On Some Properties of Cowen-Douglas Class of Operators

We will consider multiplication operators on a Hilbert space of analytic functions on a domain Ω ⊂ C. For a bounded analytic function φ on Ω, we will give necessary and sufficient conditions under which the complement of the essential spectrum of Mφ in φ(Ω) becomes nonempty and this gives conditions for the adjoint of the multiplication operatorMφ belongs to the Cowen-Douglas class of operators. Also, we characterize the structure of the essential spectrum of a multiplication operator and we determine the commutants of certain multiplication operators. Finally, we investigate the reflexivity of a Cowen-Douglas class operator.


Introduction
In this section we include some preparatory material which will be needed later.
For a positive integer  and a domain  ⊂ C, the Cowen-Douglas class   () consists of bounded linear operators  on any fixed separable infinite dimensional Hilbert space  with the following properties: Here Span denotes the closed linear span of a collection of sets in .The classes   () were introduced by Cowen-Douglas (see [1]), and each element of   () is called a Cowen-Douglas class operator.By   , we mean   () for some complex domain .For the study of Cowen-Douglas classes   , we mention [1][2][3][4][5][6][7].
Recall that a bounded linear operator  on a Hilbert space is a Fredholm operator if and only if ran  is closed and both ker  and ker  * are finite dimensional.We use () and   () to denote, respectively, the spectrum of  and the essential spectrum of .
Now let H be a separable Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. Recall that if  ∈ B(H), then Lat() is by definition the lattice of all invariant subspaces of , and AlgLat() is the algebra of all operators  in B(H) such that Lat() ⊂ Lat().An operator  in B(H) is said to be reflexive if AlgLat() = (), where () is the smallest subalgebra of B(H) that contains  and the identity  and is closed in the weak operator topology.
Also, if H is a Hilbert space of functions analytic on a plane domain Ω, a complex-valued function  on Ω for which  ∈ H for every  ∈ H is called a multiplier of H and the multiplier  on H determines a multiplication operator   on H by    = ,  ∈ H.The set of all multipliers of H is denoted by (H).Clearly (H) ⊂  ∞ (Ω), where  ∞ (Ω) is the space of all bounded analytic function on Ω.In fact ‖‖ ∞ ≤ ‖  ‖ (see [8]).
Let H be a Hilbert space of functions analytic on a domain Ω ⊂ C satisfying the following axioms: Axiom 1.For every point  ∈ Ω, the functional of point evaluation at , is a nonzero bounded linear functional on H.A space H satisfying the above conditions is called Hilbert space of analytic functions on Ω (see [3,9]).The Hardy and Bergman spaces are examples for Hilbert spaces of analytic functions on the open unit disk.
Note that, by Axiom 1, there exists a reproducing kernel   ∈ H such that () = ⟨,   ⟩ for all  ∈ H. Also, by using Axiom 2 and the closed-graph theorem, the operator of multiplication by ,   , is a bounded linear operator on H.So Axiom 2 says that (H) =  ∞ (Ω).If   is polynomially bounded on H and Ω is the open unit disk, then (H) =  ∞ (Ω) (see [9,Theorem 1]).In the rest of the paper we assume that H is a Hilbert space of analytic function on a bounded plane domain Ω.
In this paper, we want to study some properties of operators in   .We see that complement of the essential spectrum of a multiplication operator   is nonempty if and only if the adjoint of   belongs to some   .Also, we investigate the intertwining multiplication operators and reflexivity of the multiplication operator on   .For some other source on these topics one can see [10][11][12][13][14][15][16].

Multiplication Operators with Adjoint in 𝐵 𝑛 and Its Spectra
Recall that if  is a Cowen-Douglas class operator, then it should be () \   () ̸ = 0.For  ∈  ∞ (Ω), we would like to give some necessary and sufficient conditions so that (  ) \   (  ) becomes a nonempty open set.This implies a sufficient condition for the adjoint of the multiplication operator   to be a Cowen-Douglas class operator.Theorem 1.Let  be a nonconstant function in  ∞ (Ω), (  ) \   (  ) ̸ = 0, and   /‖  ‖ → 0 weakly as (, Ω) → 0. Then there exist a domain  ⊂ (Ω) and a positive integer  such that Ω ∩  −1 () consists of  points (counting multiplicity) for every  ∈ .
Proof.First note that if  ∈ (Ω), then  = () for some  ∈ Ω.But by Axiom 1, the functional of evaluation at  is a bounded point evaluation; thus the reproducing kernel   is defined and we have Now, let  be a connected component of the open set for all  in .But the index function is continuous from the set of semi-Fredholm operators into Z ∪ {±∞} with discrete topology; thus, index If  ∈ , then  = ( 0 ) for some  0 ∈ Ω and so  *    0 =   0 .Thus   0 ∈ ker (  − ) * .Since a finite subset of points  in Ω yields a linearly set independent set of functions   in H, thus Ω ∩  −1 () consist of at most  points for all  in .So for each fixed  ∈ , there exist  1 ,  2 , . . .,   in Ω and  1 ,  2 , . . .,   in N such that  ≤  and for all  ∈ Ω we have where  belongs to  ∞ (Ω) and is nonvanishing on Ω.Now by a method used in the proof of [3, Proposition 3.1] we show that the function  is also bounded below on Ω.For this choose  > 0 such that (, ) is contained in .Put  =  −1 ((, )), and thus  is a compact subset of Ω and so it has a positive distance  to Ω.Now if  is not bounded below on Ω, then there exists a sequence Since  is nonvanishing on Ω implies that (  ) → , so there exists a positive integer  such that (  ) ∈ (, ) for all  > .Hence   ∈  for all  >  that is contradiction to   → Ω.Thus the function  is indeed bounded below on Ω.Now since  is bounded below and bounded above on Ω it is an invertible element of  ∞ (Ω) and so the operator   is invertible on H because (H) =  ∞ (Ω).Thus index(  ) = 0. Note that since we get Clearly,   −   is injective; thus for  = 1, . . ., .Note that, by Axiom 3 on H, ker(  −   ) * is one-dimensional (see [17]); thus ∑  =1   =  and therefore Ω∩ −1 () consists of exactly  points (counting multiplicity) for every  ∈  and now the proof is complete.
From the proof of Theorem 1, we can conclude the following result.
Note that, by Axiom 3, for every  ∈ Ω the operator  − is bounded below on H and also the space H ⊖ ( − )H is one-dimensional (see [3]).So the Hilbert space under consideration, H, satisfies the conditions assumed by Zhu in [7].
The following result was stated by Zhu in [7, Proposition 5.2], but its proof is left to readers.For this reason we sketch a proof of this proposition and although our proof might seem more straightforward than the one stated by Zhu, we emphasise that our main idea is given from [7].Proposition 3. Suppose  ∈  ∞ (Ω) and  is a domain contained in (Ω).If there exists a positive integer  such that Ω ∩  −1 () consists of  points (counting multiplicity) for every  ∈ , then the adjoint of the operator   : H → H belongs to the Cowen-Douglas class   (), where  = { :  ∈ }.

Intertwining Multiplication Operators
The following characterization of the commutant {}  of  is given in Theorem 3.7 of [2], which is stated for the convenience of the reader.Note that  is the reproducing kernel for a coanalytic functional Hilbert space K defined in [2].

Reflexivity in Cowen-Douglas Class of Operators
Ω is a special bounded plane domain.In this section we give some sufficient conditions so that the associated canonical model is reflexive.This answers Question 5.6 in [9, p. 98].Indeed, we investigate the reflexivity of   (Ω), when Ω is an arbitrary bounded domain.
It is well known that every operator in the class   (Ω) is unitarily equivalent to the adjoint of the canonical model associated with a generalized Bergman kernel (g.B.k. for brevity)  (see [2,6]).Actually  is the reproducing kernel for a coanalytic functional Hilbert space K K (briefly K) on which we can define the operator   of multiplication by .The operator  =  *  acting on K is called the canonical model associated with .We know that, for every  in Ω,  −  is onto and and dim ker( − ) = .
Recall that a compact subset  of the plane is a spectral set for a bounded operator  if  contains () and ‖()‖ ≤ sup ∈ |()| for all rational functions  with poles off .
Also, an open connected subset  of the plane is called a Carathéodory region if its boundary equals the boundary of the unbounded component of C − .
It is proved in [4] that if  is in  1 (Ω) and  * is an injective unilateral weighted shift, then  is reflexive.Also, it has been shown that if  is in   (Ω), where Ω is a Carathéodory region such that () = Ω is a spectral set for , then  is reflexive (see [4,Theorem 2]).This implies that if  is a contraction in   (D) where D is the open unit disk, then  is reflexive.Here we want to investigate the reflexivity of  on   (Ω), where Ω is an arbitrary domain in C. Theorem 12.If  is in   (Ω), where Ω ⊂ C is an arbitrary domain, then there exists a total set  such that the weak closure of the set {():  is a polynomial,  ∈ } contains ().
Since  ∈ M, M ̸ = 0. Now clearly M is closed subspace of K  ∞ and we have Thus  ∞  ∈ M and so M ∈ Lat( ∞ ).But Lat() ⊆ Lat(),;thus Lat( ∞ ) ⊆ Lat( ∞ ) and we get M ∈ Lat( ∞ ).Therefore  ∞  ∈ M and so there exists a sequence {  }  of polynomials such that that is a total subset of K.At this time the proof is complete.
Let , {  }  , and  = {  }  be defined as in the proof of Theorem 12.At the end of the proof of Theorem 12, we saw that, for all , |(  − )(  )| → 0 as  → ∞.Now we ask the following question.If the answer of Question 13 is positive, then ‖  ‖  ≤  for some  > 0 and we may have the following corollary.Note that the special case of this corollary has been proved as Theorem 2 in [4], only whenever Ω is a Carathéodory region.Corollary 14.If  is in   (Ω) where Ω ⊂ C is a domain such that () = Ω is a spectral set for , then  is reflexive.