Abstract
In this paper we introduce and study the properties (t) and (gt), which extend properties (w) and (gw). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (t) and property (gt) hold. We also relate these properties with Weyl’s type theorems. We show that if T is a bounded linear operator acting on a Banach space \({\fancyscript{X}}\), then property (gt) holds for T if and only if property (gw) holds for T and σ(T) = σ a (T). Analogously, we show that property (t) holds for T if and only if property (ω) holds for T and σ(T) = σ a (T). We also study the properties (t) and (gt) for the operators satisfying the single valued extension property. Moreover, these properties are also studied in the framework of polaroid operators.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aiena P.: Fredholm and local spectral theory with applications to multipliers. Kluwer Acad. Publishers, Dordrecht (2004)
Aiena P., Carpintero C.: Weyl’s theorem, a-Weyl’s theorem and single-valued extension property. Extracta Math. 20, 25–41 (2005)
Aiena P.: Property (w) and perturbations II. J. Math. Anal. Appl. 342, 830–837 (2008)
Aiena P., Biondi M.T., Villafáne F.: Property (w) and perturbations III. J. Math. Anal. Appl. 353, 205–214 (2009)
Aiena P., Peña P.: Variations on Weyls theorem. J. Math. Anal. Appl. 324, 566–579 (2006)
Aiena P., Biondi M.T., Carpintero C.: On Drazin invertiblity. Proc. Amer. Math. Soc. 136, 2839–2848 (2008)
Aiena P., Aponte E., Bazan E.: Weyl type theorems for left and right polaroid operators. Integral Equations Operator Theory 66, 1–20 (2010)
Aiena P., Guillen J.R., Peña P.: Property (R) for bounded linear operator. Mediterr. J. Math. 8, 491–508 (2011)
Aiena P., Aponte E.: Polaroid type operators under perturbations. Studia Math. 214(2), 121–136 (2013)
Amouch M., Zguitti H.: On the equivalence of Browder’s and generalized Browder’s theorem. Glasg. Math. J. 48, 179–185 (2006)
Amouch M., Berkani M.: on the property (gw). Mediterr. J. Math. 5, 371–378 (2008)
Berkani M.: Index of B-Fredholm operators and gereralization of a Weyl Theorem. Proc. Amer. Math. Soc. 130, 1717–1723 (2001)
: B-Weyl spectrum and poles of the resolvent. J. Math. Anal. Appl. 272, 596–603 (2002)
Berkani M., Koliha J.: Weyl type theorems for bounded linear operators. Acta Sci. Math. 69, 359–376 (2003)
Berkani M., Castro N., Djordjević S.V.: Single valued extension property and generalized Weyl’s theorem. Math. Bohem. 131, 29–38 (2006)
Berkani M., Zariouh H.: Extended Weyl type theorems Math. Bohem. 34, 369–378 (2009)
Berkani M., Sarih M., Zariouh H.: Browder-type Theorems and SVEP Mediterr. J. Math. 8(3), 399–409 (2011)
Coburn L.A.: Weyl’s theorem for nonnormal operators Michigan Math. J. 13, 285–288 (1966)
Finch J.K.: The single valued extension property on a Banach space Pacific J. Math. 58, 61–69 (1975)
Laursen K.B.: Operators with finite ascent Pacific J. Math. 152, 323–336 (1992)
Laursen, K.B.; Neumann, M.M.: An introduction to local spectral theory Oxford. Clarendon (2000)
Oudghiri M.: Weyls and Browders theorem for operators satysfying the SVEP Studia Math. 163, 85–101 (2004)
Rakočević V.: On a class of operators Math. Vesnik 37, 423–426 (1985)
Rakočević V.: Operators obeying a-Weyl’s theorem Rev. Roumaine Math. Pures Appl. 10, 915–919 (1986)
Rashid M.H.M.: Property (w) and quasi-class (A,k) operators Revista De Le Unión Math. Argentina 52, 133–142 (2011)
Rashid M.H.M.: Weyl’s theorem for algebraically wF(p, r, q) operators with p, r > 0 and q ≥ 1 Ukrainian Math. J. 63(8), 1256–1267 (2011)
Rashid M.H.M., Noorani M.S.M.: Weyl’s type theorems for algebraically w-hyponormal operators. Arab. J. Sci. Eng. 35, 103–116 (2010)
Rashid M.H.M., Noorani M.S.M.: Weyl’s type theorems for algebraically (p, k)-quasihyponormal operators Commun. Korean Math. Soc. 27, 77–95 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rashid, M.H.M. Properties (t) and (gt) for Bounded Linear Operators. Mediterr. J. Math. 11, 729–744 (2014). https://doi.org/10.1007/s00009-013-0313-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-013-0313-x