SciELO - Scientific Electronic Library Online

vol.35 issue3Weak forms of continuity and opennessStability and boundedness in differential systems of third order with variable delay author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.35 no.3 Antofagasta Sept. 2016 


Non-linear maps preserving singular algebraic operators


Mourad Oudghiri

Uiversité Mohammed Premier - Oujda, Maroc


Khalid Souilah

Uiversité Mohammed Premier - Oujda, Maroc


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.

Subjclass [2000] : 47B49, 47L99, 47A55, 47B37.

Keywords : Non-linear preserver problems, Algebraic operators.


[1]    Z. F. BAI AND J. Hou , Linear maps and additive maps thatpreserve operators annihilated by a polynomial, J. Math. Anal. Appl. 271, pp. 139-154, (2002).         [ Links ]

[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).         [ Links ]

[3]    A. BouRHiM, J. MASHREGHI AND A. STEPANYAN, Nonlinear maps preserving the minimum and surjectivity moduli, Linear Algebra Appl. 463, pp. 171-189, (2014).         [ Links ]

[4]    J. K. HAn, H. Y. Lee And W. Y. Lee, Invertible completions of 2 X 2 upper triangular operator matrices, Proc. Amer. Math. Soc. 128, pp. 119-123, (2000).         [ Links ]

[5]    H. HAvLicek And P. Semn, From geometry to invertibility pre-servers, Studia Math. 174, pp. 99-109, (2006).         [ Links ]

[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).         [ Links ]

[7] L. K. HuA, A theorem on matrices over a field and its applications, Acta Math. Sin. (Engl. Ser.) 1, pp. 109-163, (1951).         [ Links ]

[8]    W.-L. HuANG AND P. SEMRL, Adjacencypreserving maps on Hermi-tian matrices, Canad. J. Math. 60, pp. 1050-1066, (2008).         [ Links ]

[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).         [ Links ]

[11]    V. MÜLLER, Spectral Theory ofLinear Operators and Spectral Systems in Banach Algebras. Second edition. Operator Theory: Advances and Applications, 139. Birkhauser Verlag, Basel, (2007).         [ Links ]

[12]    M. ÜMLADlC AND P. SEMRL, Additive mappings preserving operators ofrank one, Linear Algebra Appl. 182, pp. 239-256, (1993).         [ Links ]

[13]    C. PEARCY AND D. TopPiNG, Sums ofsmall numbers ofidempotents, Michigan Math. J. 14, pp. 453-465, (1967).         [ Links ]

[14]    T. PETEK AND P. SEMRL, Adjacency preserving maps on matrices and operators, Proc. Roy. Soc. Edinburgh 132A, 661-684, (2002).         [ Links ]

[15]    P. SEMRL, On Hua’sfundamental theorem ofthe geometry ofrectan-gular matrices, J. Algebra 248, pp. 366-380, (2002).

[16]    P. SEMRL, Hua’sfundamental theorem ofthe geometry ofmatrices, 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).         [ Links ]

[18]    A. E. TAYLoR AND D. C. LAY, Introduction to Functional Analysis, Wiley, New York-Chichester-Brisbane, (1980).         [ Links ]

[19] Z. WAN, Geometry ofMatrices, World Scientific Publishing Co., Sin-gapore, (1996).         [ Links ]

 Received : March 2016. Accepted : June 2016


Mourad Oudghiri
Département Math-Info,
Labo LAGA,
Faculté des Sciences d’Oujda,
Université Mohammed Premier - Oujda, 60000 Oujda,
e-mail :

Khalid Souilah
Departement Math-Info,
Labo LAGA,
Faculte des Sciences d’Oujda,
60000 Oujda,
e-mail :

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License