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Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results. [ABSTRACT FROM AUTHOR]

In the paper 'Composition operators on weighted Bergman spaces of Dirichlet series. J Math Anal Appl. 2015;426:340–363', Bailleul completely characterized the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case of symbols with $ c_0\ge 1 $ c 0 ≥ 1. But the sufficient conditions for the other case $ c_0=0 $ c 0 = 0 were unsolved. In this paper, we follow this line and study the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case $ c_0=0 $ c 0 = 0. Moreover, we also obtain the compact characterizations of composition operators with $ c_0\geq 1 $ c 0 ≥ 1. [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]

One of the aims of this paper is to characterize 3-complex symmetric weighted composition operators induced by three types of symbols on the weighted Bergman space of the right half-plane with the conjugation J f (z) = f (z ¯) ¯ . It is well known that the complex symmetry is equivalent to 2-complex symmetry for the weighted composition operators studied in the paper. However, the interesting fact that 3-complex symmetry is not equivalent to 2-complex symmetry for such operators is found in the paper. Finally, the complex normal of such operators on the weighted Bergman space of the right half-plane with the conjugation J is characterized. [ABSTRACT FROM AUTHOR]

Let ω be a doubling weight and 0 < p ≤ q < ∞ {0

It is well-known that the research of linear combination of composition operators has become a topic of increasing interest. Recently, Choe, Koo and Wang proved that the compactness of combinations composition operators induced by the symbols satisfying the condition (CNC) implies that each difference is compact on the weighted Bergman space. Motivated by that work, in this paper, we discuss which difference is compact on the weighted Bergman space when the coefficients do not satisfy the condition (CNC). [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of weighted Bergman space and obtain a general representation formula of the Bergman kernel function for the spaces on the Reinhardt domain containing the origin. As applications, we calculate the concrete forms of the Bergman kernels for some special weights on the Reinhardt domains C n , D n , m := { ( z , w ) ∈ C n × C m : ‖ w ‖ 2 < e − μ 1 ‖ z ‖ μ 2 } and V η := { ( z , z ′ , w ) ∈ C n × C m × C : ∑ j = 1 n e η j | w | 2 | z j | 2 + ‖ z ′ ‖ 2 < 1 } . [ABSTRACT FROM AUTHOR]

We study composition operators between the weighted Bergman space of the polydisk and the weighted Bergman space of the unit disk. We find conditions on the symbols to guarantee the boundedness and compactness of the composition operator. We shall also find a necessary and sufficient condition for the composition operator to be Hilbert–Schmidt. [ABSTRACT FROM AUTHOR]

We study the dominating sets and reverse Carleson measures on exponentially weighted Bergman spaces A ω p under a new metric. Then, we give some applications of reverse Carleson measure and a generalization of Theorem 1 in Luecking [Sampling measures for Bergman spaces on the unit disk. Math Ann. 2000;316:659–679]. [ABSTRACT FROM AUTHOR]

In this article, we mainly study the weighted composition-differentiation operator on the weighted Bergman space A α 2 and the derivative Hardy space S 1 2 , which characterize complex symmetric weighted composition-differentiation operator D u , φ . Moreover, the normality and self-adjointness of D u , φ are also discussed. [ABSTRACT FROM AUTHOR]

Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results. [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]

In the paper ‘Composition operators on weighted Bergman spaces of Dirichlet series. J Math Anal Appl. 2015;426:340–363’, Bailleul completely characterized the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case of symbols with c 0 ≥ 1 . But the sufficient conditions for the other case c 0 = 0 were unsolved. In this paper, we follow this line and study the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case c 0 = 0 . Moreover, we also obtain the compact characterizations of composition operators with c 0 ≥ 1 . [ABSTRACT FROM AUTHOR]

We characterize wandering subspace property of the shift operator on zero-based invariant subspaces of reproducing kernel Hilbert spaces by using the kernels of Toeplitz operators. As applications, we prove that the wandering subspace property of the shift operator holds on the weighted Bergman space A α 2 ( α > 1) for zero-based invariant subspaces when the zeros lie in a small disk centred at the origin. [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of a complex symmetric Toeplitz operator T φ on the weighted Bergman space A α 2 (D). We first give properties of complex symmetric Toeplitz operators T φ on A α 2 (D). Next, we prove that if T φ is complex symmetric with finite symbol, then T φ is hyponormal on A α 2 (D) if and only if it is hyponormal on the Hardy space H 2 (T). Finally, we consider the complex symmetric Toeplitz operator T φ on A α 2 (D) when the conjugation is a special case. [ABSTRACT FROM AUTHOR]

In Koo and Wang (Joint Carleson measure and the difference of composition operators on A α p (B n). J Math Anal Appl. 2014;419:1119–1142), the authors introduced a concept of joint Carleson measure and used it to characterize when the difference of two composition operators on weighted Bergman space over the unit ball is bounded or compact. In this paper, we extend the concept of joint Carleson measure to the polydisk setting and obtain analogue characterizations of the boundedness (compactness, resp.) of the difference of two composition operators on the weighted Bergman spaces over the unit polydisk, which may provide a unified approach for various ad hoc studies on the boundedness or the compactness of the difference of composition operators on polydisk. Moreover, we construct a concrete example to show that both the boundedness and the compactness depend on the index p when the dimension n ≥ 2 , which is in sharp contrast with the one-variable case where the boundedness and the compactness of the difference of two composition operators are independent of p>0. Due to the complexity of the Carleson measure on the unit polydisk, some new techniques are required in the polydisk setting. [ABSTRACT FROM AUTHOR]

In this paper, for 1 ≤ p < ∞ , we provide several descriptions of Schatten p -class Hankel operators H f and H f ‾ on the weighted Bergman space A ω 2 , in terms of a certain global and local mean oscillation of the symbol f ∈ L ω 2 , provided ω is in a class of regular weights. The approaches applied to rely on several classical methods, and simultaneously rely on a novel but more convenient construction associated with the atomic decomposition of A ω 2. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of the difference of composition-differentiation operators D φ − D ψ acting from Hardy spaces H p to weighted Bergman spaces A α q are completely characterized for all 0 < p , q < ∞. [ABSTRACT FROM AUTHOR]

We study the dominating sets and reverse Carleson measures on weighted Bergman spaces A ω p with invariant weights in the spirit of Lueckings results from 1981 and 1985. Then, we characterize integrating derivatives of A ω p functions, and combine this with reverse Carleson measures to characterize the reverse Carleson inequality for the derivatives of A ω p functions. [ABSTRACT FROM AUTHOR]

The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses a typical example of such domain, the space of all the geodesic segments on a hyperbolic compact Riemann surface. Our main finding is an integral formula that produces holomorphic functions on the domain from holomorphic differentials on the Riemann surface. This construction can be seen as a non-trivial example of L2 jet extension of holomorphic functions with optimal constant. As its corollary, it is shown that the weighted Bergman spaces of the domain is infinite dimensional for any weight order greater than -1 in spite of the fact that the domain does not admit any non-constant bounded holomorphic functions. [ABSTRACT FROM AUTHOR]

We characterise bounded and compact generalised weighted composition operators acting from the weighted Bergman space Apω, where 0 and ω belongs to the class D of radial weights satisfying a two-sided doubling condition, to a Lebesgue space Lqν. On the way, we establish a new embedding theorem on weighted Bergman spaces Apω which generalises the well-known characterisation of the boundedness of the differentiation operator Dnn(ƒ) = ƒ(n) from the classical weighted Bergman space Apα to the Lebesgue space Lqμ, induced by a positive Borel measure μ, to the setting of doubling weights. [ABSTRACT FROM AUTHOR]

For an analytic self-map φ of the unit disk D in the complex plane C, a nonnegative integer n, and u analytic function on D, weighted differentiation composition operator is defined by (Dφ,unƒ)(z) = u(z)ƒ(n)(φ(z)), where f is an analytic function on D and z ∊ D. In this paper, we study the boundedness and compactness of Dφ,un, from weighted Bergman spaces with admissible weights to Bloch-type spaces. [ABSTRACT FROM AUTHOR]

We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix whose entries are explicitly computable in many applications. We apply this result to the Toeplitz operator with symbol z n + c | z | s acting on certain weighted Bergman spaces and determine for what values of the constant c this operator is hyponormal. Our result strengthens an earlier result by the second author, which in turn addressed a question posed by Fleeman and Liaw in 2019. [ABSTRACT FROM AUTHOR]

For weighted Bergman spaces on the unit disk, we give trace formulas of semicommutators of Toeplitz operators with C 2 (D ‾) symbols. We generalize this formula to weighted Bergman spaces on the unit ball in higher dimensions. Applications and examples on the Hankel operators are also discussed. [ABSTRACT FROM AUTHOR]

We study the Helton-Howe trace and the Connes-Chern character for Toeplitz operators on weighted Bergman spaces via the idea of quantization. We prove a local formula for the large t -limit of the Connes-Chern character as the weight goes to infinity. And we show that the Helton-Howe trace of Toeplitz operators is independent of the weight t and obtain a local formula for the Helton-Howe trace for all weighted Bergman spaces using harmonic analysis and quantization. [ABSTRACT FROM AUTHOR]

Let G be a finite pseudoreflection group and Ω ⊆ C d be a bounded domain which is a G -space. We establish identities involving Toeplitz operators on the weighted Bergman spaces of Ω and Ω / G using invariant theory and representation theory of G. This, in turn, provides techniques to study algebraic properties of Toeplitz operators on the weighted Bergman space on Ω / G. We specialize on the generalized zero-product problem and characterization of commuting pairs of Toeplitz operators. As a consequence, more intricate results on Toeplitz operators on the weighted Bergman spaces on some specific quotient domains (namely symmetrized polydisc, monomial polyhedron, Rudin's domain) have been obtained. [ABSTRACT FROM AUTHOR]

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n -best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n -best rational approximation for the Hardy space and a recent result of n -best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations has significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations. [ABSTRACT FROM AUTHOR]

On the unit ball in the n dimensional complex Euclidean space, we introduce a new weighted Bergman space and Toeplitz operators. By Berezin transform and average functions, we estimate the essential norms of the Toeplitz operators. [ABSTRACT FROM AUTHOR]

In this paper, we characterize hyponormality of a class of operators related to a tuple of weighted shifts. As applications we construct examples of hyponormal Toeplitz operators on Hilbert spaces of holomorphic functions on the polydisk or the unit ball which include the Bergman space of the polydisk, the Hardy space of the unit ball and the Drury-Arveson space. [ABSTRACT FROM AUTHOR]

Let φ be an analytic self-map of the open unit disk D. We study the complex symmetry of composition operators Cφ on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D that is not an elliptic automorphism, we establish that if Cφ is complex symmetric, then either φ (0) = 0 or φ is linear. In the case of weighted Bergman spaces A2α, we find the non-automorphic linear fractional symbols φ such that Cφ is complex symmetric. [ABSTRACT FROM AUTHOR]

Closures of weighted Bergman spaces in Bloch type spaces are investigated in the paper. Moreover, the boundedness and compactness of the product of composition and differentiation operators from Bloch type spaces to closures of weighted Bergman spaces spaces in Bloch type spaces are characterized. [ABSTRACT FROM AUTHOR]

We extend known results on the spectra of composition operators to the weighted Bergman spaces. Our results include a study of the essential spectral radius, a determination of the spectrum when the symbol of the composition operator is univalent and non-automorphic with a fixed point in the disk, and an affirmative answer to a conjecture of MacCluer and Saxe. [ABSTRACT FROM AUTHOR]

Considering Banach Hardy spaces and weighted Bergman spaces, we find the sharp values of the Bernstein, Kolmogorov, Gelfand, and linear n-widths for the classes of analytic functions on the unit disk whose moduli of continuity of the rth derivatives averaged with weight are majorized by a given function satisfying some constraints. [ABSTRACT FROM AUTHOR]

In this paper we consider the hyponormality of Toeplitz operators T Φ on the weighted Bergman space with matrix-valued circulant symbols Φ. We give a necessary and sufficient condition for the hyponormality of T Φ with such a symbol Φ under some constraint on the entries of the matrix. [ABSTRACT FROM AUTHOR]

We extend known results on the spectra of composition operators to the weighted Bergman spaces. Our results include a study of the essential spectral radius, a determination of the spectrum when the symbol of the composition operator is univalent and non-automorphic with a fixed point in the disk, and an affirmative answer to a conjecture of MacCluer and Saxe. [ABSTRACT FROM AUTHOR]

Abstract We investigate the topological connectedness of the set of composition operators on the weighted Bergman space A α 2 under the uniform operator topology. One of our main results is to first find a necessary and sufficient condition for two composition operators on A α 2 to be linearly connected in the case of one variable. As a corollary, we show that two composition operators on A α 2 with compact difference must lie in the same path component. [ABSTRACT FROM AUTHOR]

Abstract Fix a bounded domain Ω ⊆ C n and a positive definite kernel K on Ω, both invariant under S n , the permutation group on n symbols. Let H ⊆ Hol (Ω) be the Hilbert module determined by K. We show that H splits into orthogonal direct sum of subspaces P p H indexed by the partitions p of n. We prove that each submodule P p H is a locally free Hilbert module of rank equal to square of the dimension χ p (1) of the irreducible representation corresponding to p . Given two partitions p and q , we show that if χ p (1) ≠ χ q (1) , then the sub-modules P p H and P q H are not unitarily equivalent. We prove that if H is a contractive analytic Hilbert module on Ω, then the Taylor joint spectrum of the n -tuple of multiplication operators by elementary symmetric polynomials on P p H is clos (s (Ω)) , where s : C n → C n is the symmetrization map. It is then shown that this commuting tuple of operators defines a contractive homomorphism of the ring of symmetric polynomials C [ z ] S n in n variables, equipped with the sup norm on clos (s (Ω)). [ABSTRACT FROM AUTHOR]

In this article, we consider the Schrödinger semigroup for the Laplacian Δ on ℝ n {\mathbb{R}^{n}} , and characterize the image of a Sobolev space in L 2 ⁢ (ℝ n , e u 2 ⁢ d ⁢ u) {L^{2}(\mathbb{R}^{n},e^{u^{2}}du)} under this semigroup as weighted Bergman space (up to equivalence of norms). Also we have a similar characterization for Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator H on ℝ n {\mathbb{R}^{n}}. [ABSTRACT FROM AUTHOR]

We obtain new two-sided norm estimates for the family of Bergman-type projections arising from the standard weights (1 - |z|2)α where α > -1. As α →-1, the lower bound is sharp in the sense that it asymptotically agrees with the norm of the Riesz projection. The upper bound is estimated in terms of the maximal Bergman projection, whose exact operator norm we calculate. The results provide evidence towards a conjecture that was posed very recently by the first author. [ABSTRACT FROM AUTHOR]

Let be an analytic map and ϕ an analytic self-map of the open unit disk D. Such maps induce the so-calledweighted composition operator Cϕ,Ψ : H(D) → H(D), f →Ψ (f º ϕ), where H(D) denotes the set of all analytic functions on D. We study when such an operator acting between different weighted Bergman spaces has a closed range. [ABSTRACT FROM AUTHOR]

In a recent book, the authors of this paper have studied the clas- sical heat and Laplace equations with real time variable and complex spa- tial variable by the semigroup theory methods, under the hypothesis that the boundary function belongs to the space of analytic functions in the open unit disk and continuous in the closed unit disk, endowed with the uniform norm. The purpose of the present note is to show that the semigroup theory methods works for these evolution equations of complex spatial variables, under the hypothesis that the boundary function belongs to the much larger weighted Bergman space Bpα(D) with 1 ≤ p < +∞, endowed with a Lp-norm. Also, the case of several complex variables is considered. The proofs require some new changes appealing to Jensen's inequality, Fubini's theorem for integrals and the Lp-integral modulus of continuity. The results obtained can be considered as complex analogues of those for the classical heat and Laplace equations in Lp(R) spaces. [ABSTRACT FROM AUTHOR]

An analytic self-map ϕ of the open unit disk 픻 in the complex plane induces the so-called composition operator Cϕ : H(픻) → H(픻), f ↦ f ∘ ϕ, where H(픻) denotes the set of all analytic functions on 픻. Motivated by [5] we analyze under which conditions on the weight v all composition operators Cϕ acting between the weighted Bergman space and the weighted Banach space of holomorphic functions both generated by v are bounded. [ABSTRACT FROM AUTHOR]

We study the differences of composition operators acting on weighted Bergman spaces over the upper half-plane. In this setting not all composition operators are bounded and none are compact. The idea of joint pullback measure is used to give a Carleson measure characterization of when the difference of two composition operators is bounded or compact. Alternate characterizations, not using Carleson measures, are also given for certain large classes of the inducing maps for the operators. The relationship between angular derivatives and compact differences of composition operators is also explored, which, in particular, reveals a new phenomenon due to the upper half-plane not being bounded. Our results produce a variety of examples of distinct composition operators whose difference is compact, including examples when the individual operators are not bounded. The paper closes with a characterization of when the difference of composition operators is Hilbert- Schmidt. [ABSTRACT FROM AUTHOR]

As is well known on the weighted Bergman spaces over the unit disk, compactness of differences of two composition operators is characterized by certain cancellation property of the inducing maps at every “bad” boundary point, which makes each composition operator in the difference fail to be compact. Recently, the second and third authors obtained a result implying that double difference cancellation is not possible for linear combinations of three composition operators. In this paper, we obtain a complete characterization for compact double differences formed by four composition operators. Applying our characterization, we easily recover known results on linear combinations of two or three composition operators. As another application, we also show that double difference cancellation is possible for linear combinations of four composition operators by constructing an explicit example of a compact double difference formed by two noncompact differences. In spite of such an example, our characterization also shows that double difference cancellation may occur in the global sense only, and that genuine double difference cancellation is not possible in a certain local sense. [ABSTRACT FROM AUTHOR]

We study Banach and C ⁎ -algebras generated by Toeplitz operators acting on weighted Bergman spaces A λ 2 ( B 2 ) over the complex unit ball B 2 ⊂ C 2 . Our key point is an orthogonal decomposition of A λ 2 ( B 2 ) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space A μ 2 ( D ) over the complex unit disk D . Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on A μ 2 ( D ) . The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B 2 of families S of bounded functions on D . Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L ( A μ 2 ( D ) ) are of particular interest. In this paper we discuss various examples. In the case of S = C ( D ‾ ) and S = C ( D ‾ ) ⊗ L ∞ ( 0 , 1 ) we characterize all irreducible representations of the resulting Toeplitz operator C ⁎ -algebras. Their Calkin algebras are described and index formulas are provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider the compact differences of weighted composition operators on the standard weighted Bergman spaces. Some necessary and sufficient conditions for the differences of weighted composition operators to be compact are given, which extends Moorhouse's results in (J. Funct. Anal. 219:70-92, 2005). [ABSTRACT FROM AUTHOR]

Let D be the unit disc of complex plane C, and the class of functions analytic in D. Recall that an is said to belong to the Bloch space if With the norm B is Banach space. Let be the Bloch space which consists of all satisfying Here we give a new description of Bloch spaces and weighted Bergman spaces in terms of Berezin symbols of diagonal operators associated with the Taylor coefficients of their functions. We also give in terms of Berezin symbols a characterization of the multiple shift invariant subspaces of weighted Bergman spaces. Some other questions are also discussed. [ABSTRACT FROM AUTHOR]