Weighted Bergman Spaces Research Articles

Hilbert Matrix and Its Norm on Weighted Bergman Spaces

It is well known that the Hilbert matrix $${\mathrm {H}}$$ is bounded on weighted Bergman spaces $$A^p_\alpha $$ if and only if $$1<\alpha +2<p$$ with the conjectured norm $$\pi /\sin \frac{(\alpha +2)\pi }{p}$$ . The conjecture was confirmed in the case when $$\alpha =0$$ and also in the case when $$\alpha >0$$ and $$p\ge 2(\alpha +2)$$ , which reduces the conjecture in the case when $$\alpha >0$$ to the interval $$\alpha +2<p<2(\alpha +2)$$ . In the remaining case when $$-1<\alpha <0$$ and $$p>\alpha +2$$ there has been no progress so far in proving the conjecture, moreover, there is no even an explicit upper bound for the norm of the Hilbert matrix $${\mathrm {H}}$$ on weighted Bergman spaces $$A^p_\alpha $$ . In this paper we obtain results which are better than known related to the validity of the mentioned conjecture in the case when $$\alpha >0$$ and $$\alpha +2<p<2(\alpha +2)$$ . On the other hand, we also provide for the first time an explicit upper bound for the norm of the Hilbert matrix $${\mathrm {H}}$$ on weighted Bergman spaces $$A^p_\alpha $$ in the case when $$-1<\alpha <0$$ and $$p>\alpha +2$$ .

Weighted composition operators on weighted Bergman spaces induced by doubling weights

In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_\varphi $ on Bergman type spaces $A_\omega ^p $ induced by a doubling weight ω. Let $X=\{u\in H(\mathbb{D} ): uC_\varphi \colon A_\omega ^p\to A_\omega ^p\ \text {is bounded}\}$. For some regular weights ω, we obtain that $X=H^\infty $ if and only if ϕ is a finite Blaschke product.

Algebra generated by Toeplitz operators with $${\mathbb {T}}$$-invariant symbols

We study the structure of the $$C^*$$ -algebras generated by Toeplitz operators acting on the weighted Bergman space $${\mathcal {A}}^2_{\lambda }({\mathbb {B}}^2)$$ on the two-dimensional unit ball, whose symbols are invariant under the action of the group $${\mathbb {T}}$$ . We consider three principally different basic cases of its action $$t:\,(z_1,z_2) \mapsto (tz_1,t^{k_2}z_2)$$ , with $$k_2=1,0,-1$$ . The properties of the corresponding Toeplitz operators as well as the structure of the $$C^*$$ -algebra generated by them turn out to be drastically different for these three cases.

New Characterizations for Weighted Composition Operators from Weighted Bergman Space into nth Weighted-Type Spaces

In this work, several new characterizations are given for the boundedness and compactness of weighted composition operators from weighted Bergman spaces into nth weighted-type spaces.

Blaschke Decompositions on Weighted Hardy Spaces

Recently, several authors have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either the unwinding series or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of the boundary value of an analytic function $$F: \partial {\mathbb {D}}\rightarrow {\mathbb {C}}$$ and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be efficiently approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was initiated by Coifman and Steinerberger, under the assumption that the complex valued function F has an analytic extension to $${\mathbb {D}}_{1+\epsilon }$$ for some $$\epsilon >0$$ . This provided bounds on weighted Hardy norms involving a single zero, $$\alpha \in {\mathbb {D}}$$ , of F and its Blaschke decomposition. That work also noted that in many specific examples, the unwinding series of F converges at an exponential rate to F, which when coupled with an efficient algorithm to compute a Blaschke decomposition, has led to the hope that this process will provide a new and efficient way to approximate signals. In this work, we continue the study of Blaschke decompositions on weighted Hardy Spaces for functions in the larger space $${\mathcal {H}}^2({\mathbb {D}})$$ under the assumption that the function has finitely many roots in $${\mathbb {D}}$$ . This is meaningful, since there are many functions that meet this criterion but do not extend analytically to $${\mathbb {D}}_{1+\epsilon }$$ for any $$\epsilon >0$$ , for example $$F(z)=\log (1-z)$$ . By studying the growth rate of the weights, we improve the bounds provided by Coifman and Steinerberger to obtain new estimates containing all roots of F in $${\mathbb {D}}$$ . This provides us with new insights into Blaschke decompositions on classical function spaces including the Hardy–Sobolev spaces and weighted Bergman spaces, which correspond to making specific choices for the aforementioned weights. Further, we state a sufficient condition on the weights for our improved bounds to hold for any function in the Hardy space, $${\mathcal {H}}^2({\mathbb {D}})$$ , in particular functions with an infinite number of roots in $${\mathbb {D}}$$ . These results may help to better explain why the exponential convergence of the unwinding series is seen in many numerical examples.

Complex symmetric monomial Toeplitz operators on the unit ball

Monomial Toeplitz operators Tzpz¯q on the weighted Bergman space of the unit ball in Cn are natural extension of the weighted shift operators. In higher dimensions, dim ker⁡Tzpz¯q=∞ may occur for non-trivial case, thus complex symmetric phenomena will appear. In this paper, we completely characterize when the monomial Toeplitz operator Tzpz¯q on the weighted Bergman space is JA-symmetric, where JAf(z)=f(Az‾)‾ with the symmetric unitary n×n matrix A. Moreover, we also determine all possible forms of A such that Tzpz¯q is JA-symmetric. As an application, we construct the weakly closed unital commutative algebra generated by complex symmetric Toeplitz operators.

Operator Norms and Essential Norms of Weighted Composition Operators from Analytic Function Spaces into Zygmund-type Spaces

In this work, we characterize the bounded and compact weighted composition operators from a large class of Banach spaces X of analytic functions on the open unit disk into Zygmund-type spaces. Under more restrictive conditions, we provide an approximation of the essential norm of such operators. We also show that all bounded weighted composition operators from X to the little Zygmund-type space are compact and characterize such operators. We apply our results to the cases when X is the Hardy space $$H^p$$ for $$1\le p\le \infty $$ and the weighted Bergman space $$A_\alpha ^p$$ for $$\alpha >-1$$ and $$1\le p<\infty $$ .

Weighted estimates for the Bergman projection on the Hartogs triangle

We apply modern techniques of dyadic harmonic analysis to obtain sharp estimates for the Bergman projection in weighted Bergman spaces. Our main theorem focuses on the Bergman projection on the Hartogs triangle. The estimates of the operator norm are in terms of a Bekollé-Bonami type constant. As an application of the results obtained, we give, for example, an upper bound for the Lp norm of the Bergman projection on the generalized Hartogs triangle Hm/n in C2.

Hyponormality on a Weighted Bergman Space

A bounded Hilbert space operatorTis hyponormal ifT∗T−TT∗is a positive operator. We consider the hyponormality of Toeplitz operators on a weighted Bergman space. We find a necessary condition for hyponormality in the case of a symbol of the formf+g¯wherefandgare bounded analytic functions on the unit disk. We then find sufficient conditions whenfis a monomial.

Complex Symmetry of Invertible Composition Operators on Weighted Bergman Spaces

In this article, we study the complex symmetry of compositions operators $C_{\phi}f=f\circ \phi$ induced on weighted Bergman spaces $A^2_{\beta}(\mathbb{D}),\ \beta\geq -1,$ by analytic self-maps of the unit disk. One of ours main results shows that $\phi$ has a fixed point in $\mathbb{D}$ whenever $C_{\phi}$ is complex symmetric. Our works establishes a strong relation between complex symmetry and cyclicity. By assuming $\beta\in \mathbb{N}$ and $\phi$ is an elliptic automorphism of $\mathbb{D}$ which not a rotation, we show that $C_{\phi}$ is not complex symmetric whenever $\phi$ has order greater than $2(3+\beta).$

The Commutant of Some Shift Operators

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.

Generalized Cesàro Operators: Geometry of Spectra and Quasi-Nilpotency

Abstract For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk, we prove that the spectrum of a generalized Cesàro operator $T_g$ is unchanged if the symbol $g$ is perturbed to $g+h$ by an analytic function $h$ inducing a quasi-nilpotent operator $T_h$, that is, spectrum of $T_h$ equals $\{0\}$. We also show that any $T_g$ operator that can be approximated in the operator norm by an operator $T_h$ with bounded symbol $h$ is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function $g \in \textbf{BMOA}$ to be in the $\textbf{BMOA}$ norm closure of $H^{\infty }$. This condition turns out to be equivalent to quasi-nilpotency of the operator $T_g$ on the Hardy spaces. This raises the question whether similar statement is true in the context of Bergman spaces and the Bloch space. Furthermore, we provide some general geometric properties of the spectrum of $T_{g}$ operators.

Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols

Let $$D_3$$ be the three-dimensional Siegel domain and $${\mathcal {A}}_\lambda ^2(D_3)$$ the weight-ed Bergman space with weight parameter $$\lambda >-1$$ . In the present paper, we analyse the commutative (not $$C^*$$ ) Banach algebra $${\mathcal {T}}(\lambda )$$ generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols acting on $${\mathcal {A}}_\lambda ^2(D_3)$$ . We remark that $${\mathcal {T}}(\lambda )$$ is not semi-simple, describe its maximal ideal space and the Gelfand map, and show that this algebra is inverse-closed.

Approximation in weighted Bergman spaces and Hankel operators on strongly pseudoconvex domains

Suppose D is a bounded strongly pseudoconvex domain in $${{\mathbb {C}}}^n$$ with smooth boundary, and let $$\rho $$ be its defining function. For $$1< p<\infty $$ and $$\alpha >-1$$ , we show that the weighted Bergman projection $$P_\alpha $$ is bounded on $$L^p(D, |\rho |^\alpha dV)$$ . With non-isotropic estimates for $$\overline{\partial }$$ and Stein’s theorem on non-tangential maximal operators, we prove that bounded holomorphic functions are dense in the weighted Bergman space $$A^p(D, |\rho |^\alpha dV)$$ , and hence Hankel operators can be well defined on these spaces. For all $$1<p, q<\infty $$ we characterize bounded (resp. compact) Hankel operators from p-th weighted Bergman space to q-th weighted Lebesgue space with possibly different weights. As a consequence, we generalize the main results in Pau et al. (Indiana Univ Math J 65:1639–1673, 2016) and resolve a question posed in Lv and Zhu (Integr Equ Oper Theory, 2019).

The weighted Bergman space on a sector and a degenerate parabolic equation

In this work we solve a degenerate parabolic equation for the half line with Dirichlet boundary data, and use some results from the theory of Reproducing Kernel Hilbert Spaces to show that the null reachable space of this degenerate parabolic equation is a RKHS of analytic functions on a sector, whose reproducing kernel can be written in terms of the weighted Bergman kernel on the half plane C+.

Stable convergence of inner functions

Let J be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on J where F n → F if the critical structures of F n converge to the critical structure of F. We show that this occurs precisely when the critical structures of the F n are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs the behaviour of invariant subspaces of a weighted Bergman space which are generated by a single inner function.

Bicomplex Bergman and Bloch spaces

In this article, we define the bicomplex weighted Bergman spaces on the bidisk and their associated weighted Bergman projections, where the respective Bergman kernels are determined. We study also the bicomplex Bergman projection onto the bicomplex Bloch space.

Bergman Spaces Under Maps of Monomial Type

For appropriate domains $$\Omega _{1}, \Omega _{2}$$ , we consider mappings $$\Phi _{\mathbf {A}}:\Omega _{1}\rightarrow \Omega _{2}$$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $${\mathcal {A}}^{2}(\Omega _{1})$$ into finitely many closed subspaces indexed by characters of a finite abelian group associated to the mapping $$\Phi _{\mathbf {A}}$$ . We then show that each subspace is isomorphic to a weighted Bergman space on $$\Omega _{2}$$ . This leads to a formula for the Bergman kernel on $$\Omega _{1}$$ as a sum of weighted Bergman kernels on $$\Omega _{2}$$ .

Difference of composition operators over the half-plane

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.

The dual of a generalized weighted Bergman space

The generalized weighted Bergman space $$H({\mathbb{B}}^d,\lambda )$$ is defined as a reproducing kernel Hilbert space of holomorphic functions on the open unit ball $${\mathbb{B}}^d\subseteq {\mathbb{C}}^d$$ for all $$\lambda >0$$ . When $$\lambda >d$$ , it is identical to the weighted Bergman space $${\mathcal{H}}L^2({\mathbb{B}}^d,\mu _{\lambda })$$ . We prove that the dual space $$H({\mathbb{B}}^d,\alpha )^*$$ can be identified with another generalized weighted Bergman space $$H({\mathbb{B}}^d,\beta )$$ under the pairing $$\langle f,g\rangle _{\gamma }=\int _{{\mathbb{B}}^d}A_{\lambda } f(z)\overline{B_{\lambda } g(z)} \,d\mu _{\gamma +2n}(z),$$ for $$f\in H({\mathbb{B}}^d,\alpha ),\ g\in H({\mathbb{B}}^d,\beta ),$$ where $$n=\left\lceil \frac{d}{2}\right\rceil , \gamma = \frac{\alpha +\beta }{2}$$ and $$A_{\lambda }, B_{\lambda }$$ are operators related to the number operator $$N=\sum _{i=1}^d z_i\frac{\partial }{\partial z_i}.$$