On automatic surjectivity of some additive transformations

Mustapha Ech-chérif El Kettani, El Houcine El Bouchibti


Let X be an infinite dimensional Banach space and let Φ : B(X) → B(X) be a spectrum preserving additive transformation. We show that if the image of quasi-nilpotent operators contains all quasi-nilpotent operators, then Φ is an automophism or an antiautomorphism of B(X).

Palabras clave

Banach spaces; additive transformations; quasi-nilpotent operators; automorphisms; antiautomorphism; espacios de Banach; operadores cuasi nilpotentes; automorfismos; antiautomorfismos; transformaciones aditivas.

Texto completo:



B. Aupetit, Une généralisation du théoreme de Gleason-KahaneZelazko pour les algebres de Banach, Pacific. J. Math 85, pp. 11-17, (1979).

B. Aupetit and H. du Toit Mouton, Trace and determinant in Banach algebras, Studia. Math 121, pp. 115-136, (1996).

B. Aupetit, Sur les transformations qui conservent le spectre, Banach. Algebras 97 (De Gryter, Berlin, pp. 55-78, (1998).

B. Aupetit, A Primer On Spectral Theory (Springer New-York, (1991).

M. Bresar and P. Semrl, Linear maps preserving the spectral radius, J. Funct. Anal 142, pp. 360-168, (1996).

Fillmore, Sums of operators with square-zero, Acta. Sci. Math. Szeged. 28, pp. 285-288, (1967).

A. A. Jafarian and A.R. Sourour, Spectrum preserving linear maps, J. Funct. Anal 66, pp. 255-261, (1986).

M. Omladic and P. Semrl, Spectrum preserving additive maps, Linear. Algebras. Appl 153, pp. 67-72, (1991).

W. Rudin, Functional Analysis.

P. Semrl , Spectrally bounded linear maps on B(H), Quat. J. Math. Oxford (2) 49, pp. 87-92, (1998).

P. Semrl, Linear maps that preserve the nilpotent operators, Acta. Sci. Math (szeged) 61, pp. 523-534, (1995).

S. Sakai, C∗-Algebras and W ∗-Algebras (Springer,New-York, (1971).

A.R. Sourour , Invertibility preserving linear maps on L(X), Trans. Amer. Soc 348, pp. 13-30, (1996).

DOI: http://dx.doi.org/10.4067/S0716-09172004000200004

Enlaces refback

  • No hay ningún enlace refback.