Operator norms and essential norms of weighted composition operators between Banach spaces of analytic functions

Colonna, Flavia; Tjani, Maria

doi:10.1016/j.jmaa.2015.08.073

Cited by 39 publications

(38 citation statements)

References 16 publications

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“…There is a vast literature exploring the properties of these two classes of operators, acting on various spaces of holomorphic functions. One of the most recent articles of this kind is, for example, [4], and we cite below some of its results which also follow from Theorem 2.1. Another recent relevant interesting article is [5].…”

Section: Boundednessmentioning

confidence: 99%

“…For more details and more examples of this kind for the weighted composition operators or for the integral operators with X being either the space BM OA, V M OA, the Besov, or the Bloch spaces, see further [4] and [5].…”

Section: Corollary 21[4]mentioning

confidence: 99%

“…One of the first results of this kind was the result of Madigan and Matheson in [7] for the composition operators on the little Bloch space. A more general version for weighted composition operators on initial spaces with bounded point evaluations, and mapping into the spaces H v,0 , namely part (i) of (a) in the following corollary, appears in [4].…”

Section: Compactnessmentioning

confidence: 99%

“…To address further the compactness and weak compactness of intrinsic operators T : X → H v (B v ), we look at a special class of initial spaces. The extra conditions on the space X imposed in the next few theorems are part of the restrictions on the "admissible" domains considered in [4] and [5] for the weighted composition operators and the integral operators. We start first by considering a special kind of intrinsic weakly compact operators.…”

Section: Compactnessmentioning

confidence: 99%

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Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

Zorboska

2017

Integr. Equ. Oper. Theory

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We determine the boundedness and compactness of a large class of operators, mapping from general Banach spaces of holomorphic functions into a particular type of spaces of functions determined by the growth of the functions, or the growth of the functions derivatives. The results show that the boundedness and compactness of such intrinsic operators depends only on the behaviour on the point evaluation functionals. They also generalize previous similar results about several specific classes of operators, such as the multiplication, composition and integral operators.2010 Mathematics Subject Classification. 47B38, 46E15, 47B32, 30H99.

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Section: Boundednessmentioning

confidence: 99%

Section: Corollary 21[4]mentioning

confidence: 99%

Section: Compactnessmentioning

confidence: 99%

Section: Compactnessmentioning

confidence: 99%

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Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

Zorboska

2017

Integr. Equ. Oper. Theory

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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.…”

mentioning

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

Du,

Li,

Liu

2024

Math Methods in App Sciences

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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.

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Operator norms and essential norms of weighted composition operators between Banach spaces of analytic functions

Cited by 39 publications

References 16 publications

Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

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

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

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