Some New Properties in Fredholm Theory, Schechter Essential Spectrum, and Application to Transport Theory
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Research Article Some New Properties in Fredholm Theory, Schechter Essential Spectrum, and Application to Transport Theory Boulbeba Abdelmoumen,1 Abdelkader Dehici,2 Aref Jeribi,1 and Maher Mnif1 1 2
Department of Mathematics, Faculty of Science of Sfax, Sfax 3018, Tunisia D´epartement des Sciences Exactes, Universit´e 8 Mai 1945, BP 401, Guelma 24000, Algeria
Correspondence should be addressed to Aref Jeribi, [email protected] Received 19 April 2007; Revised 11 July 2007; Accepted 24 September 2007 Recommended by Nikolaos S. Papageorgiou The theory of measures of noncompactness has many applications on topology, functional analysis, and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We give some results concerning a certain class of semi-Fredholm and Fredholm operators via the concept of measures of noncompactness. Moreover, we establish a fine description of the Schechter essential spectrum of closed densely defined operators. These results are exploited to investigate the Schechter essential spectrum of a multidimensional neutron transport operator. Copyright q 2008 Boulbeba Abdelmoumen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Let X, · be an infinite-dimensional Banach space. The open ball of X will be denoted by BX and its closure by BX . We denote by CX resp., LX the set of all closed densely defined linear operators resp., the space of all bounded linear operators on X. The set of all compact operators of LX is designed by KX. Let T ∈ CX, we write NT ⊆ X for the null space and RT ⊆ X for the range of T . We set αT : dim NT and βT : codim RT . The set of upper semi-Fredholm operators is defined by Φ X T ∈ CX such that αT < ∞, RT closed in X ,
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and the set of lower semi-Fredholm operators is defined by Φ− X T ∈ CX such that βT < ∞ then RT closed in X .
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Journal of Inequalities and Applications
ΦX : Φ X∩Φ− X is the set of Fredholm operators in CX, while Φ± X : Φ X∪Φ− X is the set of semi-Fredholm operators in CX. If T ∈ ΦX, the number iT : αT − βT is called the index of T . The spectrum of T will be denoted by σT . The resolvent set of T , ρT , is the complement of σT in the complex plane. A complex number λ is in ΦT, Φ−T, Φ±T , or ΦT if λ − T is in Φ X, Φ− X, Φ± X, or ΦX, respectively. In the next proposition we recall some well-known properties of those sets see, e.g., 11, 16, 30. Proposition 1.1. For any T ∈ CX, i ΦT , Φ−T and ΦT are open, ii iλ − T is constant on any component of ΦT . There are many ways to define the essential spectrum of a closed densely defined linear operator on a Banach space. Hence several definitions of the essential spectrum may be found in
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