In this article, we study the complex symmetry of compositions operators $C_ϕf=f\circ ϕ$ induced on weighted Bergman spaces $A^2_β(\mathbb{D}),\ β\geq -1,$ by analytic self-maps of the unit disk. One of ours main results shows that $ϕ$ has a fixed point in $\mathbb{D}$ whenever $C_ϕ$ is complex symmetric. Our works establishes a strong relation between complex symmetry and cyclicity. By assuming $β\in \mathbb{N}$ and $ϕ$ is an elliptic automorphism of $\mathbb{D}$ which not a rotation, we show that $C_ϕ$ is not complex symmetric whenever $ϕ$ has order greater than $2(3+β).$
In this paper, we characterize bounded, compact or Schatten class weighted composition operators acting on Bergman spaces with the exponential type weights. Moreover, we give the proof of the necessary part for the boundedness of composition operators on large weighted Bergman spaces given by Kriete and Maccluer in 1992.
Using an integral formula on a homogeneous Siegel domain, we show a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact. To describe the compactness of composition operators, we see a boundary behavior of the Bergman kernel.
We show that a composition operator on weighted Bergman spaces $\mathcal{A}_μ^p$ is invertible if and only if it is Fredholm if and only if it is an isometry.
The properties of Volterra-composition operators on the weighted Bergman space with exponential type weights are investigated in this paper. We state some necessary and sufficient conditions that a Volterra-composition operator from the weighted Bergman space to Bloch type space ( or little Bloch type space ) must satisfy for the Volterra-composition operator to be bounded or compact.
We study boundedness and compactness of composition operators on weighted Bergman spaces of Dirichlet series. Particularly, we obtain in some specific cases, upper and lower bounds of the essential norm of these operators and a criterion of compactness on classicals weighted Bergman spaces. Moreover, a sufficient condition of compactness is obtained using the notion of Carleson's measure.
While there have been extensive studies regarding the theory of composition operators in standard Bergman spaces, there have not been many results pertaining to large Bergman spaces due to a lack of useful tools. In this paper, we give the characterizations of the compact differences of composition operators in Bergman spaces with the exponential type weight using a newly defined Riemannian distance. Furthermore, we give a sufficient condition for the question when two composition operators lie in the same component.
Using recent characterizations of the compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces on the ball, we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H^{\infty}. Then, we prove that, for each Koranyi region Γ, there exists a map ϕtaking the ball into Γsuch that, C_ϕ is not compact on H^ψ\left(\mathbb{B}_{N}\right), when ψgrows fast. Finally, we give another characterization of the compactness of composition operator on weighted Bergman-Orlicz spaces in terms of an Orlicz "Angular derivat…
Sub-Bergman Hilbert spaces are analogues of de Branges-Rovnyak spaces in the Bergman space setting. They are reproducing kernel Hilbert spaces contractively contained in the Bergman space of the unit disk. K. Zhu analyzed sub-Bergman Hilbert spaces associated with finite Blaschke products, and proved that they are norm equivalent to the Hardy space. Later S. Sultanic found a different proof of Zhu's result, which works in weighted Bergman space settings as well. In this paper, we give a new approach to this problem and obtain a stronger result. Our method relies on the theory of reproducing ke…
It is known, from results of B. MacCluer and J. Shapiro (1986), that every composition operator which is compact on the Hardy space $H^p$, $1 \leq p < \infty$, is also compact on the Bergman space ${\mathfrak B}^p = L^p_a (\D)$. In this survey, after having described the above known results, we consider Hardy-Orlicz $H^Ψ$ and Bergman-Orlicz ${\mathfrak B}^Ψ$ spaces, characterize the compactness of their composition operators, and show that there exist Orlicz functions for which there are composition operators which are compact on $H^Ψ$ but not on ${\mathfrak B}^Ψ$.
We compute the spectra and the essential spectra of bounded linear fractional composition operators acting on the Hardy and weighted Bergman spaces of the upper half-plane. We are also able to extend the results to weighted Dirichlet spaces of the upper half-plane.
In this paper, we study Toeplitz operators on the weighted harmonic Bergman spaces with nonnegative symbols, the weights we choose here are Muckenhoupt A_2 weights. Results obtained include characterizations of bounded Toeplitz operators, compact Toeplitz operators, invertible Toeplitz operators and Toeplitz operators in the Schatten classes.
We construct an analytic self-map $ϕ$ of the unit disk and an Orlicz function $Ψ$ for which the composition operator of symbol $ϕ$ is compact on the Hardy-Orlicz space $H^Ψ$, but not compact on the Bergman-Orlicz space ${\mathfrak B}^Ψ$. For that, we first prove a Carleson embedding theorem, and then characterize the compactness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that this Carleson function is equivalent to the Nevanlinna counting function of order 2.
In this paper, we obtain a complete characterization for the compact difference of two composition operators acting on Bergman spaces with weight $ω=e^{-η}$, $Δη>0$ in terms of the $η$-derived pseudodistance of two analytic self maps. In addition, we provide simple inducing maps which support our main result. We also study the topological path component of the space of all bounded composition operators on $A^2(ω)$ endowed with the Hilbert-Schmidt norm topology.
Let $D$ be an irreducible bounded symmetric domain with biholomorphism group $G$ with maximal compact subgroup $K$. For the Toeplitz operators with $K$-invariant symbols we provide explicit simultaneous diagonalization formulas on every weighted Bergman space. The expressions are given in the general case, but are also worked out explicitly for every irreducible bounded symmetric domain including the exceptional ones.
Given $n\geq1$ and $r\in[0, 1),$ we consider the set $\mathcal{R}_{n, r}$ of rational functions having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D},$ were $\mathbb{D}$ is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in $\mathcal{R}_{n, r}\:$ (as n tends to infinity and r tends to 1-) in weighted Bergman spaces with "polynomially" decreasing weights. We also prove that this result can not be extended to weighted Bergman spaces with "super-polynomially" decreasing weights.
We identify the predual of the nonreflexive Bergman space of the upper half plane, $L_a^1(\uP,μ_{\al})$, with the little Bloch space of the upper half plane consisting of functions vanishing at $i$. We then investigate both the semigroup and spectral properties of the adjoint groups of composition operators which are naturally obtained from the duality pairing and are therefore defined on the identified predual.
We give estimates for the essential norms of a positive Toeplitz operator on the Bergman space of a minimal bounded homogeneous domain in terms of the Berezin transform or the averaging function of the symbol. Using these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.
Composition operators on weighted Bergman spaces of the polydisc
Laboratoire de Math\'ematiques Blaise Pascal UMR 6620 CNRS, Universit\'e Clermont Auvergne, Campus universitaire des C\'ezeaux, 3 place Vasarely, 63178 Aubi\`ere Cedex, France.
For $α>-1$, let $A_α^2(\mathbb{B}_N)$ be the weighted Bergman space on the unit ball $\mathbb{B}_N$ in $\mathbb{C}^N$. In this paper, we prove that the multiplication operator $M_{z^n}$ is quasi-similar to $\oplus_1^{\prod_{i=1}^N n_i}M_z$ on $A_α^2(\mathbb{B}_N)$ for the multi-index $n=(n_1,n_2,\cdots,n_N)$.
In this paper the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on $\mathcal{H}^2(\mathbb{D})$ and $\mathcal{A}_α^2(\mathbb{D}),$ the composition operator on $\mathcal{S}^2(\mathbb{D})$ induced by an analytic self map on $\mathbb{D}$ with fixed origin need not be of norm one. We have generalized the Schwartz's well known result on $\mathcal{A}_α^2(\mathbb{D})$ which characterizes the almost multiplicative operator on $\mathcal{H}^2(\mathbb{D}).$
In this paper, we show that on the weighted Bergman space of the unit disk the essential norm of a noncompact Hankel operator equals its distance to the set of compact Hankel operators and is realized by infinitely many compact Hankel operators, which is analogous to the theorem of Axler, Berg, Jewell and Shields on the Hardy space in Axler et al. (1979); moreover, the distance is realized by infinitely many compact Hankel operators with symbols continuous on the closure of the unit disk and vanishing on the unit circle.
In this paper, we consider the sum of weighted composition operator $C_{ψ_{0},φ_{0}}$ and the weighted composition--differentiation operator $D_{ψ_{n},φ_{n},n}$ on the Hardy and weighted Bergman spaces. We describe the spectrum of a compact operator $C_{ψ_{0},φ_{0}}+D_{ψ_{n},φ_{n},n}$ when the fixed point $w$ of $φ_{0}$ and $φ_{n}$ is inside the open unit disk and $ψ_{n}$ has a zero at $w$ of order at least $n$. Also the lower estimate and the upper estimate on the norm of a weighted composition--differentiation operator on the Hardy space $H^{2}$ are obtained. Furthermore, we determine the no…
In this paper, we study hyponormal weighed composition operators on the Hardy and weighted Bergman spaces. For functions $ψ\in A(\mathbb{D})$ which are not the zero function, we characterize all hyponormal compact weighted composition operators $C_{ψ,φ}$ on $H^{2}$ and $A^{2}_α$. Next, we show that for $φ\in \mbox{LFT}(\mathbb{D})$, if $C_φ$ is hyponormal on $H^{2}$ or $A^{2}_α$, then $φ(z)=λz$, where $|λ| \leq 1$ or $φ$ is a hyperbolic non-automorphism with $φ(0)=0$ and such that $φ$ has another fixed point in $\partial \mathbb{D}$. After that, we find the essential spectral radius of $C_φ$ o…
For $α>-1$, let $A^2_α$ be the corresponding weighted Bergman space of the unit ball in $\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_α$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$.
In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on $A_α^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $φ(z)=$ $z+ψ(z)$, where $ψ\in$ $H^{\infty}(\mathbb{H})$ and $\Im(ψ(z)) > ε> 0$. We especially examine the case where $ψ$ is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multiplie…
We study the complex symmetric structure of weighted composition--differentiation operators of order $n $ on the weighted Bergman spaces $A_α^2$ with respect to some conjugations. Then we provide some examples of these operators.
In this paper, some characterizations for the compact difference of composition operators on Bergman spaces $A^p_ω$ with doubling weight are given, which extend Moorhouse's characterization for the difference of composition operators on the weighted Bergman space $A^2_α$.
We establish weak factorizations for a weighted Bergman space $A^p_{\a}$, with $1<p<\infty$, into two weighted Bergman spaces on the unit ball of $\C^n$. To obtain this result, we characterize bounded Hankel forms on weighted Bergman spaces on the unit ball of $\C^n$.
If $ψ$ is analytic on the open unit disk $\mathbb{D}$ and $φ$ is an analytic self-map of $\mathbb{D}$, the weighted composition operator $C_{ψ,φ}$ is defined by $C_{ψ,φ}f(z)=ψ(z)f (φ(z))$, when $f$ is analytic on $\mathbb{D}$. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces $H^{2}(β)$, we prove that if $C_{ψ,φ}$ is cohyponormal on $H^{2}(β)$, then $ψ$ never vanishes on $\mathbb{D}$ and $φ$ is univalent, when $ψ\not \equiv 0$ and $φ$ is not a constant function. M…
The spectrum of the Cesàro operator $\mathsf{C}$ is determined on the spaces which arises as intersections $A^p_{α+}$ (resp. unions $A^p_{α-}$) of Bergman spaces $A_α^p$ of order $1<p<\infty$ induced by standard radial weights $(1-|z|)^α$, for $0<α<\infty$. We treat them as reduced projective limits (resp. inductive limits) of weighted Bergman spaces $A^p_α$, with respect to $α$. Proving that these spaces admit the monomials as a Schauder basis paves the way for using Grothendieck-Pietsch criterion to deduce that we end up with a non-nuclear Fréchet-Schwartz space (resp. a non-nuclear (DFS)-sp…
We give embedding theorems for weighted Bergman-Orlicz spaces on the ball and then apply our results to the study of composition operators in this context. As one of the motivations of this work, we show that there exist some weighted Bergman-Orlicz spaces, different from $H^{\infinity}$, on which every composition operator is bounded.
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.
A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.
In this article we characterize the boundedness and compactness of a Toeplitz-type operator on weighted Bergman spaces satisfying the so-called Bekolle-Bonami condition in terms of the Berezin transform.
In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator $W_{ϕ,ψ}$ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators.
A full description of the membership in the Schatten ideal $S_ p(A^2_ω)$ for $0<p<\infty$ of the Toeplitz operator acting on large weighted Bergman spaces is obtained.
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 $A^q_α$ are completely characterize for all $0<p,q<\infty$.
This note completely describes the bounded or compact Riemann-Stieltjes integral operators $T_g$ acting between the weighted Bergman space pairs $(A^p_α,A^q_β)$ in terms of particular regularities of the holomorphic symbols $g$ on the open unit ball of $\Bbb C^n$.
Some new characterizations on Carleson measures for weighted Bergman spaces on the unit ball involving product of functions are obtained. For these we characterize bounded and compact Toeplitz operators between weighted Bergman spaces. The above results are applied to characterize bounded and compact extended Cesàro operators and pointwise multiplication operators. The results are new even in the case of the unit disk.
We obtain ceratin estimates for the reproducing kernels of large weighted Bergman spaces. Applications of these estimates to boundedness of the Bergman projection on $L^p(\D,ω^{p/2})$, complex interpolation and duality of weighted Bergman spaces are given.
We consider 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. The condition is presented in terms of the norm of an explicit block Jacobi matrix.
In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_φ$ on Bergman type spaces $A_ω^p $ with double weight $ω$. Let $X=\{u\in H(D): uC_φ:A_ω^p\to A_ω^p \mbox{ is bounded}\}.$ For some regular weights $ω$, we obtain that $X=H^\infty$ if and only if $φ$ is a finite Blaschke product.
Bounded and compact generalized weighted composition operators acting from the weighted Bergman space $A^p_ω$, where $0<p<\infty$ and $ω$ belongs to the class $\mathcal{D}$ of radial weights satisfying a two-sided doubling condition, to a Lebesgue space $L^q_ν$ are characterized. On the way to the proofs a new embedding theorem on weighted Bergman spaces $A^p_ω$ is established. This last-mentioned result generalizes the well-known characterization of the boundedness of the differentiation operator $D^n(f)=f^{(n)}$ from the classical weighted Bergman space $A^p_α$ to the Lebesgue space $L^q_μ$,…
