / μ ^ r ∈ L ∞ or μ ~ , 1 / μ ~ ∈ L ∞ does not make μ be sampling measures on Fock spaces. [ABSTRACT FROM AUTHOR]

We prove that the Berezin transform is L p regular on a large class of domains in C n and not L 2 regular on the Hartogs triangle. [ABSTRACT FROM AUTHOR]

Given a positive Borel measure μ on the unit disk D , let K z α (w) = 1 (1 - z ¯ w) 2 + α be the reproducing kernel of A α 2 (D) at z. The Toeplitz operators with symbol μ are densely defined as follows: T μ (f) (z) = ∫ D f (w) K z α (w) ¯ d μ (w) , f ∈ H ∞ (D). Using the tools such as Carleson measures, Berezin transform and the average functions, we characterize the boundedness and compactness of Toeplitz operators T μ acting between two different Bergman–Orlicz spaces A α Φ 1 (D) and A α Φ 2 (D) for two convex growth functions Φ 1 and Φ 2 . [ABSTRACT FROM AUTHOR]

We characterize some necessary and sufficient conditions of invertible Toeplitz operators acting on the Fock space. In particular, we study the Fredholm properties of Toeplitz operators with B M O 1 symbols, where their Berezin transforms are bounded functions of vanishing oscillation. We show the Fredholm index of the Toeplitz operator via the winding of its Berezin transform along a sufficiently large circle and provide a characterization of invertible Toeplitz operators with non-negative symbols, possibly unbounded, such that the Berezin transforms of the symbols are bounded and of vanishing oscillation. [ABSTRACT FROM AUTHOR]

Functional Hilbert spaces are encountered in a variety of fields, including statistics, approximation theory, group representation theory, and so on. Smooth functions are associated with operators on Hilbert spaces of analytic functions through the Berezin transform defined by functional Hilbert space. We present new inequalities for the Berezin norm and Berezin number of limited linear operators defined on a functional Hilbert space in this study. We discovered various inequalities and their features in some papers to characterize the Berezin number and the Berezin norm. This article's originality or novelty consists of fresh Berezin number estimations for two operators. These estimates improve on the upper bounds of the Berezin numbers found in previous studies. The inequality given by El-Haddad and Kittaneh ([10]) is then improved and generalized after that. The concepts and approaches offered in this paper may serve as a starting point for future research in this field. [ABSTRACT FROM AUTHOR]

We study invertibility and compactness of positive Toeplitz operators associated to a continuous Parseval frame on a Hilbert space. As applications, we characterize compactness of affine and Weyl-Heisenberg localization operators as well as give uncertainty principles for the associated transforms. [ABSTRACT FROM AUTHOR]

The Berezin transform ˜A and the Berezin number of an operator A on the reproducing kernel Hilbert space over some set Ω with normalized reproducing kernel k̂λ are defined, respectively, by ˜A(λ) = ⟨ Ak̂λ, k̂λ ⟩, λ ∈ Ω and ber ( A ) := supλ∈Ω ∣ ˜A(λ) ∣. A straightforward comparison between these characteristics yields the inequalities ber (A) ≤ 1/2 ( ∥ A ∥ber + ∥ A 2 ∥ber1/2) . In this paper, we study further inequalities relating them. Namely, we obtained some refinement of Berezin number inequalities involving convex functions. In particular, for A ∈ B (H) and r ≥ 1 we show that ber2r (A) ≤ 1/4 ( ∥ A* A + A A* ∥berr + ∥ A* A − A A* ∥berr ) + 1/2 berr ( A² ) . [ABSTRACT FROM AUTHOR]

For a bounded linear operator A on a reproducing kernel Hilbert space H(Ω), with normalized reproducing kernel kη:= ||kη/kη||H, the Berezin transform, Berezin radius and Berezin norm are defined respectively by A(η): = D {Akηkη}H' ber(A): = supηEΩ |A(η)|and ||A||Ber = supηEΩ ||Akη||:A straightforward comparison between these characteristics yields the inequalities ber (B*A)≥ 1/2 ||(A*A)+(B*B)||ber: In this paper, we prove further inequalities relating them, and give some applications of geometrically convex functions to Berezin radius inequalities. [ABSTRACT FROM AUTHOR]

In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related results in this framework. In this case the kernel function is given by e z w ¯ + z ¯ w which can be connected to kernels of polyanalytic Fock spaces of finite order. Segal–Bargmann and Berezin type transforms are also considered in this setting. Then, we study the reproducing kernel Hilbert spaces of complex-valued functions with reproducing kernel 1 (1 - z w ¯) (1 - z ¯ w) and 1 1 - 2 Re z w ¯ . The corresponding backward shift operators are introduced and investigated. [ABSTRACT FROM AUTHOR]

We consider Toeplitz operators T a with radial symbols a on general analytic function spaces X . We pose a set of quite natural general assumptions on X , which combined with an assumption on the boundedness of the maximal Bergman projection allows us to derive sufficient conditions for the boundedness of T a . This condition concerns the behaviour of certain integral averages of the symbol a. In the case either the symbol or some its averages is nonnegative, the sufficient condition can be turned into a characterization of the boundedness of T a . [ABSTRACT FROM AUTHOR]

We characterize some necessary and sufficient conditions of invertible Toeplitz operators acting on the Fock space. In particular, we study the Fredholm properties of Toeplitz operators with BMO1 symbols, where their Berezin transforms are bounded functions of vanishing oscillation. We show the Fredholm index of the Toeplitz operator via the winding of its Berezin transform along a sufficiently large circle and provide a characterization of invertible Toeplitz operators with non-negative symbols, possibly unbounded, such that the Berezin transforms of the symbols are bounded and of vanishing oscillation.

This paper discusses the convexity of the range of the Berezin transform. For a bounded operator T acting on a reproducing kernel Hilbert space H (on a set X), this is the set B (T) : = { 〈 T k ˆ x , k ˆ x 〉 H : x ∈ X } , where k ˆ x is the normalized reproducing kernel for H at x ∈ X. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk. [ABSTRACT FROM AUTHOR]

We provide characterizations of Carleson measures on a certain class of bounded pseudoconvex domains. An example of a vanishing Carleson measure whose Berezin transform does not vanish on the boundary is given in the class of the Hartogs triangles \begin{equation*} \mathbb {H}_{k}≔\left \{ \left (z_{1},z_{2}\right)\in \mathbb {C}^{2}:\left |z_{1}\right |^{k}<\left |z_{2}\right |<1\right \},\;k\in \mathbb {Z}^{+}. \end{equation*} [ABSTRACT FROM AUTHOR]

Let ϕ be an analytic self‐map of the open unit disk D$\mathbb {D}$, and let u be an analytic function on D$\mathbb {D}$. The weighted composition operator induced by ϕ with weight u is given by (uCϕf)(z)=u(z)f(ϕ(z))$(uC_{\phi }f)(z)=u(z)f(\phi (z))$ for z in D$\mathbb {D}$ and f analytic on D$\mathbb {D}$. In this paper, we study weighted composition operators acting between two exponentially weighted Bergman spaces Aωp$A^p_{\omega }$ and Aωq$A^q_{\omega }$. We characterize the bounded, compact and Schatten class membership operators uCϕ$ uC_{\phi }$ acting from Aωp$A^p_{\omega }$ to Aωq$A^q_{\omega }$ when 0

We give some applications of Berezin transforms and Engliś C ∗ -algebras methods, namely we investigate the solution of generalized Riccati operator equation of the form X A X + ∑ i = 1 N B i X C i - D = 0 , N ≥ 2 via Berezin transforms. Also, we study finite product of operators, including finite zero-product of Toeplitz operators on the Bergman Hilbert space L a 2 D in terms of Berezin transforms. The same method is used for characterization of compact truncated operators on the reproducing kernel Hilbert space H = H Ω. [ABSTRACT FROM AUTHOR]

Let μ be a finite positive measure on the unit disk and let j ⩾ 1 be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator to be bounded or compact. We first give a necessary and sufficient condition for to be in the Schatten p-class for 1 ⩽ p < ∞ on the Bergman space A2, and then give a sufficient condition for to be in the Schatten p-class (0 < p < 1) on A2. We also discuss the generalized Toeplitz operators with general bounded symbols. If ϕ ∈ L∞ (D, dA) and 1 < p < ∞, we define the generalized Toeplitz operator on the Bergman space Ap and characterize the compactness of the finite sum of operators of the form . [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 characterize Schatten class membership of positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space in terms of averaging functions and Berezin transforms in the range of 0 < p < ∞ . [ABSTRACT FROM AUTHOR]

We construct coherent states for each eigenspace of a magnetic Laplacian on the complex projective n-space in order to apply a quantization-dequantization method. Doing so allows to define the Berezin transform for these spaces. We then establish a formula for this transform as a function of the Fubini-Study Laplacian in a closed form involving of a terminating Kampé de Fériet function. For the lowest spherical Landau level on the Riemann sphere the obtained formula reduces to the one derived by Berezin himself. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Carleson embeddings from Bergman spaces A p into L p (μ) , where μ is a positive measure on the unit disk. We describe when this injection is r -summing on A p. We complete the full characterization of such operators when p > 1 , and r ≥ 1. As an immediate application, we get the characterization of absolutely summing weighted composition operators on Bergman spaces. In passing we also prove a new connection between the boundedness of the Berezin transform and the Carleson embedding on Bergman spaces. [ABSTRACT FROM AUTHOR]

We characterize closed range composition operators on the Dirichlet space for a particular class of composition symbols. The characterization relies on a result about Fredholm Toeplitz operators with BMO1 symbols, and with Berezin transforms of vanishing oscillation.

The Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where k⌢λ=kλ‖kλ‖${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} _\lambda } = {{{k_\lambda }} \over {\left\| {{k_\lambda }} \right\|}}$ is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ber(A)=supλ∈Ω|A˜(λ)|=supλ∈Ω|〈Ak⌢λ,k⌢λ〉|${\bf{ber}}{\rm{(}}A) = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\tilde A(\lambda )} \right| = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\left\langle {A{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} }_\lambda },{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over k} }_\lambda }} \right\rangle } \right|$. In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space ℋ, then

For the open unit disc D in the complex plane, it is well known that if φ ∈ C(D) then its Berezin transform ~ φ also belongs to C(D). We say that D is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in Cn and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform. [ABSTRACT FROM AUTHOR]

Starting from a complete family for the unit sphere S n in the complex n-space C n (whose elements are coherent states attached to the Barut–Girardello space), we obtain an asymptotic expansion for the associated Berezin transform. The proof involves the computation of the asymptotic behaviour of functions in the complete family. Furthermore, in an analogous manner (slightly weaker) we obtain the asymptotic expansion of the covariant symbol of a pseudo-differential operator on L 2 (S n). [ABSTRACT FROM AUTHOR]

In this paper we introduce a class A ⊂ L∞(D) such that if ø ∈ A and satisfies certain positive-definite condition, then there exists a ψ ∈ A such that ø(z) ≤ αeψ(z), for some constant α > 0. Further, if ø(z) = hAkz, kzi, for some bounded positive, invertible operator A from the Bergman space L ² a (D) into itself then ψ(z) = h(log A)kz, kzi. Here kz, z ∈ D are the normalized reproducing kernel of L ² a (D). Applications of these results are also discussed. [ABSTRACT FROM AUTHOR]

In this note, we discuss the Bergman projection P and Toeplitz operators T μ with positive measure symbol μ between F Ψ p (C n) and F Ψ q (C n) for 1 ≤ p , q ≤ ∞ . We first show that P is a bounded projection from L Ψ p onto F Ψ p when 1 ≤ p ≤ ∞ , and then apply it to obtain results on the complex interpolation and the duality of the Fock-type spaces. Furthermore, we obtain the equivalent conditions for the boundedness and compactness of T μ in terms of the averaging function and the Berezin transform, which extend the main results about Toeplitz operators of Seip and Youssfi (J Geom Anal 23:170–201, 2013). [ABSTRACT FROM AUTHOR]