Properties (T) and (Gt) for f (T) Type Operators

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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue I Ver. VI (Jan. - Feb. 2017), PP 21-24 www.iosrjournals.org

Properties (T) and (Gt) for f (T) Type Operators Adel Babbah1, Mohamed Zohry2, MohamedArrazaki3, Anass Elhaddadi4 1

(University AbdelmalekEssaadi, Faculty of sciences, Mathematics Department,B.P. 2121, Tetouan, Morocco) (University AbdelmalekEssaadi, Faculty of sciences, Mathematics Department,B.P. 2121, Tetouan, Morocco) 3 (University AbdelmalekEssaadi, Faculty of sciences, Mathematics Department,B.P. 2121, Tetouan, Morocco) 4 (DĂŠpartement of MathĂŠmatics&Informatic, ENSA Al-Hoceima, Morocco) 2

Abstract: in this paper we establish the relation with property (t) and (gt) introduced by M. H. M. RACHID in [1] and sprectrale mapping theorem. We will special establish several sufficient and necessary conditions for which f(T) verify the (t) and (gt) as f an analytic function on the T spectrum and see the validity of these results for the semi- Browder operators. Analogously we ask question about the conditions for which the spectral theory hold for generalized a-weyl’s theorem and generalized a-browder’s theorem [18]. Keywords: generalized a-weyl’s theorem. Generalized a-browder’stheorem . Semi-Browder operators. Property (t).Property (gt).Spectral theory.

I. Introduction and Preliminary Throughout this paper, X denote an infinite-dimentional complex space, L(X) the algebra of all bounded linear operators on X. For đ?‘‡ ∈ đ??ż(đ?‘‹), let đ?‘‡ ∗ , ker đ?‘‡ , đ?‘… đ?‘‡ , đ?œ? đ?‘‡ , đ?œ?đ?‘Ž đ?‘‡ đ?‘Žđ?‘›đ?‘‘ đ?œ?đ?‘ (đ?‘‡)denote the adjoint, the null space, the range, the spectrum, the approximate point spectrum and the surjectivity spectrum of T respectively. Let đ?›ź đ?‘‡ đ?‘Žđ?‘›đ?‘‘ đ?›˝(đ?‘‡)be the nullity and deficiency of T defined by đ?›ź đ?‘‡ = dim kerâ Ą (đ?‘‡) đ?‘Žđ?‘›đ?‘‘ đ?›˝ đ?‘‡ = codim đ?‘…(đ?‘‡). Let đ?‘†đ??š+ đ?‘‹ = {đ?‘‡ ∈ đ??ż đ?‘‹ : đ?›ź(đ?‘‡) < ∞ đ?‘Žđ?‘›đ?‘‘ đ?‘… đ?‘‡ đ?‘–đ?‘ đ?‘?đ?‘™đ?‘œđ?‘ đ?‘’đ?‘‘}and đ?‘†đ??šâˆ’ đ?‘‹ = {đ?‘‡ ∈ đ??ż đ?‘‹ : đ?›˝(đ?‘‡) < ∞}denote the semi-group of upper semi-Fredholm and lower semi-Fredholm operators on X respectively. If both đ?›ź đ?‘‡ đ?‘Žđ?‘›đ?‘‘ đ?›˝ đ?‘‡ đ?‘Žđ?‘&#x;đ?‘’ đ?‘“đ?‘–đ?‘›đ?‘–đ?‘Ąđ?‘’ , then T is called Fredholm operator. If T is semi-Fredholm then the index of T is definedby đ?‘–đ?‘›đ?‘‘ đ?‘‡ = đ?›ź đ?‘‡ − đ?›˝(đ?‘‡). A bounded linear operator T acting on a Banach space X is Weyl if it is Fredholm of index zero and Browder if T is Fredholm of finite ascent and descent. Let â„‚ denote the set of complex numbers. The Weyl spectrum and Browder spectrum of T are defined by đ?œ?đ?‘¤ đ?‘‡ = {đ?œ† ∈ â„‚ : đ?‘‡ − đ?œ† đ?‘–đ?‘ đ?‘›đ?‘œđ?‘Ą đ?‘Šđ?‘’đ?‘Śđ?‘™}and đ?œ?đ?‘? đ?‘‡ = {đ?œ† ∈ â„‚ : đ?‘‡ − đ?œ† đ?‘–đ?‘ đ?‘›đ?‘œđ?‘Ą đ??ľđ?‘&#x;đ?‘œđ?‘¤đ?‘‘đ?‘’đ?‘&#x;}respectively. For T ∈ L(X), SF+− X = {T ∈ SF+ X : ind(T) ≤ 0}. Then theupper Weyl spectrum of T is defined by đ?œ?đ?‘†đ??š+− đ?‘‡ = {đ?œ† ∈ â„‚ : đ?‘‡ − đ?œ† ∉ đ?‘†đ??š+− đ?‘ż }. Let Δ đ?‘‡ = đ?œ?(đ?‘‡)\đ?œ?đ?‘¤ đ?‘‡ and Δđ?‘Ž đ?‘‡ = đ?œ?đ?‘Ž (đ?‘‡)\đ?œ?đ?‘†đ??š+− đ?‘‡ . Following Cuburn [10], we say that Weyl’s theorem holds for đ?‘ť ∈ đ?‘ł(đ?‘ż)if Δ đ?‘‡ = đ??¸ 0 đ?‘‡ , đ?‘¤đ?‘• đ?‘’đ?‘&#x;đ?‘’ E 0 T = {Îť ∈ iso Ďƒ T : 0 < đ?›ź(đ?‘‡ − đ?œ†) < ∞}. Here and elsewhere in this paper, for đ??ž ⊂ â„‚ , đ?‘–đ?‘ đ?‘œđ??ž đ?‘–đ?‘ đ?‘Ąđ?‘• đ?‘’ đ?‘ đ?‘’đ?‘Ą đ?‘œđ?‘“ đ?‘–đ?‘ đ?‘œđ?‘™đ?‘’đ?‘Ąđ?‘’đ?‘‘ đ?‘?đ?‘œđ?‘–đ?‘›đ?‘Ąđ?‘ đ?‘œđ?‘“ đ??ž. According to Rakocevic [20], an operator đ?‘‡ ∈ đ??ż đ?‘‹ is said to satisfy a-Weyl’s theorem if đ??¸đ?‘Ž0 đ?‘‡ = đ?œ?đ?‘Ž (đ?‘‡)\ đ?œ?đ?‘†đ??š+− đ?‘‡ , where đ??¸đ?‘Ž0 T = {Îť ∈ iso Ďƒa T : 0 < đ?›ź(đ?‘‡ − đ?œ†) < ∞}. It is known from [20] that an operator satisfying a-Weyl’s theorem satisfies Weyl’s theorem, but the converse does not hold in general. For đ?‘‡ ∈ đ??ż đ?‘‹ and a non negative integer n define đ?‘‡[đ?‘›] to be the restrictionđ?‘‡ đ?‘Ąđ?‘œ đ?‘…(đ?‘‡ đ?‘› ) viewed as a map for đ?‘… đ?‘‡ đ?‘› đ?‘Ąđ?‘œ đ?‘… đ?‘‡ đ?‘› (in particular đ?‘‡[0] = đ?‘‡). If for some integer n the range space đ?‘…(đ?‘‡ đ?‘› )is closed and đ?‘‡[đ?‘›] is an upper (resp., lower) semi-Fredholm operator, then T is called upper (resp., lower) semi-BFredholm operator. In this case index of T is defined as the index of semi-Fredholmoperator đ?‘‡[đ?‘›] . Moreover, if đ?‘‡[đ?‘›] is a Fredholm operator then T is called a B-Fredholm operator. An operator T is said to be B-Weyl operator if it is a B-Fredholm operator of index zero. Let đ?œ?đ??ľđ?‘Š đ?‘‡ = {đ?œ† ∈ â„‚: đ?‘‡ − đ?œ† đ?‘–đ?‘ đ?‘›đ?‘œđ?‘Ą đ??ľ − đ?‘Šđ?‘’đ?‘Śđ?‘™}.Recall that the ascent, đ?‘Ž(đ?‘‡), of an operatorđ?‘‡ ∈ đ??ż(đ?‘‹)is the smallest non negative integer p such that ker đ?‘‡ đ?‘? = kerâ Ą(đ?‘‡ đ?‘?+1 )and if such integer does not exist we put đ?‘Ž đ?‘‡ = ∞. Analogously the descent, đ?‘‘(đ?‘‡), of an operator đ?‘‡ ∈ đ??ż(đ?‘‹)is the smallest non negative integer q such that R đ?‘‡ đ?‘ž = Râ Ą (đ?‘‡ đ?‘ž+1 )and if such integer does exist we put đ?‘‘ đ?‘‡ = ∞.According to Berkani [3], an operator đ?‘‡ ∈ đ??ż(đ?‘‹)is said to be Drazin invertible if it has finite ascent and descent . The Drazin spectrum of T is defined by đ?œ?đ??ˇ đ?‘‡ = {đ?œ† ∈ â„‚: đ?‘‡ − đ?œ† đ?‘–đ?‘ đ?‘›đ?‘œđ?‘Ą đ??ˇđ?‘&#x;đ?‘Žđ?‘§đ?‘–đ?‘› đ?‘–đ?‘›đ?‘Łđ?‘’đ?‘&#x;đ?‘Ąđ?‘–đ?‘?đ?‘™đ?‘’}. Define the set đ??żđ??ˇ đ?‘‹ = {đ?‘‡ ∈ đ??ż đ?‘‹ : đ?‘Ž(đ?‘‡) < ∞ đ?‘Žđ?‘›đ?‘‘ đ?‘… đ?‘‡ đ?‘Ž đ?‘‡ +1 đ?‘–đ?‘ đ?‘?đ?‘™đ?‘œđ?‘ đ?‘’đ?‘‘ }andđ?œ?đ??żđ??ˇ đ?‘‡ = {đ?œ† ∈ â„‚: đ?‘‡ − đ?œ† ∉ đ??żđ??ˇ(đ?‘‹)}. Following [4], an operator đ?‘‡ ∈ đ??ż(đ?‘‹)is said to be left Drazin invertibleif đ?‘‡ ∈ đ??żđ??ˇ(đ?‘‹). We say that đ?œ† ∈ đ?œ?đ?‘Ž đ?‘‡ is a left pole of T if đ?‘‡ − đ?œ† ∈ đ??żđ??ˇ(đ?‘‹), and that đ?œ† ∈ đ?œ?đ?‘Ž đ?‘‡ is a left pole of T of finite rank if đ?œ†is a left pole of T and đ?›ź(đ?‘‡ − đ?œ†) < ∞[4, đ??ˇđ?‘’đ?‘“đ?‘–đ?‘›đ?‘–đ?‘Ąđ?‘–đ?‘œđ?‘› 2.6]. Let đ?œ‹đ?‘Ž (đ?‘‡)denotes the set of all left poles of T and let đ?œ‹đ?‘Ž0 (đ?‘‡)denotes set of all left poles of finite rank. It follows from [4, theorem2.8] that if đ?‘‡ ∈ đ??ż(đ?‘‹)is left Drazin invertible, then T is upper semi-BDOI: 10.9790/5728-1301062124

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