Essential Norm of Operators into Weighted-Type Spaces on the Unit Ball (original) (raw)
Weighted spaces of holomorphic functions on Banach spaces
Studia Mathematica, 2000
In this paper we study composition operators between weighted spaces of holomorphic functions defined on the open unit ball of a Banach space. Necessary and sufficient conditions are given for composition operators to be compact. We show that new phenomena appear in the infinite-dimensional setting different from the ones of the finite-dimensional one. 1. Weights. Weighted spaces The starting idea of composition operators is a simple and very natural question. Consider D the open unit disc of C and a holomorphic map φ: D −→ D . If f : D −→ C is a holomorphic function, we can compose f ◦ φ and try to analyze what happens when we let the f vary; in other words we define an operator between spaces of holomorphic functions and we want to study what properties does this operator have (continuity, compactness, . . .). This obviously depends on which are the spaces considered. First candidates are the Hardy spaces and a full study of the situation in this case can be found in [18]. In the ...
Essential Norm of Integral Type Operators Mapping into Certain Banach Spaces of Analytic Functions
Computational Methods and Function Theory, 2020
We study certain integral type operators acting on a large class of Banach spaces of analytic functions on the open unit disc. The operators map into weighted Banach spaces of analytic functions, Bloch type spaces or Zygmund type spaces. Besides characterizing boundedness, we give essential norm estimates of these operators. In order to investigate these operators, we also study weighted differentiation composition operators between these spaces. Keywords Essential norm • Integral type operator • Weighted differentiation composition operator • Weighted Banach space of analytic functions • Bloch type space • Zygmund type space Mathematics Subject Classification Primary 30E20 • 47B38; Secondary 30H99 • 46E15 1 Introduction Let H (D) denote the space of all analytic functions on the open unit disc D of the complex plane. Let u ∈ H (D) and ϕ be an analytic selfmap of D. The weighted composition operator uC ϕ is given by uC ϕ f = u • f • ϕ for all f ∈ H (D), and in Communicated by Karl-G. Grosse-Erdmann.
International Journal of Analysis and Applications, 2016
In this paper, we introduce NK-type spaces of holomorphic functions in the unit ball of C n by the help of a non-decreasing function K: (0,∞) → [0,∞). Several important properties of these spaces in the unit ball are provided. The results are applied to characterize boundedness and compactness of weighted composition operators Wu,φ from NK (B) spaces into Beurling-type classes. We also find the essential norm estimates for Wu,φ from NK(B) spaces into Beurling-type classes.
Classical Operators in Weighted Banach Spaces of Holomorphic Functions
Journal of Mathematical Sciences, 2019
We review recent results in the theory of classical operators (embedding, differentiation, and integration) in weighted Banach spaces of holomorphic functions with uniform estimates. We formulate and analyze results based on associated and essential weights.
Essential norm of a product-type operator from Bergman space to weighted-type space
The Journal of Nonlinear Sciences and Applications, 2017
In this paper, we discuss the boundedness of a product-type operator introduced by Stević, which acting from Bergman space to the weighted-type spaces or the little weighted-type spaces in the unit ball, and characterize the the essential norm of the product-type operator. From which the sufficient and necessary condition of compactness of this type operator follows immediately.
Essential norm of operators on weighted Bergman spaces of infinite order
JOURNAL OF OPERATOR THEORY
We obtain a formula for the essential norm of any operator be-tween weighted Bergman spaces of infinite order. Then we apply it to obtain or estimate essential norms of operators acting on Bloch type spaces and to differences of composition operators or Toeplitz operators on some weighted Bergman spaces. KEYWORDS: Weighted Bergman spaces of infinite order, essential norm, composition operator, Toeplitz operator. MSC (2000): 47B25.
arXiv (Cornell University), 2022
Let E be a space of holomorphic functions on the unit ball B X of a Banach space X. In this work, we introduce a Banach structure associated to E on the linear space W E(Y) containing Y-valued holomorphic functions on B X such that w • f ∈ E for every w ∈ W, a separating subspace of the dual Y ′ of a Banach Y. We establish the relation between the boundedness, the (weak) compactness of the weighted composition operators W ψ,ϕ : f → ψ • (f • ϕ) on E and W ψ,ϕ : g → ψ • (g • ϕ) on W E(Y) via some characterizations of the separating subspace W. As an application, via the estimates for the restrictions of ψ and ϕ to a m-dimensional subspace of X for some m ≥ 2, we characterize the properties mentioned above of W ψ,ϕ on Bloch-type spaces Bµ(B X) of holomorphic functions on the unit ball B X of an infinite-dimensional Hilbert space as well as their the associated spaces W Bµ(B X , Y), where µ is a normal weight on B X .
Differentiation and integration operators on weighted Banach spaces of holomorphic functions
Mathematische Nachrichten, 2016
We obtain a new natural description of the class of radial weights for which some previous results concerning the boundedness of differentiation and integration operators on corresponding spaces are valid. To do this, we develop a new elementary approach which is essentially different from the previous one and can be applied for weights and domains of general types. We also establish a new characterization of some popular classes of radial weights.
Weighted-composition operators on -spaces in the ball
Comptes Rendus Mathematique, 2013
In this Note, we introduce N p -spaces, some kind of Bergman-type spaces, of holomorphic functions in the unit ball of C n . Basic properties of these spaces are provided. We study weighted-composition operators between N p -spaces and the spaces A −q and obtain, in particular, criteria for boundedness and compactness of such operators.