×

Weyl type theorems and restrictions. (English) Zbl 1331.47005

The authors continue their studies of generalized Weyl type theorems.
Let \(T\) be a bounded, linear operator on an infinite-dimensional complex Banach space \(X\) and let \(T_n\) be the restriction of \(T\) to the range \(R(T^n)\), for \(n\geq 0\). Then \(T\) is said to be B-Weyl if, for some \(n\), the range \(R(T^n)\) is closed and \(T_n\) is a Fredholm operator having index \(0\). An operator \(T\) is said to satisfy the generalized Weyl theorem whenever any \(\lambda\) from the spectrum \(\sigma(T)\) of \(T\) is an isolated point of the spectrum of \(T\) with the dimension of the kernel of \(\lambda I-T\) being strictly positive if \(\lambda I-T\) is B-Weyl. The authors then prove that, if the zero number is not an isolated point of the spectrum of \(T\), then \(T\) satisfies the generalized Weyl theorem if and only if for some \(n\), \(R(T^n)\) is closed and \(T^n\) satisfies the generalized Weyl theorem. A similar result is claimed for operators satisfying the so-called generalized a-Weyl theorem.

MSC:

47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47A53 (Semi-) Fredholm operators; index theories
47A55 Perturbation theory of linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aiena P.: Fredholm and local spectral theory, with application to multipliers. Kluwer Academic Publishers, Dordrecht (2011)
[2] Aiena P.: Classes of operators satisfying a-Weyl’s theorem. Studia Math. 169, 105-122 (2005) · Zbl 1071.47001 · doi:10.4064/sm169-2-1
[3] Aiena P.: Quasi-Fredholm operators and localized SVEP. Acta Sci. Mat. (Szeged) 73, 251-263 (2007) · Zbl 1135.47300
[4] Aiena P., Aponte E., Balzan E.: Weyl type theorems for left and right polaroid operators. Int. Equ. Oper. Theory. 136, 2839-2848 (2010) · Zbl 1247.47002
[5] Aiena P., Berkani M.: Generalized Weyl’s theorem and quasi-affinity.. Studia Math. 198(2), 105-119 (2010) · Zbl 1192.47004 · doi:10.4064/sm198-2-1
[6] Aiena P., Biondi MT., Carpintero C.: On Drazin invertibility. Proc. Am. Math. Soc. 136, 2839-2848 (2008) · Zbl 1142.47004 · doi:10.1090/S0002-9939-08-09138-7
[7] Aiena P., Garcia O.: Generalized Browder’s theorem and SVEP. Mediterr. J. Math. 4(2), 215-228 (2007) · Zbl 1136.47002 · doi:10.1007/s00009-007-0113-2
[8] Aiena P., Peña P.: A variation on Weyl’s theorem. J. Math. Anal. Appl. 324, 566-579 (2006) · Zbl 1101.47001 · doi:10.1016/j.jmaa.2005.11.027
[9] Amouch M.: Weyl type theorems for operators satisfying the single-valued extension property. J. Math. Anal. Appl. 326, 1476-1484 (2007) · Zbl 1117.47007 · doi:10.1016/j.jmaa.2006.03.085
[10] Amouch M., Berkani M.: On the property (gw). Mediterr. J. Math 5(3), 371-378 (2008) · Zbl 1188.47011 · doi:10.1007/s00009-008-0156-z
[11] Berkani M.: On a class of quasi-Fredholm operators. Int. Equ. Oper. Theory 34(1), 244-249 (1999) · Zbl 0939.47010 · doi:10.1007/BF01236475
[12] Berkani M.: Restriction of an operator to the range of its powers. Studia Math. 140(2), 163-175 (2000) · Zbl 0978.47011
[13] Berkani M., Sarih M.: On semi B-Fredholm operators. Glasgow Math. J. 43, 457-465 (2001) · Zbl 0995.47008 · doi:10.1017/S0017089501030075
[14] Berkani M., Koliha J.: Weyl type theorems for bounded linear operators. Acta Sci. Math. (Szeged) 69, 359-376 (2003) · Zbl 1050.47014
[15] Cao X.: Topological uniform descent and Weyl type theorem. Linear Algebra Appl. 420, 175-182 (2007) · Zbl 1111.47014 · doi:10.1016/j.laa.2006.07.002
[16] Carpintero C., García O., Rosas E., Sanabria J.: B-Browder spectra an localized SVEP. Rendiconti del Circolo Matematico di Palermo 57, 241-255 (2008) · Zbl 1162.47002 · doi:10.1007/s12215-008-0017-4
[17] Carpintero C., Muñoz D., Rosas E., García O., Sanabria J.: Generalized Weyl’s theorems for polaroid operators. Carpathian J. Math 27(1), 24-33 (2011) · Zbl 1259.47007
[18] Carpintero, C., Muñoz, D., Rosas, E., García, O., Sanabria, J.: Weyl type theorems and restrictions for bounded linear operators. Extractha Mathematicae (to appear) (2013) · Zbl 1308.47001
[19] Coburn, L.A.: Weyl’s theorem for nonnormal operators. Res. Notes Math. 51, (1981) · Zbl 0173.42904
[20] Curto R., Han YM.: Generalized Browder’s and Weyl’s theorem for Banach space operators. J. Math. Anal. Appl. 418, 385-389 (2007)
[21] Duggal BP.: Polaroid operators satisfying Weyl’s theorem. Linear Algebra Appl. 414, 271-277 (2006) · Zbl 1096.47039 · doi:10.1016/j.laa.2005.10.002
[22] Finch JK.: The single valued extension property on a Banach space. Pac. J. Math. 58, 61-69 (1975) · Zbl 0315.47002 · doi:10.2140/pjm.1975.58.61
[23] Heuser H.: Functional analysis. Marcel Dekker, New York (1982) · Zbl 0465.47001
[24] Mbekhta M., Müller V.: On the axiomatic theory of the spectrum II. Studia Math. 119, 129-147 (1996) · Zbl 0857.47002
[25] Labrousse JP.: Les opérateurs quasi Fredholm: une généralization des opérateurs semi Fredholm. Rend. Circ. Mat. Palermo 29(2), 161-258 (1980) · Zbl 0474.47008 · doi:10.1007/BF02849344
[26] Rakočević V.: Operators obeying a-Weyl’s theorem. Rev. Roumaine Math. Pures Appl. 34(10), 915-919 (1989) · Zbl 0686.47005
[27] Zguitti H.: A note on generalized Weyl’s theorem. J. Math. Anal. Appl. 316(1), 373-381 (2006) · Zbl 1101.47002 · doi:10.1016/j.jmaa.2005.04.057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.