THEOREM OF SPECTRAL APPLICATION FOR THE E SSENTIAL QUASI-FREDHOLM SPECTRUM

In 1958, T. Kato proved that a closed semi-Fredholm operator A in a Ba nach space can be written A = A(1) + A(0) where A(0) is a nilpotent op erator and A(1) is a regular one. J. P. Labrousse studied and characte rised this class of operators in the case of Hilbert spaces. He also d efined a new spectrum named ''essential quasi-Fredholm spectrum'' and denoted sigma(e)(A). In this paper we prove that the essential quasi-F redholm spectrum defined by J. P. Labrousse satisfies the mapping spec tral theorem, i.e.: If A is a bounded operator in a Hilbert space and f an analytic function in a neighbourhood of the spectrum sigma(A) of A, then f(sigma(e)(A)) = sigma(e)(f(A)).