Mathematica Bohemica

Mathematica Bohemica, Vol. 145, No. 2, pp. 113-126, 2020

A note on preservation of spectra for two given operators

Carlos Carpintero, Alexander Gutiérrez, Ennis Rosas, José Sanabria

Received March 24, 2018. Published online March 6, 2019.

Abstract: We study the relationships between the spectra derived from Fredholm theory corresponding to two given bounded linear operators acting on the same space. The main goal of this paper is to obtain sufficient conditions for which the spectra derived from Fredholm theory and other parts of the spectra corresponding to two given operators are preserved. As an application of our results, we give conditions for which the above mentioned spectra corresponding to two multiplication operators acting on the space of functions of bounded $p$-variation in Wiener's sense coincide. Additional illustrative results are given too.

Keywords: restriction of an operator; spectral property; semi-Fredholm spectra; multiplication operator

Classification MSC: 47A10, 47A11, 47A53, 47A55

DOI: 10.21136/MB.2019.0038-18

PDF available at: Institute of Mathematics CAS Digital Mathematics Library

References:
[1] P. Aiena: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Academic Publishers, Dordrecht (2004). DOI 10.1007/1-4020-2525-4 | MR 2070395 | Zbl 1077.47001
[2] P. Aiena: Quasi-Fredholm operators and localized SVEP. Acta Sci. Mat. 73 (2007), 251-263. MR 2339864 | Zbl 1135.47300
[3] P. Aiena, M. T. Biondi, C. Carpintero: On Drazin invertibility. Proc. Am. Math. Soc. 136 (2008), 2839-2848. DOI 10.1090/S0002-9939-08-09138-7 | MR 2399049 | Zbl 1142.47004
[4] F. R. Astudillo-Villaba, R. E. Castillo, J. C. Ramos-Fernández: Multiplication operators on the spaces of functions of bounded {$p$}-variation in Wiener's sense. Real Anal. Exch. 42 (2017), 329-344. DOI 10.14321/realanalexch.42.2.0329 | MR 3721805 | Zbl 06870333
[5] B. A. Barnes: The spectral and Fredholm theory of extensions of bounded linear operators. Proc. Am. Math. Soc. 105 (1989), 941-949. DOI 10.2307/2047057 | MR 0955454 | Zbl 0673.47003
[6] B. A. Barnes: Restrictions of bounded linear operators: Closed range. Proc. Am. Math. Soc. 135 (2007), 1735-1740. DOI 10.1090/S0002-9939-06-08624-2 | MR 2286083 | Zbl 1124.47002
[7] M. Berkani: On a class of quasi-Fredholm operators. Integral Equations Oper. Theory 34 (1999), 244-249. DOI 10.1007/BF01236475 | MR 1694711 | Zbl 0939.47010
[8] M. Berkani, M. Sarih: On semi B-Fredholm operators. Glasg. Math. J. 43 (2001), 457-465. DOI 10.1017/S0017089501030075 | MR 1878588 | Zbl 0995.47008
[9] C. Carpintero, D. Muñoz, E. Rosas, J. Sanabria, O. García: Weyl type theorems and restrictions. Mediterr. J. Math. 11 (2014), 1215-1228. DOI 10.1007/s00009-013-0369-7 | MR 3268818 | Zbl 1331.47005
[10] J. K. Finch: The single valued extension property on a Banach space. Pac. J. Math. 58 (1975), 61-69. DOI 10.2140/pjm.1975.58.61 | MR 0374985 | Zbl 0315.47002
[11] H. G. Heuser: Functional Analysis. A Wiley-Interscience Publication. John Wiley & Sons, Chichester (1982). MR 0640429 | Zbl 0465.47001

Affiliations: Carlos Carpintero, Universidad de Oriente, Av. Gran Mariscal-Edif. Rectorado, Cumaná, Venezuela and Universidad Autónoma del Caribe, Calle 90 # 46-112, Barranquilla, Atlántico, Colombia, e-mail: carpintero.carlos@gmail.com; Alexander Gutiérrez, Universidad Autónoma del Caribe, Calle 90 # 46-112, Barranquilla, Atlántico, Colombia, e-mail: algutierrez10@gmail.com; Ennis Rosas, Departamento de Matemáticas, Universidad de Oriente, Av. Gran Mariscal-Edif. Rectorado, Cumaná, Venezuela and Departamento de Ciencias Naturales y Exactas, Universidad de la Costa, Calle 58 # 55-66, Barranquilla, Colombia, e-mail: ennisrafael@gmail.com, erosas@cuc.edu.co; José Sanabria, Departamento de Matemáticas, Universidad de Oriente, Av. Gran Mariscal-Edif. Rectorado, Cumaná and Venezuela and Facultad de Ciencias Básicas, Universidad del Atlántico, Carrera 43 # 50-53, Barranquilla, Colombia, e-mail: jesanabri@gmail.com

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