Banach Spaces Of Analytic Functions Research Articles

Composition operators on vector-valued analytic function spaces: a survey

We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinite-dimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BMOA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated.

Invertible Toeplitz products, weighted norm inequalities, and $\mathrm{A}_p$ weights

In this paper, we characterize invertible Toeplitz products on a number of Banach spaces of analytic functions, including weighted Bergman space $L^p_a (\mathbb{B}_n, dv_\gamma)$, the Hardy space $H^p(\partial \mathbb{D})$, and the weighted Fock space F${}_\alpha ^p$ for $p > 1$. The common tool in the proofs of our characterizations will be the theory of weighted norm inequalities and A${}_p$ type weights. Moreover, we analyze and compare the various A${}_p$ type conditions that arise in our characterizations. Finally, we extend the reverse H\older inequality of Zheng and Stroethoff \cite{SZ1, SZ2} for $p = 2$ to the general case of $p > 1$.

A class of integral operators on spaces of analytic functions

We determine the spectrum and essential spectrum as well as resolvent estimates for a class of integral operatorsTμ,νf(z)=zμ−1(1−z)−ν∫0zf(ξ)ξ−μ(1−ξ)ν−1dξ acting on either the analytic Besov spaces or other Banach spaces of analytic functions on the unit disc, including the classical Hardy and weighted Bergman spaces as well as certain Dirichlet spaces and generalized Bloch spaces. Our results unify and extend recent work by Aleman and Persson, [2], the current authors, [4], and Albrecht and Miller, [1].

Volterra operators and semigroups in weighted Banach spaces of analytic functions

We characterize the boundedness, compactness and weak compactness of Volterra operators $$V_{g}(f)(z) := \int _{0}^{z}f({\zeta })g^{\prime }({\zeta })\,d{\zeta }$$ acting between different weighted spaces of type $$H^{\infty }_{v}$$ in terms of the symbol function $$g$$ , for the case when $$v$$ is a quasi-normal weight, a notion weaker than normality. Then we apply the characterization of compactness to analyze the behavior of semigroups of composition operators on $$H^{\infty }_{v}$$ .

Solid extensions of the Cesàro operator on the Hardy space [formula omitted

We introduce and study the largest Banach space of analytic functions on the unit disc which is solid for the coefficient-wise order and to which the classical Cesàro operator C:H2→H2 can be continuously extended, while still maintaining its values in H2. Properties of this Banach space H(ces2) as well as a characterization of individual analytic functions which belong to H(ces2) are presented. In addition, both the multiplier space of H(ces2) and the spectrum of C:H(ces2)→H(ces2) are determined.

Commutants and hyper-reflexivity of multiplication operators

We characterize the commutants of some multiplication operators on a Banach space of analytic functions defined on a bounded domain in the plane. Under certain conditions on the symbol of a multiplication operator, we show that its commutant is a set of multiplication operators. This partially answers a question of Axler, Cuckovic and Rao. Next, the hyper-reflexivity of these multiplication operators are proved. The paper is concluded by proving the hyper-reflexivity of the multiplication operators with symbols \varphi (z) = z^k, k=1, 2,... .

Bounded superposition operators between weighted Banach spaces of analytic functions

We characterize all entire functions that transform a weighted Banach spaces of analytic functions Hμ1∞ into another space of the same kind Hμ2∞ by superposition for very general weights μ1 and μ2.

Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

We investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H∞. Applications for composition operators on weighted Bloch spaces are given.

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type \(H^{\infty }\) into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered.

Essential norms of integral operators on spaces of analytic functions

This note aims at investigating the essential norms of integral operators on Banach spaces of analytic functions. We study the essential norm of Ig on some classical Banach spaces (the Bloch space, BMOA and the Dirichlet space). The essential norms of the Volterra type operator Tg on the Bloch space are given by the distance from the logarithmic Bloch function to the little logarithmic Bloch space. The essential norms of the little Hankel operator are also investigated.

Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

The logarithmic Bloch spaceBlog⁡is the Banach space of analytic functions on the open unit disk𝔻whose elementsfsatisfy the condition∥f∥=sup⁡z∈𝔻(1-|z|2)log⁡ (2/(1-|z|2))|f'(z)|<∞. In this work we characterize the bounded and the compact weighted composition operators from the Hardy spaceHp(with1≤p≤∞) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mappingHpinto the little logarithmic Bloch space defined as the subspace ofBlog⁡consisting of the functionsfsuch thatlim⁡|z|→1(1-|z|2)log⁡ (2/(1-|z|2))|f'(z)|=0.

The Essential Norm of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

For a wide class of radial weights we calculate the essential norm of a weighted composition operator \({uC_\varphi}\) on the weighted Banach spaces of analytic functions in terms of the analytic function \({u \colon \mathbb D \to \mathbb C}\) and the nth power of the analytic selfmap \({\varphi}\) of the open unit disc \({\mathbb D}\) . We also apply our result to calculate the essential norm of composition operators acting on Bloch type spaces with general radial weights.

Coefficient multipliers on Banach spaces of analytic functions

Motivated by an old paper of Wells (J. London Math. Soc. 2 (1970), 549{556) we dene the space X Y , where X and Y are \homogeneous Banach spaces of analytic functions on the unit disk D; by the requirement that f can be represented as f = P 1=0 gn hn; with gn 2 X; hn 2 Y and P 1 n=1 kgnkXkhnkY < 1: We show that this construction is closely related to coecient multipliers. For example, we prove the formula (( X Y );Z) = (X; (Y;Z)); where (U;V ) denotes the space of multipliers from U to V , and as a special case (X Y ) = (X;Y ); where U = (U;H 1 ). We determine H 1 X for a class of spaces that contains H p and ' p (1 p 2), and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.

Mapping properties for the Bargmann transform on modulation spaces

We investigate mapping properties for the Bargmann transform and prove that this transform is isometric and bijective from modulation spaces to convenient Banach spaces of analytic functions. We also present some consequences. For example we prove that the spectrum of the Harmonic oscillator is the same for all modulation spaces.

Extensions of the classical Cesaro operator on Hardy spaces

For each $1\le p&lt;\infty$, the classical Cesàro operator $\mathcal C$ from the Hardy space $H^p$ to itself has the property that there exist analytic functions $f\notin H^p$ with ${\mathcal C}(f)\in H^p$. This article deals with the identification and properties of the (Banach) space $[{\mathcal C}, H^p]$ consisting of all analytic functions that $\mathcal C$ maps into $H^p$. It is shown that $[{\mathcal C}, H^p]$ contains classical Banach spaces of analytic functions $X$, genuinely bigger that $H^p$, such that $\mathcal C$ has a continuous $H^p$-valued extension to $X$. An important feature is that $[{\mathcal C}, H^p]$ is the largest amongst all such spaces $X$.

Compact and Weakly Compact Composition Operators on BMOA

Any analytic map φ of the unit disc \({\mathbb{D}}\) into itself induces a composition operator C φ on BMOA, mapping \({f \mapsto f \circ \varphi}\), where BMOA is the Banach space of analytic functions \({f\colon \mathbb{D} \to \mathbb{C}}\) whose boundary values have bounded mean oscillation on the unit circle. We show that C φ is weakly compact on BMOA precisely when it is compact on BMOA, thus solving a question initially posed by Tjani and by Bourdon, Cima and Matheson in the special case of VMOA. As a crucial step of our argument we simplify the compactness criterion due to Smith for C φ on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. Additional equivalent compactness criteria are established. Furthermore, we prove the unexpected result that compactness of C φ on VMOA implies compactness even from the Bloch space into VMOA.

Remarks on mapping properties for the Bargmann transform on modulation spaces

We investigate the mapping properties for the Bargmann transform and prove that this transform is isometric and bijective from modulation spaces to convenient Banach spaces of analytic functions.

Fredholm composition operators on analytic function spaces

We give a very short proof of a characterization of Fredholm composition operators acting on a large class of Banach spaces of analytic functions defined on the open unit ball of the Euclidian n-complex space.

Multiplicative Isometries and Isometric Zero-Divisors

AbstractFor some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

Absolutely summing multipliers on abstract Hardy spaces

We study summing multipliers from Banach spaces of analytic functions on the unit disc of the complex plane to the complex Banach sequence lattices. The domain spaces are abstract variants of the classical Hardy spaces generated by the complex symmetric spaces. Applying interpolation methods, we prove the Hausdorff-Young and Hardy-Littlewood type theorems. We show applications of these results to study summing multipliers from the Hardy-Orlicz spaces to the Orlicz sequence lattices. The obtained results extend the well-known results for the H p spaces.