Vector-valued Bergman Spaces Research Articles

OPERATORS ON BERGMAN SPACES

Bergman space theory has been at the core of complex analysis research for many years. Indeed, Hardy spaces are related to Bergman spaces. The popularity of Bergman spaces increased when functional analysis emerged. Although many researchers investigated the Bergman space theory by mimicking the Hardy space theory, it appeared that, unlike their cousins, Bergman spaces were more complex in different aspects. The issue of invariant subspace constitutes one common problem in mathematics that is yet to be resolved. For Hardy spaces, each invariant subspace for shift operators features an elegant description. However, the method for formulating particular structures for the large invariant subspace of shift operators upon Bergman spaces is still unknown. This paper aims to characterize bounded Hankel operators involving a vector-valued Bergman space compared to other different vector value Bergman spaces.

Difference of composition operators on spaces of vector-valued holomorphic functions

In this paper, we completely characterize the boundedness of difference of weighted composition operators between weak and strong vector-valued Bergman spaces in three terms: one is a function theoretic characterization of Julia-Carathéodory type, the second is a power type characterization and the other is a measure theoretic characterization of Carleson type. Furthermore, the bounded difference of composition operators is investigated for corresponding vector-valued Fock space case, which is in sharp contrast with some phenomenon on the setting of vector-valued Bergman space.

Hyponormality of Toeplitz operators with polynomial symbols on the vector valued Bergman space

Hyponormality of Toeplitz operators with polynomial symbols on the vector valued Bergman space

Vector-Valued Kernels of Bergman Type

Bergman reproducing integral formulas can be obtained for holomorphic mappings \(f{:}\,{\mathbb {B}}\rightarrow {\mathbb {C}}^n,\,{\mathbb {B}}\) the open unit ball of \({\mathbb {C}}^n\), by applying the well-known formulas for scalar-valued functions on \({\mathbb {B}}\) to each coordinate function of f, provided those coordinate functions each lie in an appropriate Bergman space. Here, we consider an alternative formulation whereby f is reproduced as the integral of the product of a fixed vector-valued kernel and the scalar expression \(\langle f(z),z \rangle ,\,z\in {\mathbb {B}}\), where \(\langle \cdot ,\cdot \rangle \) is the Hermitian inner product in \({\mathbb {C}}^n\). We provide two different classes of vector-valued kernels that reproduce holomorphic mappings lying in spaces properly containing the weighted vector-valued Bergman spaces. An analysis of these larger spaces is given. The first set of kernels arises naturally from the scalar-valued Bergman kernels, while the second yields the orthogonal projection onto an isomorphic space of scalar-valued functions in the unweighted case.

Little Hankel operators on the Bergman space

In this paper we obtain a characterization of little Hankel operators defined on the Bergman space of the unit disk and then extend the result to vector valued Bergman spaces. We then derive from it certain asymptotic properties of little Hankel operators.

Weighted composition operators on weak vector-valued bergman spaces and Hardy spaces

In this paper we investigate weighted composition operators between weak and strong vector-valued Bergman spaces and Hardy spaces, and give some estimates of their norms.

THE HYPONORMAL TOEPLITZ OPERATORS ON THE VECTOR VALUED BERGMAN SPACE

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators <TEX>$T_{\Phi}$</TEX>, where <TEX>${\Phi}$</TEX> = <TEX>$F+G^*$</TEX>, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space <TEX>$L^2_a(\mathbb{D},\mathbb{C}^n</TEX><TEX>)$</TEX>. We also show some necessary conditions for the hyponormality of <TEX>$T_{F+G^*}$</TEX> with <TEX>$F+G^*{\in}h^{\infty}{\otimes}M_{n{\times}n}$</TEX> on <TEX>$L^2_a(\mathbb{D},\mathbb{C}^n)$</TEX>.

Products of Toeplitz Operators on a Vector Valued Bergman Space

We give a necessary and a sufficient condition for the boundedness of the Toeplitz product T F T G* on the vector valued Bergman space $${L_a^2(\mathbb{C}^n)}$$ , where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [13]. We also characterize boundedness and invertibility of Toeplitz products T F T G* in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space [17].

A joint similarity problem for n-tuples of operators on vector-valued Bergman spaces

We investigate n-tuples of commuting Foias–Williams/Peller type operators acting on vector-valued weighted Bergman spaces. We prove that a commuting n-tuple of such operators is jointly (completely) polynomially bounded if and only if it is similar to an n-tuple of contractions, if and only if each of the n operators is polynomially bounded.

A joint similarity problem on vector-valued Bergman spaces

We investigate pairs of commuting Foias–Williams/Peller type operators acting on vector-valued weighted Bergman spaces. We prove that a commuting pair of such operators is jointly polynomially bounded if and only if it is similar to a pair of contractions, if and only if both operators are polynomially bounded.

Discretizations of Integral Operators and Atomic Decompositions in Vector-Valued Weighted Bergman Spaces

We prove a general atomic decomposition theorem for weighted vector-valued Bergman spaces, which applies to duality problems and to the study of compact Toeplitz type operators.

Hankel operators on Bergman spaces and similarity to contractions

We consider Foguel-Hankel operators on vector-valued Bergman spaces. Such operators defined on Hardy spaces play a central role in the famous example by Pisier of a polynomially bounded operator which is not similar to a contraction. On Bergman spaces we encounter a completely different behaviour; power boundedness, polynomial boundedness, and similarity to a contraction are all equivalent for this class of operators.

Composition operators with linear fractional symbols on vector-valued Bergman spaces

Let φ and ϕ be linear fractional self-maps of the unit diskD andX a separable Hilbert space. In this paper we completely characterize the weak compactness of the product operators of a composition operationCφ with another one’s adjointCρ* on the vector-valued Bergman spaceB1 (X) for formsCφCφ* andCφ*Cφ.

Multipliers on Vector Valued Bergman Spaces

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p &lt; ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.