Fredholm Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions of TypeH∞

Let X,Y be Banach spaces (infinite dimensional spaces). We denote by B(X, Y) the space of bounded linear operators from X to Y. A bounded linear operator T : X → Y is said to be Fredholm if the spaces KerT and CoKerT = Y/ImT are finite dimensional. Every Fredholm operator has closed range. It is known that T is Fredholm if and only if T is invertible modulo the compact operators and if and only if its dual map is Fredholm. For background on Fredholm operators we refer to [1]. Let φ and ψ be analytic functions on the open unit disk D of the complex plane C such that φ(D) ⊆ D. These maps define, on the space H(D) of analytic functions on D, the socalledweighted composition operatorWφ,ψ byWφ,ψ(f) = ψ(f∘ φ). It combines the classical composition operator φ : f 󳨃→ f∘φwith the pointwisemultiplication operatorMψ : f 󳨃→ ψ⋅ f. These operators have been studied by different authors on various types of function spaces. For more information about composition operators, we refer the reader to the excellent monographs of Cowen and MacCluer [2] and Shapiro [3]. In his thesis Schwartz [4] showed that a composition operator φ on the Hardy space H 2 (D) is invertible if and only if φ is a conformal automorphism of the unit disk. Cima et al. [5] proved that only the invertible operators on H2(D) are Fredholm. Various authors have considered the composition operator φ on several Banach spaces of analytic functions and have studied when φ is Fredholm. See, for example, [6] for theHardy and Bergman spaces inD, [7] for various spaces on domain inC, [8] for a variety of Hilbert spaces of analytic functions on domains in C, [9] for the space H∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E, and [10] for Banach spaces of analytic functions on D satisfying certain conditions. Recently, Zhao has given a characterization of bounded Fredholm weighted composition operator on Dirichlet spaces [11], on Hardy spaces [12], and on a class of weighted Hardy spaces [13]. In this paper we consider the weighted composition operator defined on the weighted Banach spaces of holomorphic functionsH V and on the smaller spacesH 0 V (see Section 2 for the definition). In this framework, Contreras andHernándezDı́az [14] and Montes-Rodŕıguez [15], continuing work by [16], characterized the boundedness and compactness of φ,ψ between weighted Banach spaces of analytic functions


Introduction
Let ,  be Banach spaces (infinite dimensional spaces).We denote by B(, ) the space of bounded linear operators from  to .A bounded linear operator  :  →  is said to be Fredholm if the spaces Ker and CoKer = /Im are finite dimensional.Every Fredholm operator has closed range.It is known that  is Fredholm if and only if  is invertible modulo the compact operators and if and only if its dual map is Fredholm.For background on Fredholm operators we refer to [1].
Let  and  be analytic functions on the open unit disk D of the complex plane C such that (D) ⊆ D. These maps define, on the space (D) of analytic functions on D, the socalled weighted composition operator  , by  , () = (∘ ).It combines the classical composition operator   :   → ∘ with the pointwise multiplication operator   :   → ⋅ .These operators have been studied by different authors on various types of function spaces.For more information about composition operators, we refer the reader to the excellent monographs of Cowen and MacCluer [2] and Shapiro [3].In his thesis Schwartz [4] showed that a composition operator   on the Hardy space  2 (D) is invertible if and only if  is a conformal automorphism of the unit disk.Cima et al. [5] proved that only the invertible operators on  2 (D) are Fredholm.Various authors have considered the composition operator   on several Banach spaces of analytic functions and have studied when   is Fredholm.See, for example, [6] for the Hardy and Bergman spaces in D, [7] for various spaces on domain in C  , [8] for a variety of Hilbert spaces of analytic functions on domains in C  , [9] for the space  ∞ (  ) of bounded analytic functions on the open unit ball of a complex Banach space , and [10] for Banach spaces of analytic functions on D satisfying certain conditions.Recently, Zhao has given a characterization of bounded Fredholm weighted composition operator on Dirichlet spaces [11], on Hardy spaces [12], and on a class of weighted Hardy spaces [13].
In this paper we consider the weighted composition operator defined on the weighted Banach spaces of holomorphic functions  ∞ V and on the smaller spaces  0 V (see Section 2 for the definition).In this framework, Contreras and Hernández-Díaz [14] and Montes-Rodríguez [15], continuing work by [16], characterized the boundedness and compactness of  , between weighted Banach spaces of analytic functions  ∞ V and  0 V .We are interested in finding a characterization of Fredholm weighted composition operators in terms of properties of the symbol  and of the multiplier function .
Our paper is motivated mainly by the works [10,17].In the first one, Bonet et al. characterized when   is a Fredholm operator if it is considered between  ∞ V or  0 V .Recently, Galindo and Lindström considered in [10] composition operators on a class of Banach spaces of analytic functions defined on D and , satisfying certain conditions, and they proved that the composition operator   :  →  associated with the analytic self-map  : D → D is Fredholm if and only if  is a biholomorphic map and if and only if   is invertible.We will present a characterization of Fredholm composition operators and Fredholm weighted composition operators when they are considered between  ∞ V or  0 V , for a general typical weight function V (see Theorem 10).Similar results to the ones presented here for more concrete weights are already established in Section 3 of [18].Although some of the techniques are known and have already been used in the mentioned paper, we present here proofs obtained independently in a more general setting with some applications.Some particular cases of Banach spaces  considered in [10,18] are the standard weighted Bergman spaces    with  > −1 and  ≥ 1, the little Bloch space, and the weighted Banach spaces  0 V  , 0 <  < ∞, where V  () = (1 − || 2 )  .For any weight,  0 V does not satisfy in general condition (C1) of [10] or conditions (C1) and (C1)  of [18].Hence, we complement recent work by Contreras, Galindo, Hernández-Daz, Hyvärinen, Lindström, Nieminen, and Saukko, among others.

Notation and Preliminaries
Let D be the open unit disk in the complex plane C and let us denote by (D) the set of analytic functions from D into C.A weight function V : D → R + is a radial bounded continuous function that is nonincreasing with respect to || such that lim || → 1 − V() = 0.In the literature these types of weight functions are called typical weight functions.The weighted Banach spaces of analytic functions are defined as follows: endowed with the norm ‖ ⋅ ‖ V .The function V ≡ 1 is not a weight function according to our assumptions.In this case  ∞ V =  ∞ and  0 V = {0}.Many results on weighted spaces of analytic functions are formulated in terms of the so-called associated weight function which is defined by the formula where   denotes the point evaluation of .By [19], we know that Ṽ is also a weight function; we have that Ṽ ≥ V > 0, and for each  ∈ D we can find where the inclusion map is the canonical injection from a Banach space into its bidual.We refer to [19][20][21] for more information about these spaces.
For ,  ∈ (D) with (D) ⊆ D and  =  ∞ V or  0 V , we consider the composition operator and the weighted composition operator is defined as follows: provided it is well defined.We always assume that given a weighted composition operator  , there exists a point  ∈ D such that () ̸ = 0.In case the composition operator   is bounded and  is a bounded analytic function on the unit disk,  ∈  ∞ , the multiplication operator V is also bounded and can be decomposed as  , =     , where then Theorem 2.3 in [16] ensures that   is bounded on  ∞ V and  0 V .For instance, the standard weights V  () = (1 − ||)  ,  > 0, are weight functions which have (1).
The point evaluations play an important role in our results.Our first two lemmas are known.We give a short proof for the reader's convenience.

Lemma 1. For every 𝑧 ∈ D, the point evaluation 𝛿
V . ( Then Under the hypothesis that V is a typical weight we know that the closed unit ball of  0 V is dense in the closed unit ball of  ∞

V
for the compact open topology (see [20]).
Lemma 2. If V is a weight function, then From formula (2) and Lemma 1 we obtain the conclusion.
(ii) Let  be a polynomial; then lim Since For ,  Banach spaces (infinite dimensional spaces) a bounded linear operator  :  →  is said to be Fredholm if the spaces Ker and CoKer = /Im are finite dimensional.Note that a consequence of the Fredholmness is that the condition codim(Im) = dim(/Im) < ∞ implies that the range of  is closed.It is known that  :  →  is Fredholm if and only if  * :  * →  * is Fredholm and if and only if there are  ∈ B(, ),  1 ∈ K(), and  2 ∈ K() such that  =   + 1 ,  =   + 2 , where K() denote the set of compact operators on .As a consequence, invertible operators are Fredholm and compact operators cannot be Fredholm.We refer to [1, Chapter III] for the proofs of these results.

Fredholm Weighted Composition Operators
Recall that the weighted composition operator  , : Proof.It is a consequence of the fact that ( 0 V ) * * =  ∞ V and that  , :  ∞ V →  ∞ V coincides with the biadjoint map of  , :  0 V →  0 V whenever both operators are well defined.

Proposition 5. Let 𝜑 : D → D be analytic function and 𝜓 ∈ 𝐻
Proof.Our proof starts with the observation that for every point in D there is a neighborhood where  is injective.
Otherwise, there exists  ∈ D and there are disjoint infinite sequences {  }  and {V  }  ⊆ D such that lim    = lim  V  =  and (  ) = (V  ).By Lemma 4,  has only a finite number of zeros in D; then we can assume (  )(V  ) ̸ = 0.For every  ∈ N, we may define Moreover, {  }  is an infinite linearly independent sequence in ( ∞ V ) * .Indeed, if we suppose then for all polynomial  we obtain For every  ∈ {1, 2, . . ., } it is sufficient to take a polynomial   such that to conclude   = 0.
By assuming that  , is Fredholm, then  * , is Fredholm and dim(Ker * , ) is finite.This fact contradicts the fact that {  }  is an infinite linearly independent sequence in Ker * , .Note that  cannot be constant.We are now in position to show that  is injective.Assume  is not injective.Then there are ,  ∈ D,  ̸ = , such that () = () =: .By our observation above there are   = (,  1 ) and   = (, As in the first part of the proof, we can assume that (  )(V  ) ̸ = 0. We define {  } as above.It is an infinite linearly independent sequence in Ker * , .A contradiction since  * , is Fredholm.
We observe that Lemma 8 is also true if we replace  0 V by  ∞ V .
Proposition 9. Let  : D → D be an analytic function and  ∈  0 V .If   and  , are bounded on  0 V or  ∞ V and  , is also Fredholm, then   and   are Fredholm.
is Fredholm if and only if  is an automorphism, which is equivalent to  being an automorphism.