A new approach in index theory
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Mathematische Zeitschrift
A new approach in index theory M. Berkani1 Received: 5 February 2020 / Accepted: 15 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we define an analytical index for a continuous family of Fredholm operators parameterized by a topological space X into a Hilbert space H , as a sequence of integers, extending naturally the usual definition of the index and we prove the homotopy invariance of the index. We give also an extension of the Weyl theorem for normal continuous families and we prove that if H is separable, then the space of B-Fredholm operators on H is path connected. Keywords B-Fredholm · Connected components · Fredholm · Homotopy · Index Mathematics Subject Classification 47A53 · 58B05
1 Introduction Let L(H ) be the Banach algebra of all bounded linear operators defined from an infinite dimensional separable Hilbert space H to H , K (H ) the closed ideal of compact operators on H and L(H )/K (H ) the Calkin Algebra. We write N(T ) and R(T ) for the nullspace and the range of an operator T ∈ L(H ). An operator T ∈ L(H ) is called [5, Definition 1.1] a Fredholm operator if both the nullity of T , n(T ) = dim N(T ) and the defect of T , d(T ) = codim R(T ), are finite. The index ind(T ) of a Fredholm operator T is defined by ind(T ) = n(T ) − d(T ). It is well known that if T is a Fredholm operator, then R(T ) is closed. Definition 1.1 [6] Let T ∈ L(H ) and let (T ) = {n ∈ N : ∀m ∈ N, m ≥ n ⇒ R(T n ) ∩ N(T ) ⊆ R(T m ) ∩ N(T )}. Then the degree of stable iteration of T is defined as dis(T ) = inf (T ) (with dis(T ) = ∞ if (T ) = ∅). Define an equivalence relation R on the set f dim(H )× f cod(H ), where f dim(H ) is the set of finite dimensional vector subspaces of H and f cod(H ) is the set of finite codimension
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M. Berkani [email protected] Science Faculty of Oujda, University Mohammed I, Laboratory LAGA, Oujda, Morocco
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M. Berkani
vector subspaces of H by: (E 1 , F1 )R(E 1 , F1 ) ⇔ dim E 1 − codim F1 = dim E 1 − codim F1 , Since H is an infinite dimensional vector space, then the map: : [ f dim(H ) × f cod(H )]/R → Z, defined by ((E 1 , F1 )) = dim E 1 − codim F1 , where (E 1 , F1 ) is the equivalence class of the couple (E 1 , F1 ), is a bijection. Moreover generate a commutative group structure on the set [ f dim(H ) × f cod(H )]/R and ψ is then a group isomorphism, where dim (resp. codim) is set for the dimension (resp. codimension) of a vector space. Reformulating [3, Proposition 2.1], we obtain. Proposition 1.2 Let H be a Hilbert space and let T ∈ L(H ). If there exists an integer n such that (N (T ) ∩ R(T n ), R(T ) + N (T n )) is an element of f dim(H ) × f cod(H ), then d = dis(T ) is finite and for all m ≥ d, (N (T ) ∩ R(T m ), R(T ) + N (T m )) is an element of f dim(H ) × f cod(H ) and (N (T ) ∩ R(T m ), R(T ) + N (T m )) R (N (T ) ∩ R(T d ), R(T ) + N (T d )) Definition 1.3 [3, Defintion 2.2] Let H be a Hilbert space and let T ∈ L(H ). Then T is called a B-Fredholm operator if there exists an integer
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