Abstract
Let \({\mathscr {L}}(X)\) be the algebra of all bounded linear operators on a complex Banach space X. Complete descriptions are given of the nonlinear maps of \({\mathscr {L}}(X)\) preserving local invertibility of \(T*S\) for different kinds of binary operations \(*\) on operators such as the sum \(T+S\), the difference \(T-S\), and the product TS. Extensions of these results to the case of different Banach spaces are also established. As application, mappings from \({\mathscr {L}}(X)\) onto itself that preserve the inner local spectral radius zero of such binary operations on operators are described.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aiena, P.: Fredholm and Local Spectral Theory with Applications to Multipliers. Kluwer, Boston (2004)
Aupetit, B.: A Primer on Spectral Theory. Springer, New York (1991)
Bendaoud, M.: Preservers of local spectra of matrix sums. Linear Algebra Appl. 438, 2500–2507 (2013)
Bendaoud, M.: Preservers of local spectrum of matrix Jordan triple products. Linear Algebra Appl. 471, 604–614 (2015)
Bendaoud, M., Douimi, M., Sarih, M.: Maps on matrices preserving local spectra. Linear Multilinear Algebra 61, 871–880 (2013)
Bendaoud, M., Jabbar, M., Sarih, M.: Additive local invertibility preservers. Publ. Math. Debr. 85, 467–480 (2014)
Bendaoud, M., Jabbar, M., Sarih, M.: Preservers of local spectra of operator products. Linear Multilinear Algebra 63, 806–819 (2015)
Bendaoud, M., Sarih, M.: Surjective linear maps preserving certain spectral radii. Rocky Mountain J. Math. 41, 727–735 (2011)
Bendaoud, M., Sarih, M.: Locally spectrally bounded linear maps. Math. Bohem. 136, 81–89 (2011)
Bendaoud, M., Sarih, M.: Additive local spectrum compressors. Linear Algebra Appl. 435, 1473–1478 (2011)
Bendaoud, M., Sarih, M.: Additive maps preserving the inner local spectral radius. Oper. Theory Adv. Appl. 236, 95–102 (2014)
Benhida, C., Zerouali, E.H.: Local spectral theory of linear operators \(RS\) and \(SR\). Integr. Equ. Oper. Theory 73, 1–8 (2006)
Bhatia, R., Šemrl, P., Sourour, A.R.: Maps on matrices that preserve the spectral radius distance. Studia Math. 134, 99–110 (1999)
Bourhim, A., Mashreghi, J., Stepanyan, A.: Nonlinear maps preserving the minimum and surjectivity moduli. Linear Algebra Appl. 463, 171–189 (2014)
Bračič, J., Müller, V.: Local spectrum and local spectral radius at a fixed vector. Studia Math. 194, 155–162 (2009)
Brešar, M., Šemrl, P.: On locally linearly dependent perators and derivations. Trans. Amer. Math. Soc. 351, 1257–1275 (1999)
Chan, J.T., Li, C.K., Sze, N.S.: Mappings preserving spectra of products of matrices. Proc. Amer. Math. Soc. 135, 977–986 (2007)
Costara, C.: Linear maps preserving operators of local spectral radius zero. Integr. Equ. Oper. Theory 73, 7–16 (2012)
Cui, J., Li, C.-K.: Maps preserving peripheral spectrum of Jordan products of operators. Oper. Matrices 6, 129–146 (2012)
Dolinar, G., Houb, J., Kuzma, B., Qi, X.: Spectrum nonincreasing maps on matrices. Linear Algebra Appl. 438, 3504–3510 (2013)
Gao, H.: \(*\)-Jordan-triple multiplicative surjective maps on B(H). J. Math. Anal. Appl. 401, 397–403 (2013)
González, M., Mbekhta, M.: Linear maps on \(M_{n}(\mathbb{C})\) preserving the local spectrum. Linear Algebra Appl. 427(2–3), 176–182 (2007)
Havlicek, H., Šemrl, P.: From geometry to invertibility preservers. Studia Math. 174, 99–109 (2006)
Hou, J.C., Li, C.-K., Wong, N.C.: Maps preserving the spectrum of generalized Jordan product of operators. Linear Algebra Appl. 432, 1049–1069 (2010)
Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. Oxford University Press, New York (2000)
Miller, T.L., Miller, V.G., Neumann, M.: Local spectral properties of weighted shifts. J. Operator Theory 51, 71–88 (2004)
Molnár, L.: Some characterizations of the automorphisms of \(B(H)\) and \(C(H)\). Proc. Amer. Math. Soc. 130, 111–120 (2001)
Müller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory: Advances and Applications, vol. 39. Springer, Berlin (2007)
Zhang, W., Hou, J.: Maps preserving peripheral spectrum of Jordan semi-triple products of operators. Linear Algebra Appl. 435, 1326–1335 (2011)
Acknowledgements
The first author thank the hospitality of the organizers of MatTriad’2015, Coimbra, Portugal, September 6–11, 2015, where the main results of this chapter were announced. This work was partially supported by a grant from MIU-SRA, Morocco.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Bendaoud, M., Jabbar, M., Sarih, M. (2017). Nonlinear Local Invertibility Preservers. In: Bebiano, N. (eds) Applied and Computational Matrix Analysis. MAT-TRIAD 2015. Springer Proceedings in Mathematics & Statistics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-49984-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-49984-0_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49982-6
Online ISBN: 978-3-319-49984-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)