Fredholm Weighted Composition Operator on Weighted Hardy Space

It is natural to ask for which ψ and φ the weighted composition operator Cψ,φ is bounded and compact on H. In the past ten years, this problem on various Hilbert spaces of analytic functions has been studied extensively. It seems there is no unified ways to solve the problem since different spaces have very distinguished properties; see [1–10] for the characterization of bounded and compact weighted composition operators on different function spaces. In [11], the invertibility of weighted composition operator on the classical Hardy space is characterized, as an extension, in [12, 13]; the Fredholmness of weighted composition operator on Dirichlet space andweightedDirichlet space is characterized, respectively. The results show that the Fredholmness and invertibility of weighted composition operator are closely related to properties of the reproducing kernel functions. Inspired by the characterization of Fredholm composition operator in [6], in this paper, we give a unified characterization of Fredholmweighted composition operator on a class of weighted Hardy space. Now we follow some notations as in [14]. H is called a weighted Hardy space if the monomials


Introduction
Let H be a Hilbert space of analytic functions on the open unit disk D. For  ∈ H and analytic self-map  of D, the weighted composition operator  , on H is defined as  ,  =  ( ∘ ) ,  ∈ H. ( It is natural to ask for which  and  the weighted composition operator  , is bounded and compact on H.In the past ten years, this problem on various Hilbert spaces of analytic functions has been studied extensively.It seems there is no unified ways to solve the problem since different spaces have very distinguished properties; see [1][2][3][4][5][6][7][8][9][10] for the characterization of bounded and compact weighted composition operators on different function spaces.In [11], the invertibility of weighted composition operator on the classical Hardy space is characterized, as an extension, in [12,13]; the Fredholmness of weighted composition operator on Dirichlet space and weighted Dirichlet space is characterized, respectively.The results show that the Fredholmness and invertibility of weighted composition operator are closely related to properties of the reproducing kernel functions.
Inspired by the characterization of Fredholm composition operator in [6], in this paper, we give a unified characterization of Fredholm weighted composition operator on a class of weighted Hardy space.Now we follow some notations as in [14].
H is called a weighted Hardy space if the monomials 1, ,  2 , . . .constitute a complete orthogonal set of nonzero vectors in H. Denote () = ‖  ‖; then for  ∈ H with So we use  2 () to denote the weighed Hardy space with weighted sequence {()}. 2 () is called automorphism invariant if, for any automorphism map  of D, That is, the composition operator   is bounded on  As an application, in the end of this paper, we give a complete characterization of Fredholm and invertible weighed composition operator on D  (−∞ <  < ∞).Recall that D  is a class of important weighed Hardy space with () = ( + 1)  .

Fredholm Weighted Composition
Operator on  2 () Recall that  2 () has reproducing kernel function That is, for  ∈  2 (), Let then [] is the th derivative evaluation kernel function at  in  2 () [14], that is, for  ∈  2 (), Firstly we give some discussion about weighted composition operator  , acting on []  .The following lemmas is well known and easy to verify.
By Fań di Bruno's formula, (  ∘ ) () () can be expressed as linear combination of where  = ∑  =1   and ∑  =1   =  with nonnegative integer   . Since it follows from Lemma 3 and ( 9) that  * , [] can be expressed as where   (, ) is algebraic combination of  () and  () with 0 ≤ ,  ≤ . Since the degree of nonzero coefficient  []  () () is at least  and the coefficient of so by (12) the coefficient of On the other hand, So by (10) we have Comparing ( 12) and ( 17) with the aid of (15), we obtain By the above reasoning, we obtain the following lemma.
Note that in [ for some nonnegative integer .If  , is a bounded Fredholm operator on  2 (), then  is an automorphism of D.
Theorem 6 gives the necessary condition of  for the weighted composition operator  , to be Fredholm on  2 () with condition (22).Furthermore, if  2 () is automorphism invariant, then the multiplication operator   on  2 () must be bounded Fredholm since in this case   is invertible and So in the following, we consider the condition for a multiplication operator   on  2 () to be Fredholm.Denote M = { is analytic in D,  ∈  2 (),  ∈  2 ()}.For  ∈ M, the multiplication operator   on  2 () is bounded.M is called the multiplier space of  2 ().
Proof.If   is a Fredholm operator, then there exist bounded operator  and compact operator  such that where  is identity.Acting on []  /‖  ‖)‖ → 0.
The following example shows that generally  2 () is not automorphism invariant even if  2 () satisfies condition (22).

Application
In the section, we give a complete characterization of bounded Fredholm and bounded invertible weighted composition operator on D  .First, we cite some well-known results of D  .Denotes M  by the multiplier space of D  .