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Some Properties of Multiplication Operators Acting on Banach Spaces of Measurable Functions

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Some Properties of Multiplication Operators Acting on Banach Spaces of Measurable Functions Bolet´ınde Matem´aticas 23(2) 221{237 (2016) 221 UN-EMAT UN Encuentro de Matematicas´ 2016 Some properties of Multiplication Operators acting on Banach spaces of measurable functions Algunas propiedades del Operador Multiplicaci´onactuando sobre espacios de Banach de funciones medibles Julio C. Ramos Fern´andez1;a Abstract. The purpose of this article is to survey the recent results about the properties of multiplication operators acting on Banach spaces of measurable functions. Most of them was presented by the author in a conference held in Bogot´a2016, on the occasion of celebrating the \IV UN Encuentro de Matem´aticas"at the \Universidad Nacional de Colombia". Keywords: Multiplication Operator, measurable functions, K¨othe spaces. Resumen. El prop´ositode este art´ıculoes el de divulgar resultados recientes sobre las propiedades del operador multiplicaci´onactuando en espacios de Ba- nach de funciones medibles. La mayor´ıade estos resultados fueron presentados por el autor en una conferencia, la cual tuvo lugar en Bogot´a2016, en ocasi´on de la celebraci´ondel \IV UN Encuentro de Matem´atica"en la Universidad Nacional de Colombia. Palabras claves: Operador Multiplicaci´on,funciones medibles, Espacios de K¨othe. Mathematics Subject Classification: 47B38, 46E30. Recibido: septiembre de 2016 Aceptado: octubre de 2016 1Departamento de Matem´aticas, Universidad de Oriente, Cuman´a,Venezuela [email protected] 222 Julio C. Ramos Fern´andez 1. Introduction and Preliminaries This note is an overview of the one talk given by the author at \Departamento de Matem´aticas" of the \Universidad Nacional de Colombia" in Bogot´aon the occasion of celebrating the \IV UN Encuentro de Matemtica - 2016". This talk was an invitation to Professors and Students of that Department to make some research in the topic of Operator Theory on Banach spaces of measurable functions. Through this note, we consider real extended measurable functions whose domain of definition is a σ-finite measure space (Ω; Σ; µ). We recall that a σ- algebra Σ is a non-empty collection of subsets of the set Ω, which is closed under complement and it is closed under union of countably infinite many subsets of Ω. An element of Σ is called measurable set. A measure µ defined on the σ-algebra Σ is a real extended and nonnegative function such that µ(;) = 0 and 1 ! 1 [ X µ = µ(Ek); k=1 k=1 for all fEkgk2N of pairwise disjoint sets of Σ. Furthermore, we are going to suppose that the measure µ is complete, that is, if A ⊂ B, B 2 Σ and µ(B) = 0, then A 2 Σ and therefore µ(A) = 0. The measure space is called σ-finite if Ω may be expressed as disjoint enumerable union of measurable subsets all of them having finite measure. The set of all measurable functions on (Ω; Σ; µ) is denote as L0(µ) = L0(Ω; Σ; µ). In this space, two functions f and g are equal if they are equal almost everywhere (a.e.), that is, if µ(ft 2 Ω: f(t) 6= g(t)g) = 0. Hence L0(µ) consists of equivalence classes defined by the relation f ∼ g if and only if f = g a.e. Since multiplication of measurable functions is also a measurable function, we can propose the following problems: • Problem 1. We fix a function u 2 L0(µ) and now we take functions f belonging to a subspace X of L0(µ). The problem, in this case, consists in to describe the set Y = fu · f : f 2 Xg: Clearly, Y is a subspace of L0(µ). But, what other properties does Y have? • Problem 2. We consider fixed two subspaces X and Y of L0(µ). The question is: What properties must the function u 2 L0(µ) have in order to u · f belongs to Y for all function f 2 X?. The set Mult(X; Y ) = fu 2 L0(µ): u · f 2 Y 8f 2 Xg is called multipliers from X to Y . We can see that Mult(X; Y ) is a subspace of L0(µ). Bolet´ınde Matem´aticas 23(2) 221-237 (2016) Some properties of Multiplication Operators 223 In both cases, given a function u 2 L0(µ), the relation Mu which maps each function f 2 X to the element u · f 2 Y , defines a linear operator which it is known as multiplication operator with symbol u. In the case that X = Y = L0(µ) we have the following very useful properties: Proposition 1.1. The operator Mu : L0(µ) ! L0(µ) is injective or 1 − 1 if and only if µ(Zu) = 0, where Zu = ft 2 Ω: u(t) = 0g. Proof. Let us suppose first that µ(Zu) > 0. Since Ω is σ-finite, we can assume that 0 < µ(Zu) < 1, then the measurable function 1Zu, the characteristic function of the set Zu, is not the null function and furthermore it satis es Mu1Zu = u·1Zu = 0. This means that ker(Mu) = ff 2 L0(µ): Mu(f) = 0g 6= f0g and Mu is not 1 − 1 on L0(µ). Conversely, if µ(Zu) = 0 and Muf = 0, then u · f = 0µ−a.e. This implies that Af := ft 2 Ω: f(t) 6= 0g ⊂ Zu [ ft 2 Ω: u(t)f(t) 6= 0g and since the measure µ is complete, we can conclude that µ(Af ) = 0; that is, f = 0µ−a.e. and ker(Mu) = f0g. Proposition 1.2. If Mu : L0(µ) ! L0(µ) is surjective or onto, then it is injective. Proof. If Mu : L0(µ) ! L0(µ) is not 1 − 1, then by Proposition 1, the set Zu has positive measure. Since Ω is a σ-finite space, we can assume that 0 < µ(Zu) < 1. Thus, the non-null function 1Zu 2 L0(µ) does not belong to Ran (Mu) = fMu(f): f 2 L0(µ)g, because in other case, we can find a measurable function f 2 L0(µ) such that 1Zu = u · f. Hence for any t 2 Zu we have 1 = 1Zu(t) = u(t) · f(t) = 0 which is impossible. This shows that Mu : L0(µ) ! L0(µ) is not onto. As a direct consequence of the above proposition, we have: Corollary 1.3. The operator Mu : L0(µ) ! L0(µ) is bijective or invertible if and only if Ran (Mu) = L0(µ). The objective is to study other topological properties of multiplication op- erator. We consider the operator Mu : X ! Y , where (X; k·kX ) and (Y; k·kY ) are Banach subspaces of L0(µ). The goal is to characterize the symbols or functions u 2 L0(µ) for which the operator Mu : X ! Y will be continuous, compact, Fredholm, has closed range or has range finite. Also, it is interesting to estimate the norm and the essential norm of the operator Mu : X ! Y . Allow us recall that a linear transformation T : X ! Y is said to be continuous or bounded if there exists M > 0 such that kTxkY ≤ MkxkX for all x 2 X. The norm of the operator T : X ! Y is defined as kT k = supfkTxkY : kxkX = 1g: (1) Bolet´ınde Matem´aticas 23(2) 221-237 (2016) 224 Julio C. Ramos Fern´andez The collection of all bounded operators from X into Y is denoted by B(X; Y ), which is a Banach space with the norm of operators defined in (1). When X = Y , we write B(X) instead B(X; X). An operator T : X ! Y has closed range if T (X) is a closed subset of (Y; k · kY ). The operator T 2 B(X; Y ) is an isomorphism or bounded below if there exists M > 0 such that kTxkY ≥ MkxkX for all x 2 X. It is known that T 2 B(X; Y ) is bounded below if and only if T is 1 − 1 and has closed range. An operator T 2 B(X; Y ) has finite range if dim(T (X)) < 1; while the transformation T : X ! Y is compact if T (B) is a compact subset of (Y; k · kY ), where B = fx 2 X : kxkX ≤ 1g. It is known that all operator having finite range is a compact operator. The class of all compact operators from X into Y is denoted by K(X; Y ). The essential norm of an operator T 2 B(X; Y ), denoted as kT ke, is the distance from T to K(X; Y ), that is, kT ke = inffkK − T k : K : X −! Y is compactg: The operator T 2 B(X; Y ) is called Fredholm if dim(ker(T )) < 1 and dim(Y = Ran(T )) < 1. The reader interested in studying the properties of the operators de ned in this section may consult the texts of Conway 1990 [12], Kato 1995 [18] and M¨uller 2007 [25]. 2. Multiplication Operators on Lp(µ) Spaces In this section we gather some studies about the properties of multiplication operators acting between Lp(µ) spaces. In order to put the notations accord- ingly with the various articles that we cite later, we fix a σ-finite measure space (Ω; Σ; µ), where the measure µ is suppose to be complete. The multiplication operator with symbol u 2 L0(µ) will be denoted by Mu. Let us recall the definition and some properties of the classical Lebesgue spaces Lp(µ), where p > 0 is a parameter fixed. A function f 2 L0(µ) belongs to the space Lp(µ), with 0 < p ≤ 1, if kfkp < 1, where 8 1= R jf(t)jpdµ(t) p ; if 0 < p < 1; <> Ω kfkp = :>inffλ > 0 : jfj ≤ λµ − a:e:g; p = 1: The following facts are well known.
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