Weighted Bergman Spaces Research Articles

On the radicality property for spaces of symbols of bounded Volterra operators

In [1] it is shown that the Bloch space B in the unit disc has the following radicality property: if an analytic function g satisfies that gn∈B, then gm∈B, for all m≤n. Since B coincides with the space T(Aαp) of analytic symbols g such that the Volterra-type operator Tgf(z)=∫0zf(ζ)g′(ζ)dζ is bounded on the classical weighted Bergman space Aαp, the radicality property was used to study the composition of paraproducts Tg and Sgf=Tfg on Aαp. Motivated by this fact, we prove that T(Aωp) also has the radicality property, for any radial weight ω. Unlike the classical case, the lack of a precise description of T(Aωp) for a general radial weight, induces us to prove the radicality property for Aωp from precise norm-operator results for compositions of analytic paraproducts.

Toeplitz operators between weighted Bergman spaces over the half-plane

Toeplitz operators between weighted Bergman spaces over the half-plane

Sums of Generalized Weighted Composition Operators from Weighted Bergman Spaces Induced by Doubling Weights into Bloch-Type Spaces

The single generalized weighted composition operator Du,ψn on various spaces of analytic functions has been investigated for decades, i.e., Du,ψnf=u·(f(n)∘ψ), where f∈H(D). However, the study of the finite sum of generalized weighted composition operators with different orders, i.e., PU,ψkf=u0·f∘ψ+u1·f′∘ψ+⋯+uk·f(k)∘ψ, is far from complete. The boundedness, compactness and essential norm of sums of generalized weighted composition operators from weighted Bergman spaces with doubling weights into Bloch-type spaces are investigated. We show a rigidity property of PU,ψk. Specifically, the boundedness and compactness of the sum PU,ψk is equivalent to those of each Dun,ψn, 0≤n≤k.

On the norm of the Hilbert matrix operator on weighted Bergman spaces

It is known that the norm of the Hilbert matrix operator on weighted Bergman spaces Aαp was conjectured by Karapetrović to be πsin⁡(α+2)πp when α>−1 and p>α+2. The conjecture has been confirmed by Božin and Karapetrović in the case α=0. In this paper we prove the conjecture for the cases both α=1 and 0<α≤147. Moreover, we also show that the conjecture is valid when −1<α<0 and p≥2(α+2).

Projections onto Lp-Bergman spaces of Reinhardt domains

For 1<p<∞, we emulate the Bergman projection on Reinhardt domains by using a Banach-space basis of Lp-Bergman space. The construction gives an integral kernel generalizing the (L2) Bergman kernel. The operator defined by the kernel is shown to be an absolutely bounded projection on the Lp-Bergman space on a class of domains where the Lp-boundedness of the Bergman projection fails for certain p≠2. As an application, we identify the duals of these Lp-Bergman spaces with weighted Bergman spaces.

Words of analytic paraproducts on Hardy and weighted Bergman spaces

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=g(z)f(z). We are concerned with the study of the boundedness of operators in the algebra Ag generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in Ag are finite linear combinations of finite products (words) of Tg,Sg,Mg which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of Ag can be characterized by one of the conditions g∈H∞, or gn belongs to BMOA or the Bloch space, for some integer n>0. However, it is also proved that there are many operators, even single words in Ag whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in Ag in terms of a “fractional power” of the symbol g, that only depends on the number of appearances of each of the letters Tg,Sg,Mg in the given word.

Korenblum’s Principle for Bergman Spaces with Radial Weights

AbstractWe show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces $$A^p_w$$ A w p with arbitrary (non-negative and integrable) radial weights w in the case $$1\le p&lt;\infty $$ 1 ≤ p &lt; ∞ . We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption $$\liminf _{r\rightarrow 0^+} w(r)&gt;0$$ lim inf r → 0 + w ( r ) &gt; 0 , we show that the principle fails whenever $$0&lt;p&lt;1$$ 0 &lt; p &lt; 1 .

Composition operators on weighted Bergman spaces induced by doubling weights

Composition operators on weighted Bergman spaces induced by doubling weights

Analytic representations of backward shifts on weighted Bergman and Dirichlet spaces

The backward shifts (the adjoints of multiplication by z) on weighted Bergman spaces of the unit disk are important in operator theory. For example, Agler [1] shows their restrictions to invariant subspaces of vector-valued weighted Bergman spaces are model operators for hypercontractions. We present analytic representations of backward shifts on weighted Bergman spaces and Dirichlet type spaces. Since these analytic representations are not space specific, it is interesting to study these operators on Hilbert spaces of holomorprhic functions on the unit disk. We initiate this study by developing several properties of these operators. In particular, we obtain simple proofs and extend the invariant subspace theorem of shift plus Volterra operator on the Hardy space by Čučković and Paudyal [16]. Furthermore, we describe invariant subspaces of shift plus Volterra operator on the Bergman space by relating them to the invariant subspaces of the Dirichlet shift. Analytic representations of the tuple of backward shifts on weighted Bergman spaces of the unit ball are also discussed.

Weighted composition operators that preserve p-frames

We establish the equivalence between being invertible and preserving p-frames for weighted composition operators on the unit disk. Moreover, we obtain the symbol properties of the bounded invertible operators. Based on those results, we prove that for the weighted Bergman spaces Aap(dAα), Besov spaces Bp and weighted Dirichlet spaces Dα2, weighted composition operators preserve p-frames on X if and only if they preserve q-Riesz bases on X⁎, and obtain related application on dynamical sampling of weighted composition operators. Furthermore, we characterize the equivalence between Fredholmness and the invertibility of weighted composition operators.

Some results for weighted Bergman space operators via Berezin symbols

Some results for weighted Bergman space operators via Berezin symbols

Stević–Sharma‐type operators between Bergman spaces induced by doubling weights

Using Khinchin's inequality, Geršgorin's theorem, and the atomic decomposition of Bergman spaces, we estimate the norm and essential norm of Stević–Sharma‐type operators between weighted Bergman spaces and and the sum of weighted differentiation composition operators with different symbols from the weighted Bergman spaces to . The estimates of those between Bergman spaces remove all the restrictions of a result of Stevic, Sharma, and Bhat. As a by‐product, we also get an interpolation theorem for Bergman spaces induced by doubling weights.

C*-Algebras Generated by Radial Toeplitz Operators on Polyanalytic Weighted Bergman Spaces

C*-Algebras Generated by Radial Toeplitz Operators on Polyanalytic Weighted Bergman Spaces

3-Complex Symmetric and Complex Normal Weighted Composition Operators on the Weighted Bergman Spaces of the Half-Plane

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 Jf(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.

Product type operators acting between weighted Bergman spaces and Bloch type spaces

Product type operators acting between weighted Bergman spaces and Bloch type spaces

Mixed de Branges–Rovnyak and sub-Bergman spaces

In this paper we obtain some general results about mixed second order de Branges–Rovnyak spaces defined by an operator A such that both A and A⁎ are 2-hypercontractions. As applications of these results, we study mixed second order sub-Bergman spaces since the analytic Toeplitz operator Tb on weighted Bergman space Aα2 for α≥0 on the unit disk is such that both Tb and Tb⁎ are 2-hypercontractions.

A CLASS OF SYMBOLS THAT INDUCE BOUNDED COMPOSITION OPERATORS FOR DIRICHLET-TYPE SPACES ON THE DISC

Abstract We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.

Schatten class Hankel operators on weighted Bergman spaces induced by regular weights

In this paper, for 1≤p<∞, we provide several descriptions of Schatten p-class Hankel operators Hf and Hf‾ on the weighted Bergman space Aω2, in terms of a certain global and local mean oscillation of the symbol f∈Lω2, 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 Aω2.

Sharp inequalities for holomorphic function spaces

In this paper we establish some sharp inequalities for holomorphic Fock spaces in Cn and weighted Bergman spaces in the unit polydisk, which generalizes some results of Burbea. As an application, we prove Furdui's conjecture about the norm of composition operators between Fock spaces.

Difference of composition-differentiation operators from Hardy spaces to weighted Bergman spaces via harmonic analysis

In this paper, the boundedness and compactness of the difference of composition-differentiation operators Dφ−Dψ acting from Hardy spaces Hp to weighted Bergman spaces Aαq are completely characterized for all 0<p,q<∞.