Unbounded B-Fredholm Operators on Hilbert Spaces

Abstract

This paper is concerned with the study of a class of closed linear operators densely defined on a Hilbert space $H$ and called B-Fredholm operators. We characterize a B-Fredholm operator as the direct sum of a Fredholm closed operator and a bounded nilpotent operator. The notion of an index of a B-Fredholm operator is introduced and a characterization of B-Fredholm operators with index $0$ is given in terms of the sum of a Drazin closed operator and a finite-rank operator. We analyse the properties of the powers $T^m$ of a closed B-Fredholm operator and we establish a spectral mapping theorem.

A characterization of unbounded generalized meromorphic operators

“…The aim of this section is to give some characterizing properties of unbounded generalized meromorphic operators by mean of B-Fredholm spectrum and polynomially meromorphic operators. First, we start with the following theorem proved in [4] Proof. Since, λ i λ j , then the polynomials p i (λ) = (λ − λ i ) m i , 1 ≤ i ≤ n, are relatively prime, and hence there exist polynomials q i (λ) and q j (λ) such that q i (λ)p i (λ) + q j (λ)p j (λ) = 1, with 1 ≤ i j ≤ n. Then, we have for sufficiently large m…”

On the index of pseudo B-Fredholm operator

“…For proving [4,Theorem 2.4], the authors used the characterization of B-Fredholm operators in the case of Hilbert spaces based on the Kato's decomposition of quasi-Fredholm operators [15].…”

Unbounded generalized B-Fredholm operators

“…The class of B-Fredholm operators was first introduced by M. Berkani in [2] in the case of bounded operators acting on a Banach space. This notion of operators was generalized by Berkani and Castro-González in [4] to unbounded operators in Hilbert space. Recently, this class of operators was extended and studied by O. García et al in [12] to generalized bounded B-Fredholm operators acting on a Banach space.…”