On Additive Preservers of Certain Classes of Algebraic Operators
DOI:
https://doi.org/10.17398/Keywords:
Algebraic operators, Linear preserver problemsAbstract
In this article we provide a complete description of all additive surjective unital maps in the algebra of all bounded linear operators acting on an in nite-dimensional Hilbert space, preserving in both directions the set of non-invertible algebraic operators or the set of invertible algebraic operators.
Downloads
References
Z.F. Bai, J.C. Hou, Additive maps preserving nilpotent operators or spectral radius, Acta Math. Sin. (Engl. Ser.) 21 (5) (2005), 1167-1182.
M. Bresar, P. Semrl, Mappings which preserve idempotents, local automorphisms, and local derivations, Canad. J. Math. 45 (3) (1993), 483-496.
M. Bresar, P. Semrl, On local automorphisms and mappings that preserve idempotents, Studia Math. 113 (2) (1995), 101-108.
M. Burgos, A. Kaidi, M. Mbekhta, M. Oudghiri, The descent spectrum and perturbations, J. Operator Theory 56 (2) (2006), 259-271.
L.B. Hadwin, Local multiplications on algebras spanned by idempotents, Linear and Multilinear Algebra 37 (4) (1994), 259-263.
J.K. Han, H.Y. Lee, W.Y. Lee, Invertible completions of 2 × 2 upper triangular operator matrices, Proc. Amer. Math. Soc. 128 (1) (2000), 119-123.
J. Hou, J. Cui, Additive maps on standard operator algebras preserving invertibilities or zero divisors, Linear Algebra Appl. 359 (2003), 219-233.
J. Hou, S. Hou, Linear maps on operator algebras that preserve elements annihilated by a polynomial, Proc. Amer. Math. Soc. 130 (8) (2002), 2383-2395.
A.A. Jafarian, A.R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (2) (1986), 255-261.
M. Kuczma, “An introduction to the Theory of Functional Equations and Inequalities”, Pa´nstwowe Wydawnictwo Naukowe, Warszawa, 1985.
L. Molnar, “Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces”, Lecture Notes in Mathematics 1895, Springer-Verlag, Berlin, 2007.
V. Muller, “Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras”, Second edition, Operator Theory: Advances and Applications 139, Birkh¨auser Verlag, Basel, 2007.
M. Omladi c, P. Semrl, Additive mappings preserving operators of rank one, Linear Algebra Appl. 182 (1993), 239-256.
M. Omladi c, P. Semrl, Linear mappings that preserve potent operators, Proc. Amer. Math. Soc. 123 (4) (1995), 1069-1074.
C. Pearcy, D. Topping, Sums of small numbers of idempotents, Michigan Math. J. 14 (1967), 453-465.
P. Semrl, Linear maps that preserve the nilpotent operators, Acta Sci. Math. (Szeged) 61 (1-4) (1995), 523-534.
P. Semrl, Linear mappings that preserve operators annihilated by a polynomial, J. Operator Theory 36 (1) (1996), 45-58.
A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1980.
