Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Go
Turkish Journal of Analysis and Number Theory. 2016, 4(5), 146-154
DOI: 10.12691/tjant-4-5-5
Open AccessArticle

Generalized α-Browder's and Generalized α-Weyl's Theorems for Banach Space

M. H. M. RASHID1,

1Department of Mathematics& Statistics, Faculty of Science P.O.Box(7), Mu'tah University, Al-Karak, Jordan

Pub. Date: November 03, 2016

Cite this paper:
M. H. M. RASHID. Generalized α-Browder's and Generalized α-Weyl's Theorems for Banach Space. Turkish Journal of Analysis and Number Theory. 2016; 4(5):146-154. doi: 10.12691/tjant-4-5-5

Abstract

In this paper, we give necessary and sufficient conditions for a Banach space T to satisfy the generalized α-Browder’s theorem. We also prove that the spectral mapping theorem holds for the left Drazin invertible and for analytic functions on a neighborhood of σ(T). As applications, we show that if T* is algebraically ωF(p,r,q) for each p,r>0 and q≥1, or if T* is algebraically quasi-class A, then the generalized α-Weyl’s theorem hold for f(T), where fHol(σ(T)), the space of functions analytic on an open neighborhoods of σ(T).

Keywords:
Weyl’s theorem α-Weyl’s theorem generalized α-Weyl’s theorem α-Browder’s theorem generalized α-Browder’s theorem reduction -isoloid reduced subspace

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  P. Aiena, Fredholm and local spectral theory with applications to multipliers, Kluwer, 2004.
 
[2]  I. J. An, Y. M. Han, Weyls theorem for algebraically quasi-class A Operators. Integral Equation Operator Theory 62(2008): 1-10.
 
[3]  S. K. Berberian, An extension of Weyl’s theorem to a class of not necessarily normal operators. Michigan Math. J. 16(1969): 273-279.
 
[4]  S.K. Berberian, The Weyl spectrum of an operator. Indiana Univ. Math. J. 20(1970): 529-544.
 
[5]  M. Berkani, On a class of Quasi-Fredholm operators. Integral Equation Operator Theory 34 (1999): 244-249.
 
[6]  M. Berkani, Index of B-Fredholm operators and generalization of a Weyl theorem. Proc. Amer. Math. Soc. 130(2002): 1717-1723.
 
[7]  M. Berkani, B-Weyl spectrum and poles of the resolvent. J. Math. Anal. Appl. 272(2002): 596-603.
 
[8]  M. Berkani, On the equivalence of Weyl theorem and generalized Weyl theorem. Acta Math. Sinica 272 (1)(2007): 103-110.
 
[9]  M. Berkani, A. Arroud, Generalized weyl’s theorem and hyponormal operators. J. Austral. Math. Soc. 76(2004): 1-12.
 
[10]  M. Berkani, J. Koliha, Weyl type theorems for bounded linear operators. Acta Sci. Math. (Szeged) 69 (1-2)(2003): 359-376.
 
[11]  M. Berkani, M. Sarih, An atkinson-type theorem for B-Fredholm operators, Studia Math. 148(2001) 251-257.
 
[12]  L. A. Coburn, Weyl’s theorem for nonnormal operators, Michigan Math. J. 13(1966) 285-288.
 
[13]  R. E. Curto, Y. M. Han, Weyl’s theorem, a-Weyl’s theorem, and local spectral theory, J. Londan Math. Soc.(2) 67(2003): 499-509.
 
[14]  B. P. Duggal, S. V. Djordjevic, Generalized Weyl’s theorem for a class of operators satisfying a norm condition II, Math. Proc. Royal Irish Acad. 104A(2006) 1-9.
 
[15]  B. P. Duggal, I. H. Jeon, I. H. Kim, On Weyl’s theorem for quasi-class A operators, J. Korean Math. Soc. 43(4)(2006) 899-909.
 
[16]  J. K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58(1975) 61-69.
 
[17]  T. Furuta, M. Ito and Yamazaki T., A subclass of paranormal operators including class of log-hyponormal and several related classes, Sci. math. 1 (1998) 389-403.
 
[18]  I. H. Jeon, I. Kim, On operators satisfying T*|T2|T*|T2|T*. Linear Algebra Appl. 418(2006): 854-862.
 
[19]  J. J. Koliha, Isolated spectral points, Proc. Amer. Math. Soc. 124(1996) 3417-3424.
 
[20]  M. Lahrouz, M. Zohry, Weyl type theorems and the approximate point spectrum, Irish Math. Soc. Bulletin 55(2005)41-51.
 
[21]  K. B. Laursen, Operators with finite ascent, Pacific J. Math. 152(1992) 323-336.
 
[22]  V. Rakocević, On a class of operators, Mat. Vesnik. 37 (1985) 423-426.
 
[23]  V. Rakocević, Operators obeying a-Weyl’s theorem, Rev. Roumaine Math. Pures Appl. 10(1986) 915-919.
 
[24]  V. Rakocevic, Operators Obeying a-Weyl’s theorem, Publ. Math. Debrecen 55(3-4)(1999) 283-298.
 
[25]  M.H.M. Rashid, M.S.M. Noorani and A.S. Saari, Weyl’s type theorems for quasi-Class A operators, J. Math. Stat. 4 (2)(2008) 70-74.
 
[26]  M.H.M. Rashid, M.S.M. Noorani and A.S. Saari, Generalized Weyl’s theorem for log-hyponormal, Malaysian J. Math. Soc. 2 (1)(2008): 73-82.
 
[27]  H. Weyl, Uber beschrankte quadratische Formen, deren Differenze vollsteting ist, Rend. Circ. Math. Palermo 27(1909): 373-392.
 
[28]  J. Yuan and Z. Gao, Spectrum of Class wF(p,r,q) Operators, J. Ineq. Appl. Article ID 27195, 10 pages, 2007.
 
[29]  H. Zguitti, A note on generlized Weyl’s theorem, J. Math. Anal. Appl. 316(2006) 373-381.