International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2016): 79.57 | Impact Factor (2015): 6.391 Volume 7 Issue 2, February 2018 www.ijsr.net Licensed Under Creative Commons Attribution CC BY New Properties and Their Relation to type Weyl- Browder Theorems Buthainah A.A. Ahmed 1 , Zaman Adel Rashid 2 Department of Mathematics, College of Science, University of Baghdad, Iraq Abstract: In this paper we introduce and study the new spectral properties (o), (ao), (sz) and (asz). Our main goal in this paper is to study relationship between these properties and other Weyl – Browder type theorems. 1. Introduction Let be a Banach pace, and let () be the Banach algebra of all bounded linear operators acting on . For an operator ∈() we denote by () the dimension of the kernel of , ( = . ()), and () the codimension of the range , ( = dim ( ∖ )). If ()< ∞ and () closed then is said to be upper semi-Fredholm, while ∈() is said to be lower semi-Fredholm if < ∞. Let + and − denote the class of all upper semi-Fredholm operators and the class of all lower semi-Fredholm operators, respectively. If ∈() is either an upper or lower semi-Fredholm operator, then is called a semi- Fredholm operator, while if ()< ∞ and () are finite then is called a Fredholm operator. The index of given by = − . For ∈(), let + − = ∈ + : ≤ 0. Then the Weyl essential approximate spectrum of is defined by + − ={∈ ℂ: − ∉ + − }. An operator ∈() is said to be a Weyl operator if it is a Fredholm operator of index zero. The Weyl spectrum ()of is defined by ={∈ℂ: − }. The ascent = () of an operator is defined as the smallest non- negative integer such that = +1 and the descent = () defined as the smallest non-negative integer such that = +1 . If () and () are both finite then = .Now we will define the class of all upper semi-Browder operators ℬ + ={∈ + : ()< ∞}, and the class of all lower semi-Browder operators ℬ − = ∈ − : < ∞. The class of all Browder operators is defined by ℬ = ℬ + ∩ ℬ − . Browder spectrum of is defined by = ∈ ℂ: − ∉ ℬ. Recall that the operator is said to have the single valued extension property at 0 ∈ℂ ( for short), only analytic function : → which satisfies the equation − =0 for all ∈ is the function≡ 0. An operator is said to be have if has at every point ∈ℂ, (see, [8]). For ∈() and non-negative integer, define [0] to be restriction of to ( )viewed as a map from ( ) into ( ) [ in particular = ]. Abounded linear operator is said to be an upper (resp. lower) semi-B-Fredholm operator if for some integer the range space ( )is closed and [ ] is an upper (resp. lower ) semi-Fredhlom operator, and in this case the index of is defined as the index of semi-Fredholm operator [ ] ,(see, [5]). If [ ] is a Fredholm operator, then is said to be a B-Fredholm. Lemma 1.1[2] Let ∈(). Then ( ) is upper semi-B-Fredholm and ()< ∞ if and only if ∈ + . (ii) is lower semi-B-Fredholm and < ∞ if and only if ∈ − . An operator ∈() is called a B-Weyl operator, if it is a B-Fredholm operator of index zero. The B-Weyl spectrum ()of is defined by = ∈ℂ: − − .If ∈() has finite ascent and descent then is called Drazin invertible. The Drazin spectrum of is defined by = ∈ ℂ: − . Define also the set () by ={∈()< ∞ +1 } and = ∈ℂ: − ∉ . From [1], ∈ℂ is pole of resolvent of if and only if 0 < max ( − , − )< ∞. Moreover, if this is true, then ( − = − . While, ∈ () is a left pole of if − ∈ () and is a left pole of of finite rank if is a left pole of and − < ∞. Let + − () be the class of all upper semi-B-Fredholm operators, + − ={ ∈ + : () ≤ 0}. The upper B-Weyl spectrum of is defined by + − = ∈ℂ: − ∉ + − . In the following table we will provide some symbol and notations that we will use: symbol and notations : spectrum of , : approximate point spectrum of , : Weyl spectrum of , : B-Weyl spectrum of , + −: upper semi-Weyl spectrum of , + −:upper semi-B-Weyl spectrum of , Paper ID: ART20179269 DOI: 10.21275/ART20179269 1502