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We study Laplace equation and heat equation with the Gaussian white noise as the boundary/initial condition. Using the stochastic pre-orthogonal adaptive Fourier decomposition on the Bergman space, we give sparse expansions of the solutions of the boundary/initial value problems. The proposed methodology breaks away from the H - H K structure. The key point to the solution is to use the Wiener isometry property of the Gaussian white noise integral and the corresponding Green's function to represent the covariance of the solution. For the heat equation with the Gaussian white noise initial value, we define a class of parabolic Bergman space following E. Stein, prove existence of Bergman kernel, and obtain its expression, and further verify the boundary vanishing property needed for generating a sparse expansion. In order to make the heat flow satisfy a long-time decay property, we induce a subspace of the parabolic Bergman space, which corresponds to a class of homogeneous Sobolev spaces. [ABSTRACT FROM AUTHOR]

Given a doubling weight ω on the unit disk D , let Aωp be the space of all the holomorphic functions f, where ∥ f ∥ A p := (∫ D ∣ f (z) ∣ p (z) d A (z)) 1 / p < ∞. We completely characterize the topological connectedness of the set of composition operators on Aωp. As an application, we construct an interesting example which reveals that two composition operators on Aαp in the same path component may fail to have a compact difference and give a negative answer to the Shapiro-Sundberg question in the (standard) weighted Bergman space. In addition, we completely describe the central compactness of any finite linear combinations of composition operators on Aωp in three terms: a Julia-Carathéodory-type function-theoretic characterization, a power-type characterization, and a Carleson-type measure-theoretic characterization. [ABSTRACT FROM AUTHOR]

We obtain a new boundedness criterion for the difference of two composition operators from a weighted Bergman space A α p into a Lebesgue space L q (μ) , where 0 < q < p and α > - 1 . As a consequence, we provide a direct proof that such a bounded difference operator is necessarily compact. We also characterize compact differences of composition operators from A α p into A β q explicitly for 0 < p ≤ q and α , β > - 1 . [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]

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]

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]

Motivated by the recent work of Galindo et al. (J. Funct. Anal. 265, 629–643, 2013), in this paper, we give an elegant compactness criterion for any finite linear combination of composition operators on the weighted Bergman space in terms of power type characterization. More precisely, let T be any finite linear combination of composition operators, then T is compact on A α p (D) if and only if lim n → ∞ ∥ T z n ∥ A − γ ∥ z n ∥ A − γ = 0 , which reveals that the compactness of T on the weighted Bergman space A α p (D) and Korenblum space A−γ(D) are equivalent. [ABSTRACT FROM AUTHOR]

We obtain the norm and essential norm estimates for the difference of two weighted composition operators from a weighted Bergman space A α p into a Lebesgue space L q (μ) , where 0 < p ≤ q , α > - 1 and μ is a positive Borel measure on the unit disk of C . The weights of the weighted maps in this paper are not necessarily analytic or bounded. Consequently, characterizations for the boundedness and compactness of the difference operator are established. [ABSTRACT FROM AUTHOR]

We give a concrete description of complex symmetric monomial Toeplitz operators T z p z ¯ q on the weighted Bergman space A2(Ω), where Ω denotes the unit ball or the unit polydisk. We provide a necessary condition for T z p z ¯ q to be complex symmetric. When p,q ∈ ℕ2, we prove that T z p z ¯ q is complex symmetric on A2(Ω) if and only if p1 = q2 and P2 = q1. Moreover, we completely characterize when monomial Toeplitz operators T z p z ¯ q on A 2 (D n) are JU-symmetric with the n × n symmetric unitary matrix U. [ABSTRACT FROM AUTHOR]

Holomorphic functions on the maximal Grauert tube of a hyperbolic compact Riemann surface are studied. It is shown that their weighted Bergman spaces are infinite dimensional for arbitrary weight order greater than - 1 in spite of the fact that they do not admit any non-constant bounded holomorphic functions. The key ingredient of the proof is a computation of weighted norms of analytic continuations of eigenfunctions of the Laplacian in terms of the hypergeometric function. This result complements our previous work (Adachi in Trans Am Math Soc 374(10):7499–7524, 2021) where it was shown that the space of geodesic segments on the Riemann surface has exactly the same property. [ABSTRACT FROM AUTHOR]

We obtain conditions that imply reverse Carleson inequalities for weighted Bergman spaces with Békollé weights. Our main theorem generalizes the results in the unweighted case by Luecking. [ABSTRACT FROM AUTHOR]

The spectrum of a weighted composition operator C ψ , φ that is induced by an automorphism has been investigated for over 50 years. However, many results are got only under the condition that the weight function ψ is continuous up to the boundary. In this paper we study the spectra and essential spectra of C ψ , φ on weighted Bergman spaces when φ is an elliptic automorphism, without the assumption that ψ is continuous up to the boundary. [ABSTRACT FROM AUTHOR]

We completely describe the boundedness and compactness of Hankel operators with general symbols acting on Bergman spaces with exponential type weights. [ABSTRACT FROM AUTHOR]

In this work, we introduce a family of weighted Bergman spaces { A α , n } n ∈ N . This family satisfies the continuous inclusions A α , n ⊂ ⋯ ⊂ A α , 2 ⊂ A α , 1 ⊂ A α , 0 = A α , where A α is the classical weighted Bergman space. Next, we define and study the derivative operator ∇ = d d z and its adjoint operator L α = z 2 d d z + (α + 2) z on the weighted Bergman space A α , and we establish an uncertainty inequality of Heisenberg-type for this space. A more general uncertainty inequality for the space A α , n is also given when we considered the operators ∇ n = ∇ n and L α , n : = (L α) n . Afterward, we give Heisenberg-type and Laeng-Morpurgo-type uncertainty inequalities for the Bargmann transform B α , which is an isometric isomorphism between the space A α and the Lebesgue space L 2 (R + , d μ α) , where d μ α is an appropriate measure. [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]

We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure d ω (y) d x with the doubling property ω (0 , 2 t) ≤ C ω (0 , t) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the weighted Bergman space A ω p in the unit ball B of C n with doubling weight ω satisfying ∫ r 1 ω (t) d t < C ∫ 1 + r 2 1 ω (t) d t , 0 ≤ r < 1. The q-Carleson measures for A ω p are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for A ω p are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator T g : A ω p → A ω q are also investigated in this paper with 0 < p ≤ q < ∞ , where T g f (z) = ∫ 0 1 f (t z) R g (t z) dt t , f ∈ H (B) , z ∈ B. [ABSTRACT FROM AUTHOR]

To overcome the unboundedness of the half-plane, we use Khinchine's inequality and atom decomposition techniques to provide joint Carleson measure characterizations when the difference of composition operators is bounded or compact from standard weighted Bergman spaces to Lebesgue spaces over the half-plane for all index choices. For applications, we obtain direct analytic characterizations of the bounded and compact differences of composition operators on such spaces. This paper concludes with a joint Carleson measure characterization when the difference of composition operators is Hilbert-Schmidt. [ABSTRACT FROM AUTHOR]

In this paper, we study the hyponormal Toeplitz operators T φ , where φ is a trigonometric polynomial symbol with finite degree. We present necessary and sufficient conditions for the hyponormality of T φ under some assumptions about the coefficients of φ . Next, we find the necessary condition for hyponormality of T φ . [ABSTRACT FROM AUTHOR]

We study a certain operator of multiplication by monomials in the weighted Bergman space both in the unit disk of the complex plane and in the polydisk of the n-dimensional complex plane. Characterization of the commutant of such operators is given. [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]

We obtain a characterization for the commutant of certain weighted shift operators of higher multiplicity. The characterization is based on the realization of the operator as multiplication by monomials on weighted Bergman spaces of the unit disk. [ABSTRACT FROM AUTHOR]

It is known that the image of a Sobolev space W m , 2 (R) , m ∈ N in L 2 (R , e u 2 d u) under the Schrödinger semigroup e i t Δ is a Hilbert space of entire functions HL 2 (C , u t m (z) d z) . In this article, it is shown that a lattice Z in the complex plane to be a sampling for the space HL 2 (C , u t m (z) d z) if and only if the lower density D - (Z) > 1 8 t 2 π . [ABSTRACT FROM AUTHOR]

In the paper we consider the polyharmonic Bergman space for the union of the rotated unit Euclidean balls. Using so called zonal polyharmonics we derive the formulas for the kernel of this space. Moreover, we study the weighted polyharmonic Bergman space. By the same argument we get the Bergman kernel for this space. [ABSTRACT FROM AUTHOR]

Let X be a Banach space of holomorphic functions on the unit ball B n in C n whose point-evaluation functionals are bounded. In this work, we characterize the bounded weighted composition operators from X into a weighted-type Banach space H μ ∞ (B n) , where the weight μ is an arbitrary positive continuous function on B n . We determine the norm of such operators in terms of the norm of the point-evaluation functionals. Under some restrictions on X, we characterize the compact weighted composition operators mapping X into H μ ∞ (B n) . Under an alternative set of conditions, we provide essential norm estimates. We apply our results to the cases when X is the Hardy space H p (B n) , the weighted Bergman space A α p (B n) for α > - 1 and 1 ≤ p < ∞ , the Bloch space B and the little Bloch space B 0 . In all these cases we obtain precise formulas of the essential norm. [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 completely characterize the finite rank commutator and semicommutator of two monomial-type Toeplitz operators on the weighted Bergman space or on the weighted pluriharmonic Bergman space of the unit polydisk. As a consequence, some interesting new phenomenons appear in operator theory on the unit polydisk which are different from that on the unit disk and the unit ball. [ABSTRACT FROM AUTHOR]

In this article, we consider the Schrödinger semigroup related to the Dunkl-Laplacian Δμ (associated to finite reflection group G) on Rn. We characterize the image of L2(Rn,eu2hμ(u)du) under the Schrödinger semigroup as a reproducing kernel Hilbert space. We define Dunkl-Sobolev space in L2(Rn,eu2hμ(u)du) and characterize it's image under the Schrödinger semigroup associated to G=Z2n as a reproducing kernel Hilbert space up to equivalence of norms. Also we provide similar results for Schrödinger semigroup associated to Dunkl-Hermite operator. [ABSTRACT FROM AUTHOR]

Given a bounded strongly pseudoconvex domain D in C n with smooth boundary, we give a characterization through products of functions in weighted Bergman spaces of (λ , γ) -skew Carleson measures on D, with λ > 0 and γ > 1 - 1 n + 1 . [ABSTRACT FROM AUTHOR]

In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of causal weighted composition operators on these spaces are related to each other and use it to show that an (unweighted) composition operatorCφ is bounded on a Zen space if and only if φ has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces. [ABSTRACT FROM AUTHOR]

Consider Hankel operators Hf on the weighted Bergman space La2(B,dvα). In this paper we characterize the membership of Hf∗Hfs/2=|Hf|s in the norm ideal CΦ, where 0 and the symmetric gauge function Φ is allowed to be arbitrary. [ABSTRACT FROM AUTHOR]

In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space. [ABSTRACT FROM AUTHOR]

Recently, Gao et al. (Chin Ann Math 37B:585-594, 2016) proved a sufficient condition for order boundedness of a weighted composition operator acting between Dirichlet spaces. In this paper, we prove that their condition is a necessary and sufficient condition for order boundedness of a weighted composition operator acting between these spaces. [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 $$\varphi $$ be an arbitrary linear-fractional self-map of the unit disk $${\mathbb {D}}$$ and consider the composition operator $$C_{-1, \varphi }$$ and the Toeplitz operator $$T_{-1,z}$$ on the Hardy space $$H^2$$ and the corresponding operators $$C_{\alpha , \varphi }$$ and $$T_{\alpha , z}$$ on the weighted Bergman spaces $$A^2_{\alpha }$$ for $$\alpha >-1$$ . We prove that the unital C $$^*$$ -algebra $$C^*(T_{\alpha , z}, C_{\alpha , \varphi })$$ generated by $$T_{\alpha , z}$$ and $$C_{\alpha , \varphi }$$ is unitarily equivalent to $$C^*(T_{-1, z}, C_{-1, \varphi }),$$ which extends a known result for automorphism-induced composition operators. For maps $$\varphi $$ that are not automorphisms of $${\mathbb {D}}$$ , we show that $$C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })$$ is unitarily equivalent to $$C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})$$ , where $${\mathcal {K}}_{\alpha }$$ and $${\mathcal {K}}_{-1}$$ denote the ideals of compact operators on $$A^2_{\alpha }$$ and $$H^2$$ , respectively, and apply existing structure theorems for $$C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})/{\mathcal {K}}_{-1}$$ to describe the structure of $$C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })/\mathcal {K_{\alpha }}$$ , up to isomorphism. We also establish a unitary equivalence between related weighted composition operators induced by maps $$\varphi $$ that fix a point on the unit circle. [ABSTRACT FROM AUTHOR]

In this work, we provide an approximation of the essential norm of the weighted composition operators acting on a Hilbert function space and mapping into a Zygmund-type space, and give characterizations of the boundedness and compactness of such operators. Our results apply to a large class of weighted Hardy spaces, including the Hardy space H, the weighted Bergman space $${A^2_\alpha \,\, (\alpha \geq 1)}$$ , and the logarithmic Bergman space $${A^2_{\rm log}}$$ . [ABSTRACT FROM AUTHOR]

It has been known that the difference of two composition operators induced by linear fractional self-maps of a ball cannot be nontrivially compact on either the Hardy space or any standard weighted Bergman space. In this paper we extend this result in two significant directions: the difference is extended to general linear combinations and inducing maps are extended to linear fractional maps taking a ball into another possibly of different dimension. [ABSTRACT FROM AUTHOR]

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product. [ABSTRACT FROM AUTHOR]

Extending our results in Bauer and Vasilevski (J Funct Anal 265(11):2956-2990, ) the present paper gives a detailed structural analysis of a class of commutative Banach algebras $$\mathcal {B}_k(h)$$ generated by Toeplitz operators on the standard weighted Bergman spaces $$\mathcal {A}_{\lambda }^2(\mathbb {B}^n)$$ over the complex unit ball $$\mathbb {B}^n$$ in $$\mathbb {C}^n$$ . In the most general situation we explicitly determine the set of maximal ideals of $$\mathcal {B}_k(h)$$ and we describe the Gelfand transform on a dense subalgebra. As an application to the spectral theory we prove the inverse closedness of algebras $$\mathcal {B}_k(h)$$ in the full algebra of bounded operators on $$\mathcal {A}_{\lambda }^2(\mathbb {B}^n)$$ for certain choices of $$h$$ . Moreover, it is remarked that $$\mathcal {B}_k(h)$$ is not semi-simple. In the case of $$k=(n)$$ we explicitly describe the radical $$\hbox {Rad}\, \mathcal {B}_n(h)$$ of the algebra $$\mathcal {B}_n(h)$$ . This result generalizes and simplifies the characterization of $$\hbox {Rad}\,\mathcal {B}_2(1)$$ , which was given in Bauer and Vasilevski (Integr Equ Oper Theory 74:199-231, ). [ABSTRACT FROM AUTHOR]

We study Toeplitz operators on the weighted harmonic Bergman space on the upper half-plane. Two classes of symbols are considered here: symbols that depend only on the vertical variable and symbols that depend only on the angular variable. For the first case, we prove that Toeplitz operators with such kind of symbols generate a commutative $$C^*$$ -algebra in every weighted harmonic Bergman space. This algebra is isomorphic to the algebra of all very slowly oscillating functions. On the other hand, Toeplitz operators whose symbols depend only on the angular variable generate a non commutative $$C^*$$ -algebra which is isomorphic to the $$C^*$$ -algebra of all $$2\times 2$$ matrix-valued continuous functions $$(f_{ij}(t))$$ defined on $$[-\infty ,\infty ]$$ and such that they satisfy $$f_{12}(\pm \infty )=f_{21}(\pm \infty )=0$$ and $$f_{11}(\pm \infty )=f_{22}(\mp \infty )$$ . [ABSTRACT FROM AUTHOR]

In this note we consider the hyponormality of Toeplitz operators Tϕ on weighted Bergman space A2α(𝔻) with symbol in the class of functions f + g̅ with polynomials f and g. [ABSTRACT FROM AUTHOR]

This paper shows that the boundedness of a weighted composition operator on the Hardy-Hilbert space on the disc or half-plane implies its boundedness on a class of related spaces, including weighted Bergman spaces. The methods used involve the study of lower-triangular and causal operators. [ABSTRACT FROM AUTHOR]

The weighted Weiss conjecture states that the system theoretic property of weighted admissibility can be characterized by a resolvent growth condition. For positive weights, it is known that the conjecture is true if the system is governed by a normal operator; however, the conjecture fails if the system operator is the unilateral shift on the Hardy space $${H^2(\mathbb{D})}$$ (discrete time) or the right-shift semigroup on $${L^2(\mathbb{R}_+)}$$ (continuous time). To contrast and complement these counterexamples, in this paper, positive results are presented characterizing weighted admissibility of linear systems governed by shift operators and shift semigroups. These results are shown to be equivalent to the question of whether certain generalized Hankel operators satisfy a reproducing kernel thesis. [ABSTRACT FROM AUTHOR]

For α > −1, let dνα denote the weighted Lebesgue measure on the bidisk and μ a complex measure satisfying some Carleson-type conditions. In this paper, we show a sufficient and necessary condition for the Toeplitz operator $$T_{\bar \mu }^\alpha$$ to be bounded or compact on weighted Bergman space La1( dνα). [ABSTRACT FROM AUTHOR]

The compact differences of composition operators acting on the weighted L-Bergman space over the unit disk is characterized by the angular derivative cancellation property and due to Moorhouse. In this paper we extend Moorhouse's characterization, as well as some related results, to the ball and, at the same time, to the weighted L-Bergman space for the full range of p. [ABSTRACT FROM AUTHOR]

We prove that, for every $$\alpha > -1$$, the pull-back measure $$\varphi ({\mathcal A }_\alpha )$$ of the measure $$d{\mathcal A }_\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha \, d{\mathcal A } (z)$$, where $${\mathcal A }$$ is the normalized area measure on the unit disk $$\mathbb D $$, by every analytic self-map $$\varphi :\mathbb D \rightarrow \mathbb D $$ is not only an $$(\alpha \,{+}\, 2)$$-Carleson measure, but that the measure of the Carleson windows of size $$\varepsilon h$$ is controlled by $$\varepsilon ^{\alpha + 2}$$ times the measure of the corresponding window of size $$h$$. This means that the property of being an $$(\alpha + 2)$$-Carleson measure is true at all infinitesimal scales. We give an application by characterizing the compactness of composition operators on weighted Bergman-Orlicz spaces. [ABSTRACT FROM AUTHOR]

In the spaces of analytic functions f in the unit disk with mixed norm and measure satisfying the Δ-condition, sharp necessary conditions on subsequences of zeros $\{ z_{n_k } (f)\} $ of the function f are obtained in terms of subsequences of numbers { n}. These conditions can be used to define, in the spaces with mixed norm, subsets of functions with certain extremal properties; these subsets provide answers to a number of questions about the zero sets of the spaces under consideration and, in particular, about weighted Bergman spaces. [ABSTRACT FROM AUTHOR]