On α-Weyl Operators

The purpose of this article is to present Schechter’s manner to introduce α-Wayl operators and compare this definition with another one given by Yadav and Arora. Moreover, we introduce generalized Weyl operator in the way that we keep many properties of the class of Weyl operators.


Introduction
Let H be a (complex) Hilbert space and ( ) denote the Hilbert dimension of the space H, where 0 h > ℵ .Then any nonzero proper closed two-side ideal in ( ) for some 0 h α ≤ < .Denote this ideal with α  (For more details see [1]).In the case when 0 α = ℵ (cardinal- ity of the natural numbers), we have ( ) , the ideal of all compact operators.We denote the kernel of an operator by ( ) N T , ( ) ( ) , where ( ) , Theorem 3.2.8).Another way to introduce Fredholm ope- rators is using the dimensions of the kernel and the codimension of the rang of an operator: The set of all upper (respectively, lower) semi-Fredholm oper- ators will be denoted by ( ) The set of all semi-Fredholm operators is defined by and the set of all Fredholm operators is defined by The index of a semi-Fredholm operator is defined as ( ) ( ) ( ) and the Weyl operators with The Weyl operator still conserves one of the basic properties for the operators between finite dimensional spaces: Fredholm alternative.Moreover, with some extra conditions (like finite ascent or descent), such operators have very nice property: there are Drazin invertible (For more details about generalized invertibility we suggest [3]).Now, the natural question appears: it is necessary to observe only finite dimensional situation for kernel, or co-dimension of range, or ascent and descent, etc.
We can find the very first investigation in this direction in the works of G. Edgar, J. Ernest and S. G. Lee.In papers [4] [5], they introduce the definition of an α-closed subspace which allowed them to give a new definition of an α-Fredholm operator.Accordingly ([4], Definition 2.7), ( ) S.C. Arora and P. Dharmarha, beside that observed joint weighted spectrum, introduced α-Weyl operators like α-Fredholm operator with abstract index equal to zero, or like intersection of perturbations with the operators of rank α (for more details see [6]- [11]).Beside that these two definitions of α-Weyl operators are not equivalent, the notion of abstract index is not very applicable, such that we need new ways of generalization of the index of an operator that widen Weyl operator theory in infinite dimensional case.
The main results of the paper are present in remaining two sections.In the next section we define β-index of an α-Fredholm operator, for 1 h β α ≤ < < , which we use in definition of α-Weyl operator and α-Weyl spectrum (Definition 2).In the theorems 3, 5 and 7 we give some basic properties of such operators.In the last section we define the generalized Weyl operator in the way that we widespread the definition given by D.S. Djordjević in [12].The new class of the generalized Weyl operator conserves many properties of the class of Weyl operators (see Theorem 9).

α-Weyl Operators
For Hilbert spaces, L. A Coburn, in [13], defined the Weyl spectrum of an operator as S B H ∈ .On the other hand, in the same year, M. Schechter (see [14]) defined the Weyl spectrum of T by [15] established the equivalence between both definitions and we will use ( ) The notion of Fredholm operators can be extended to an arbitrary dimension (less then the dimension of the space H) of the null space of T and * T using the α-closedness.In this way, in [4], G. Edgar, J. Ernest If ( ) d T denotes the approximate nullity of T (for the definition and basic property see [4]), then, by ([4], Theorem 2.6) and ( [16], Theorem 3.1), we have nice (Atkison type) characterization for an α-Fredholm operator.

Theorem 1. Let T be an operator on ( )
B H and α be a cardinal number such that 0 h α ℵ ≤ < .Then the following conditions are equivalent: For more properties of α-Fredholm operators we specially refer to [4] [16] [17].Later, Yadav and Arora, in [11], for non separable Hilbert spaces, defined the Weyl spectrum of wight α for some operator ( ) T B H ∈ be an α-Fredholm operator, 1 h α ≤ < , then we can extend the definition of index, for all β α < , using slightly modification of definition in [18] (for more details see [16]): In the way of Schechter definition of the Weyl operators, we defined α-Weyl operators next.Definition 2. For an operator ( ) We can define the α-Weyl spectrum in (one of the usual) way(s): is not an -Weyl operator : or ind 0, for , .
Now arise natural question about equivalency of two ways of definition of α-Weyl spectrums.It is easy to see that ( ) ( ) ( ) The next theorem gives us the answer for all α, 0 h α ℵ ≤ < .
In the same decomposition of H we can present T λ − in the way: ( ) Additionally, by ([16], Lemma 4.8), ( : is an arbitrary isomorphism between γ-dimensional (closed sub)spaces.Then T  is an invertible operator and . Hence, ( ) is an invertible operator, i.e.
( ) (1) In the future, we will use the notation ( ) for the α-Weyl spectrum of T. (2) In the case when 0 α ℵ < , if we slightly modificated the proof of Theorem 3 with additional condition that Iso , we can see that α-Weyl operator that is not invertible can be approximated (in the norm) with an invertible operator, i.e. its belong to ( ) , where ( ) is a closed subspace of H.For T denotes the rank of an operator.Using duality we have classical (Atkinson) characterization of Fredholm operators is invertibility in Calkin algebra, i.e.( )

Theorem 3 .
Let T be an operator on ( ) B H and let α be a cardinal number such that 0 h , then the results follows from the classical Fredholm theory.17], Theorem 2.6) and ([18], Corollary 1), T λ − is an α-Fredholm operator and λ − is an α-Fredholm operator and, for every cardinal 0 β α ℵ ≤ < , we have .Suppose that T λ − has closed range.Let i be an isometry from ( ) same dimension γ ) and let ( ) In the case when T λ − is α-Fredholm operator with no closed range, by ([4], Theorem 2.6), for each 0 >  small enough, there exists a closed subspace W  of H that contains

Theorem 5 . 6 .Theorem 7 .
Let T be an operator on ( ) B H and let α be a cardinal number such that 0 h α ℵ ≤ < .Then the following conditions are equivalent: Proof.(1) ⇔ (2) follows directly from definition of α-Fredholm operator and definition of ⇔ (3) follows from ([4], Theorem 2.6) and ([16], Theorem 3.1 and Lemma 4.8).Remark Any of equivalent condition (1)-(3) of Theorem 5 implies that there exists a closed subspaces M and N of H such that M H =  from definition of d  for T and N we define in similar way, only using operator *T .Let T be an operator on ( ) B H and let α be a cardinal number such that 0