Multiplication operators between different Wiener-type variation spaces
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Multiplication operators between different Wiener‑type variation spaces Franklin R. Astudillo‑Villalba1 · Julio C. Ramos‑Fernández2 · Margot Salas‑Brown3 Received: 9 August 2019 / Accepted: 29 September 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract In this article, we characterize all of the multipliers u which define continuous, invertible, finite and closed range, compact, and Fredholm multiplication operators Mu acting on two different spaces of functions of bounded p-variation in the sense of Wiener. Keywords Multipliers · Multiplication operators · Bounded variation · Total variation Mathematics Subject Classification 26B30 · 47B38 · 46E40
1 Introduction During the past decade, there has been increasing interest in properties of multipliers between functional Banach spaces. Given two Banach spaces X and Y, whose elements are real or complex measurable functions with the same domain D, a multiplier for X and Y is defined to be a measurable function u defined on D such that u ⋅ f ∈ Y for all f ∈ X , where ⋅ denotes pointwise function multiplication and is dropped henceforward, with no confusion resulting. The set of all multipliers from X to Y is denoted by M(X, Y) ; that is, we let { } M(X, Y) = u ∈ L0 (𝜇) ∶ uf ∈ Y ∀f ∈ X ,
* Franklin R. Astudillo‑Villalba [email protected] Julio C. Ramos‑Fernández [email protected] Margot Salas‑Brown [email protected] 1
Instituto Superior de Formación Docente Salomé Ureña, Recinto Urania Montás, Investigation groups: GIIEUM, GIAM, San Juan de la Maguana, República Dominicana
2
Proyecto Curricular de Tecnología en Electrónica, Facultad Tecnológica, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia
3
Matemáticas Aplicadas y Ciencias de la Computación, Escuela de Ingeniería, Ciencia y Tecnología, Universidad del Rosario, Bogotá, Colombia
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where L0 (𝜇) denotes the vector space of all measurable functions on the measure . Each multiplier u ∈ M(X, Y) induces a linear operator Mu ∶ X → Y space (D, 𝛴, 𝜇) given by Mu (f ) = uf . In case Mu is continuous, we call it the multiplication operator from X to Y with symbol u. From properties of multiplication operator, one can obtain spaces X and Y. For instance, if we let { beautiful relations between the Banach } I = u ∶ D → ℂ|Mu ∶ X → Y is bijective , then all h ∈ Y can be written uniquely as the product of an f ∈ X and an u ∈ I ; that is, we have a factorization Y of the form I ⋅ X . This topic has been the subject of several recent papers (see e.g [21, 26]). Multiplication operators are classical and continue to be widely studied. In particular, multipliers of spaces of measurable functions were thoroughly examined during the mid-twentieth century. For example, Halmos’s monograph [15, Chap. 7] contains important information about multiplication operators on the Hilbert space L2 (𝜇) of square integrable measurable functions with respect to a given measure 𝜇. We also mention here
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