Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

Zorboska, Nina

doi:10.1007/s00020-017-2361-2

Cited by 6 publications

(2 citation statements)

References 13 publications

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“…There is a vast literature on the multiplication, composition, differentiation, integration or weighted composition operators between specific holomorphic function spaces. Recently, much attention has been paid to the study of these operators acting from general classes of Banach spaces of holomorphic functions mapping into weighted-type or Bloch-type spaces (see [1,2,4,23]). Motivated by these work, we provide the following framework that unified the settings studied in several papers.…”

Section: Introductionmentioning

confidence: 99%

Products of Composition, Multiplication and Iterated Differentiation Operators Between Banach Spaces of Holomorphic Functions

Wang¹,

Wang²,

Guo³

2020

Taiwanese J. Math.

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Let H(D) denote the space of holomorphic functions on the unit disk D of C, ψ, ϕ ∈ H(D), ϕ(D) ⊂ D and n ∈ N ∪ {0}. Let C ϕ , M ψ and D n denote the composition, multiplication and iterated differentiation operators, respectively. To treat the operators induced by products of these operators in a unified manner, we introduce a sum operator n j=0 M ψj C ϕ D j . We characterize the boundedness and compactness of this sum operator mapping from a large class of Banach spaces of holomorphic functions into the kth weighted-type space W, and give its estimates of norm and essential norm. Our results show that the boundedness and compactness of the sum operator depend only on the symbols and the norm of the point-evaluation functionals on the domain space. Our results cover many known results in the literature. Moreover, we introduce the order boundedness of the sum operator and turn its study into that of the boundedness and compactness.

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Section: Introductionmentioning

confidence: 99%

Products of Composition, Multiplication and Iterated Differentiation Operators Between Banach Spaces of Holomorphic Functions

Wang¹,

Wang²,

Guo³

2020

Taiwanese J. Math.

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“…For arbitrary α, β ∈ R, the bounded and compact operators C g ϕ : Λ β (D) → Λ α (D) were characterized in [3]. See also [2,12] for general approaches to related problems.…”

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confidence: 99%

Essential norms of weighted composition operators between Lipschitz spaces of arbitrary order

Doubtsov

2018

Complex Variables and Elliptic Equations

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Let D denote the unit disk of C and let Λ α (D) denote the scale of holomorphic Lipschitz spaces extended to all α ∈ R. For arbitrary α, β ∈ R, we characterize the bounded weighted composition operators from Λ β (D) into Λ α (D) and estimate their essential norms. IntroductionLet H(D) denote the space of holomorphic functions on the unit disk D of C.It is well known that the definition of the space Λ α (D) does not depend on J whenever J > α. For α ∈ R, J ∈ Z + and J > α, Λ α (D) is a Banach space with respect to the following norm:where the first summand is defined to be zero for J = 0. The above norms are equivalent for different parameters J, J > α, so in what follows, we use the brief notation · Λ α in the place of · Λ α,J (D) with the smallest J, J > α.Holomorphic Lipschitz spaces. If α > 0, then each f ∈ Λ α (D) extends to a Lipschitz function on the unit circle ∂D. So, we say that Λ α (D), α > 0, is a holomorphic Lipschitz space. The standard holomorphic Lipschitz spaces Λ α (D) are those with 0 < α < 1. Also, Λ α (D) with α < 1 are often called Bloch type spaces because the Bloch space Λ 0 (D) is defined with J = 1 (see, for example, [7,9]). One has J = 2 in the definition of the classical Zygmund space Λ 1 (D), so Λ α (D), 1 ≤ α < 2 or α < 2, are sometimes called Zygmund type spaces (see, for example, [4]).The scale Λ α (D) splits at the point α = 0. Indeed, Λ 0 (D) is the classical Bloch space; Λ α (D), α < 0, is the growth space defined by the following condition: |f (z)| ≤ C(1 − |z|) α , z ∈ D.1991 Mathematics Subject Classification. Primary 47B33; Secondary 30H30, 30H99, 46E15, 47B38.

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Volterra Type Operators Between Bloch Type Spaces and Weighted Banach Spaces

Lin

2019

Integr. Equ. Oper. Theory

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The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, we completely characterize the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces D p α to D q β (−1 < α, β and 0 < p < q < ∞), which essentially complete their works. Furthermore, we investigate the order boundedness of Volterra type operators between weighted Dirichlet spaces.2010 Mathematics Subject Classification. 47G10, 31C25, 47B38.

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Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

Cited by 6 publications

References 13 publications

Products of Composition, Multiplication and Iterated Differentiation Operators Between Banach Spaces of Holomorphic Functions

Products of Composition, Multiplication and Iterated Differentiation Operators Between Banach Spaces of Holomorphic Functions

Essential norms of weighted composition operators between Lipschitz spaces of arbitrary order

Volterra Type Operators Between Bloch Type Spaces and Weighted Banach Spaces

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