# A new approach in index theory

###### Abstract

In this paper, we define an analytical index for a continuous family of B-Fredholm operators parameterized by a topological space as a sequence of integers, extending naturally the usual definition of the index and we prove the homotopy invariance of the index. In the case of a continuous family of Fredholm operators, we prove that if is a compact locally connected space, the analytical index establishes an isomorphism between the homotopy equivalence classes of families of Fredholm operators and the group where is the cardinal of the connected components of

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^{0}footnotetext: 2010 Mathematics Subject Classification: 47A53, 58B05

Key words and phrases: B-Fredholm, connected components, Fredholm, homotopy, index

## 1 Introduction

Let be the Banach algebra of all bounded linear operators defined from an infinite dimensional separable Hilbert space to the closed ideal of compact operators on and the Calkin Algebra. We write and for the nullspace and the range of an operator . An operator is called [6, Definition 1.1] a Fredholm operator if both the nullity of and the defect of , are finite. The index of a Fredholm operator is defined by . It is well known that if is a Fredholm operator, then is closed.

###### Definition 1.1.

Define an equivalence relation on the set where is the set of finite dimensional vector subspaces of and is the set of finite codimension vector subspaces of by:

Since is an infinite dimensional vector space, then the map:

defined by where is the equivalence class of the couple is a bijection. Moreover generate a commutative group structure on the set [ and is then a group isomorphism, where (resp. ) is set for the dimension (resp. codimension) of a vector space.

Reformulating [3, Proposition 2.1], we obtain.

###### Proposition 1.2.

Let be a Hilbert space and let If there exists an integer such that is an element of then is finite and for all is an element of and

where

###### Definition 1.3.

[3, Defintion 2.2] Let be a Hilbert space and let Then is called a B-Fredholm operator if there exists an integer such that is an element of In this case the index is defined by:

From Proposition 1.2, the definition of the index of a B-Fredholm operator is independent of the choice of the integer Moreover, it extends the usual definition of the index of Fredholm operators, which are obtained if

Note also that if is an integer such that is an element of then from [4, Theorem 3.1], is closed and the operator defined by is a Fredholm operator whose index is equal to the index of

In the papers [1] and [2], the analytical index of a single or a family of elliptic operators is expressed in terms of K-theory. The aim of this paper is to give a construction of an analytical index (called here simply index), for a continuous family of Fredholm operators parameterized by a topological space without using K-theory. It consists of a sequence of integers and gives a natural extension of the usual definition of the index of a single Fredholm operator. An extension to the case of continuous families of B-Fredholm operators is also considered.

## 2 Index of families of Fredholm operators

Consider now a family of Fredholm operators parametrized by a topological space that is a continuous map where is the set of Fredholm operators, endowed with the norm topolgy of We denote by the image of an element

Define an equivalence relation c on the space by setting that if and only if and belongs to the same connected component of Let be the quotient space associated to this equivalence relation and let be the space of continuous maps from the topological space into the topological space and let the map:

defined by for all Define also the map

by setting for all Here stands for the cardinal of the connected components of the topological space assuming that the space has at most a countable connected components.

###### Definition 2.1.

The analytical index (or simply the index) of a family of Fredholm operators parameterized by a topological space is defined by

Explicitly, we have:

Thus the index of a family of B-Fredholm operators is a sequence of integers in which may be a finite sequence or infinite sequence, depending on the cardinal of the connected components of

###### Theorem 2.2.

The index of a continuous family of Fredholm operators parameterized by a topological space is well defined as an element of In particular if is reduced to a single element, then the index of is equal to the usual index of the Fredholm operator

Proof. From the usual properties of the index [6, Theorem 3.11], we know that two Fredholm operators located in the same connected component of the set of Fredholm operators have the same index. Moreover, as is continuous, the image of a connected component of the topological space is included in a connected component of the set of Fredholm operators. This shows that the index of a family of Fredholm operators is well defined, and it is clear that if is reduced to a single element, the index of defined here is equal to the usual index of the single Fredholm operator

###### Example 2.3.

Let and such that for all Then from Definition 2.1, the index of the family is simply But since the dimension of the kernel presents a discontinuity in to build an index using K-theory needs more tools.

###### Definition 2.4.

A continuous family from to is said to be compact if is compact for all

###### Proposition 2.5.

i) Let and let be a continuous compact family from to Then is a Fredholm family and

ii) Let be two Fredholm families, then the family defined by is a Fredholm family and

Proof. This is clear from the usual properties of Fredholm operators.

###### Theorem 2.6.

Assume that is a compact topological space. Then the set of continuous compact families from to is a closed ideal in the Banach algebra

Proof. Recall that is a unital algebra with the usual properties of addition, scalar multiplication and multiplication defined by:

The unit element of is the constant function defined by the identity of for all Moreover as is compact, then if we set then equipped with this norm is a Banach algebra. Similarly equipped with the norm is a unital Banach algebra, where is the usual projection from onto the Calkin algebra

It is clear that is an ideal of Assume now that is a sequence in converging in to Then converges to as each is compact, then

###### Remark 2.7.

In the same way as in the case of the Calkin algebra, Theorem 2.6 generates a new Banach algebra which is Moreover, there is a natural injection defined by where is the equivalence class of the element of in and is the natural projection.

Open question: Given an element does there exist a continuous family such that

###### Theorem 2.8.

Assume that is a compact topological space and let Then is a Fredholm family if and only if is invertible in the Banach algebra

Proof. Assume that is a Fredholm family, then for all is a Fredholm operator. Thus is invertible in Let be its inverse, then the family defined by is a continuous family, because the inversion is a continuous map in the Banach algebra and is the inverse of in the Banach algebra

Conversely if is invertible in the Banach algebra then there exists such that where is defined by for all being the identity of Thus Thus is invertible in the Calkin algebra is a Fredholm operator and

###### Definition 2.9.

Let be in We will say that and are Fredholm homotopic, if there exists a map such for all and is a Fredholm operator.

B-Fredholm homotopic elements are defined in the same way.

###### Theorem 2.10.

Let be two Fredholm homotopic elements of Then

Proof. Since and are Fredholm homotopic, there exists a continuous map such that such that and for all For a fixed the map defined by is a continuous path in linking to Thus So and then

###### Theorem 2.11.

Let be a compact topological space. Then the index is a continuous locally constant function from into the group

Proof. Let then such that because is open in Then the index is constant on because is connected. We have Since is compact, there exists in such that Let the minimum of the and let such that If then and there exists such that Then So Hence the index is a locally constant function, in particular it is a continuous function.

###### Theorem 2.12.

Let be a compact topological space. Then the set is an open subset of the Banach algebra endowed with the uniform norm

Proof. Let then such that We have Since is compact, there exists in such that Let the minimum of the Let such that If then and there exists such that Then and is a Fredholm operator. Therefore and is open in

Alternatively, we can see that where is the open group of invertible elements of the unital Banach algebra and is the map defined by for all

In the case of a locally connected compact topological space, we give a result similar to the theorem of Atiyah-Jnich [6, Theorem 3.40].

###### Theorem 2.13.

Let be a locally connected compact topological space. Then the index is a bijective monoid isomorphism map from the homotopy equivalence classes of continuous families of Fredholm operators into the group

Proof. As is a locally connected compact topological space, its connected components are clopen subsets of and their cardinal is finite. Let be those connected components. To prove that the index is surjective, let On each connected component of , take a Fredholm operator such that , for and define the map by if Then it is clear that is continuous and that Thus the index is surjective.

Now let such that and let be an orthonormal basis of Let also be the Hilbet space generated by the family and be the orthogonal projection from H onto Then is a Fredholm operator and Consider the set induced with the topology of the product topological space

Since is compact, we can choose large enough so that for all and there exists an homeomorphism such that for each induces a linear isomorphism between and See [6, p.89] for more details about the construction of

Moreover is a Fredholm homotpy linking the family to the family and is a continuous family of invertible operators. As the orthogonal projection is of index the constant Fredholm family can be connected to the constant Fredholm family Thus the Fredholm family is Fredholm homotopic to a continuous family of invertible operators.

## 3 Index of families of B-Fredholm operators

In this section, we consider a continuous family of B-Fredholm operators parametrized by a topological space that is a continuous map where is the set of B-Fredholm operators, endowed with the norm topolgy of As before denotes the image of the element by the map

We begin by extending the Definition 2.1 to the case of continuous B-Fredholm families.

###### Definition 3.1.

The index of a continuous family of B-Fredholm operators parameterized by a topological space is defined by where:

defined by

for all where

This definition is clearly independent of the choice of the sequence since if and then Explicitly, we have:

Thus the index of a family of B-Fredholm operators is a sequence of integers in As before stands for the cardinal of the connected components of the topological space assuming that the space has at most a countable connected components.

###### Proposition 3.2.

Let be two B-Fredholm operators acting on a Hilbert space having equal index. Then and are B-Fredholm path connected.

Proof. Let such that Then From [4, Remark A] there exists such if both of and are Fredholm operator and

Let such that Since and are Fredholm operators having the same index, then they are Fredholm path-connected. As is B-Fredholm path-connected to and is B-Fredholm path connected to then and are B-Fredholm path-connected.

We characterize now the connected components of the topological space of B-Fredholm operators

###### Theorem 3.3.

The connected components of the topological space of B-Fredholm operators are the sets

Proof. Let from the properties of Fredholm opertors, we know that is connected in Let such that and let be a Fredholm operator of index Then From Proposition 3.2, is B-Fredholm connected to Thus is path connected and so it is connected. Thus the connected components of are the sets

###### Theorem 3.4.

Let be two B-Fredholm homotopic elements of Then

Proof. Since and are B-Fredholm homotopic, there exists a continuous map such that such that and for all For a fixed the map defined by is a continuous path in linking to Thus because and must be in the same connected component of So and then

In the case of B-Fredholm operators, we cannot except to have a similar theorem to Theorem 2.13, because as shown by the following example, the homotopy equivalence classes of families of B-Fredholm operators is not a monoid.

###### Example 3.5.

Let and let be the operators defined on by:

It’s easy to see that is the projection on the closed subspace of generated by the orthogonal family:

and is therefore a B-Fredholm operator.

Similarly is a projection on the closed subspace of generated by the orthogonal family:

and is a B-Fredholm operator.

However

Moreover, the following example shows that there exists B-Fredholm operators such that is B-Fredholm, but

###### Example 3.6.

. Let , and let be the operators defined on by:

Then is a B-Fredholm operator of index is a B-Fredholm operator with index But is a B-Fredholm operator with index

###### Theorem 3.7.

If are commuting families of operators in If and where is the unit element of the algebra then and

Proof. For all are commuting operators satisfying where is the identity operator on As and are B-Fredholm operators, then From [5, Theorem 1.1], is a B-Fredholm operator and As is continuous, then and

###### Definition 3.8.

A B-Fredholm family is called a uniformly B-Fredholm family, if there exists such that is a Fredholm operator and for all and all such that

###### Theorem 3.9.

Let be a locally connected compact topological space. If is a uniformly B-Fredholm family of index then is B-Fredholm homotopic to a family of invertible operators.

Proof. Since is a uniformly B-Fredholm family of index there exists such that is a Fredholm operator and for all and all such that Thus is a continuous Fredholm family of index Moreover the map defined by is a B-Fredholm homotopy such that and From Theorem 2.13, is Fredholm homotopic to a family of invertible operators. As the family is B-Fredholm homotopic to then is B-Fredholm homotopic to a family of invertible operators.

###### Corollary 3.10.

Let be a locally connected compact topological space. If is a B-Fredholm family of index such that is isolated in the spectrum of in the Banach algebra then is B-Fredholm homotopic to a family of invertible operators.

Proof. Since is isolated in the spectrum of in the Banach algebra there exists such that if then is invertible in Thus for all and all scalar such that is a Fredholm operator of index because it is clearly B-Fredholm path connected to Thus is a uniformly B-Fredholm family of index The corollary is then a consequence of Theorem 3.9.

## References

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Mohammed Berkani,

Science faculty of Oujda,

University Mohammed I,

Laboratory LAGA,

Morocco

,