Abstract
In this note we obtain some characterizations of Banach spaces which are not isomorphic to any of their proper subspaces. In particular, we show that a Banach space X is not isomorphic to any of its proper subspaces if and only the equality σRD(LT) = σLD(T) holds for every bounded linear operator T on X if and only if int(σ(T)) ⊆ σLD(T) holds for every bounded linear operator T on X, where LT denotes the left multiplication operator by T.
References
[1] AIENA, P.: Fredholm and Local Spectral Theory, with Application to Multipliers, Kluwer Academic Publishers, Dordrecht, 2004.Search in Google Scholar
[2] AIENA, P.: Quasi-Fredholm operators and localized SVEP, Acta Sci. Math. (Szeged) 73 (2007), 251-263.Search in Google Scholar
[3] AIENA, P.-BIONDI, M. T.-CARPINTERO, C.: On Drazin invertibility, Proc. Amer. Math. Soc. 136 (2008), 2839-2848.10.1090/S0002-9939-08-09138-7Search in Google Scholar
[4] AIENA, P.-MILLER, T. L.: On generalized a-Browder’s theorem, Studia Math. 180 (2007), 285-300.10.4064/sm180-3-7Search in Google Scholar
[5] AIENA, P.-SANABRIA, J. E.: On left and right poles of the resolvent, Acta Sci. Math. (Szeged) 74 (2008), 667-685.Search in Google Scholar
[6] BEL HADJ FREDJ, O.-BURGOS,M.-OUDGHIRI,M.: Ascent spectrum and essential ascent spectrum, Studia Math. 187 (2008), 59-73.10.4064/sm187-1-3Search in Google Scholar
[7] BERKANI,M.: Restrition of operator to the range of its power, Studia Math. 140 (2000), 163-175.10.4064/sm-140-2-163-175Search in Google Scholar
[8] BERKANI, M.-KOLIHA, J. J.: Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), 359-376.Search in Google Scholar
[9] BERKANI,M.-SARIH, M.: An Atkinson-type theorem for B-Fredholm operators, Studia Math. 148 (2001), 251-257.10.4064/sm148-3-4Search in Google Scholar
[10] BOASSO, E.: Drazin spectra of Banach space operators and Banach algebra elements, J. Math. Anal. Appl. 359 (2009), 48-55.10.1016/j.jmaa.2009.05.036Search in Google Scholar
[11] BURGOS, M. ET AL: The descent spectrum and perturbations, J. Operator Theory 56 (2006), 259-271.Search in Google Scholar
[12] GOWERS, W. T.: A solution to Banach’s hyperplane problem, Bull. Lond. Math. Soc. 26 (1994), 523-530. 13] GOWERS, W. T.-MAUREY, B.: The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874.10.1090/S0894-0347-1993-1201238-0Search in Google Scholar
[14] GRABINER, S.: Uniform ascent and descent of bounded operators, J. Math. Soc. Japan 34 (1982), 317-337.10.2969/jmsj/03420317Search in Google Scholar
[15] HAÏLY, A.-KAIDI, A.-RODRÍGUEZS PALACIOS, A.: Algebra descent spectrum of operators, Israel J. Math. 177 (2010), 349-368.10.1007/s11856-010-0050-9Search in Google Scholar
[16] LAUSTSEN, N. J.: On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces, Glasg. Math. J. 45 (2003), 11-19.10.1017/S0017089502008947Search in Google Scholar
[17] MBEKHTA, M.-MÜLLER, V.: On the axiomatic theory of spectrum II, Studia Math. 119 (1996), 129-147.10.4064/sm-119-2-129-147Search in Google Scholar
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