Théorème de l’application spectrale pour le spectre essentiel quasi-Fredholm

Abstract

Abstract. In 1958, T. Kato proved that a closed semi-Fredholm operator A in a Banach space can be written A = A 1 ⊕ A 0 where A 0 is a nilpotent operator and A 1 is a regular one.J. P. Labrousse studied and characterised this class of operators in the case of Hilbert spaces. He also defined a new spectrum named "essential quasiFredholm spectrum" and denoted σe (A).In this paper we prove that the essential quasi-Fredholm spectrum defined by J. P. Labrousse satisfies the mapping spectral theorem, i.e.: If A is a bounded operator in a Hilbert space and f an analytic function in a neighbourhood of the spectrum σ(A) of A, then f(σe(A)) = σe(f (A)). Dans ce travail nous allons démontrer que le spectre essentiel quasi-Fredholm défini par J. P. Labrousse vérifie le théorème de l'application spectrale, c'està dire: Si A est un opérateur bourné d'un espace de Hilbert dans lui même et f une fonction analytique au voisinage du spectre σ(A) de A, alors f(σe(A)) = σe(f (A)).

Unbounded B-Fredholm Operators on Hilbert Spaces

“…From [4,Lemma 2.3] it follows that each B-Fredholm operator is a quasi-Fredholm operator of degree d 0 = dis(T ). We also observe that the definition of the index of a B-Fredholm operator is independent of the integer d ∈ ∆(T ) chosen.…”

“…Hence, we have proved that T d is a Fredholm closed operator. However, the fact that, for some integer n, the range space R(T n ) is closed and the restriction operator T n is a Fredholm closed operator does not allow us to conclude that T is a B-Fredholm operator in the sense of Definition 2.1, as in the case of bounded operators [4].…”