Non-linear maps preserving singular algebraic operators
DOI:
https://doi.org/10.4067/S0716-09172016000300007Keywords:
Non-linear preserver problems, algebraic operators, problemas de preservación no-lineal, operadores algebraicos.Abstract
Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space H. We prove that if Φ is a surjective map on B(H) such that Φ(I) = I + Φ(0) and for every pair T, S ∈ B(H), the operator T — S is singular algebraic if and only if Φ(T) — Φ(S) is singular algebraic, then Φ is either of the form Φ(T) = ATA-1 + Φ(0) or the form Φ(T) = AT*A-1 + Φ(0) where A : H → H is an invertible bounded linear, or conjugate linear, operator.References
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[2] Z. F. Bai and J. Hou, Additive maps preserving nilpotent operators or spectral radius, Acta Math. Sin. (Engl. Ser.) 21, pp. 1167-1182, (2005).
[3] A. Bourhim, J. Mashreghi and A. Stepanyan, Nonlinear maps preserving the minimum and surjectivity moduli, Linear Algebra Appl. 463, pp. 171-189, (2014).
[4] J. K. Han, H. Y. Lee and W. Y. Lee, Invertible completions of 2×2 upper triangular operator matrices, Proc. Amer. Math. Soc. 128, pp. 119-123, (2000).
[5] H. Havlicek and P. Semrl, From geometry to invertibility preservers, Studia Math. 174, pp. 99-109, (2006).
[6] J. Hou and L. Huang, Characterizing isomorphisms in terms of completely preserving invertibility or spectrum, J. Math. Anal. Appl. 359, pp. 81-87, (2009).
[7] L. K. Hua, A theorem on matrices over a field and its applications, Acta Math. Sin. (Engl. Ser.) 1, pp. 109-163, (1951).
[8] W.-L. Huang and P. Semrl, Adjacency preserving maps on Hermitian matrices, Canad. J. Math. 60, pp. 1050-1066, (2008).
[9] A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66, pp. 255-261, (1986).
[10] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Panstwowe Wydawnictwo Naukowe, Warszawa, (1985)
[11] V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Second edition. Operator Theory: Advances and Applications, 139. Birkhäuser Verlag, Basel, (2007).
[12] M. Omladic and P. Semrl, Additive mappings preserving operators of rank one, Linear Algebra Appl. 182, pp. 239-256, (1993).
[13] C. Pearcy and D. Topping, Sums of small numbers of idempotents, Michigan Math. J. 14, pp. 453-465, (1967).
[14] T. Petek and P. Semrl, Adjacency preserving maps on matrices and operators, Proc. Roy. Soc. Edinburgh 132A, 661-684, (2002).
[15] P. Semrl, On Hua’s fundamental theorem of the geometry of rectangular matrices, J. Algebra 248, pp. 366-380, (2002).
[16] P. Semrl, Hua’s fundamental theorem of the geometry of matrices, J. Algebra 272, pp. 801-837, (2004).
[17] K. Souilah, On additive preservers of certain classes of algebraic operators, Extracta Math. 30, pp. 207-220, (2015).
[18] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, Wiley, New York-Chichester-Brisbane, (1980).
[19] Z. Wan, Geometry of Matrices, World Scientific Publishing Co., Singapore, (1996).
Published
2017-03-23
How to Cite
[1]
M. Oudghiri and K. Souilah, “Non-linear maps preserving singular algebraic operators”, Proyecciones (Antofagasta, On line), vol. 35, no. 3, pp. 301-316, Mar. 2017.
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