Nonlinear fixed points preservers

• Published in: Rendiconti del Circolo matematico di Palermo 2021-12, Vol.70 (3), p.1269-1276
• Main Authors: Bouramdane, Y., Ech-Cherif El Kettani, M., Lahssaini, A.
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Banach spaces; Fixed points (mathematics); Geometry; Linear operators; Mathematics; Mathematics and Statistics; Operators (mathematics)
• ISSN: 0009-725X; 1973-4409
• DOI: 10.1007/s12215-020-00558-7

Let B ( X ) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X . For A ∈ B ( X ) , let F ( A ) be the set of all fixed points of A . For an integer k ≥ 2 , let ( i 1 , ⋯ , i m ) be a finite sequence with terms chosen from { 1 , ⋯ , k } and assume that at least one of the terms in ( i 1 , ⋯ , i m ) appears exactly once. The generalized product of k operators A 1 , ⋯ , A k ∈ B ( X ) is defined by A 1 ∗ A 2 ∗ ⋯ ∗ A k = A i 1 A i 2 ⋯ A i m and includes the usual product and the triple product. In this paper we characterize the form of surjective maps from B ( X ) into itself satisfying dim F ( ϕ ( A 1 ) ∗ ⋯ ∗ ϕ ( A k ) ) = dim F ( A 1 ∗ ⋯ ∗ A k ) for all A 1 , ⋯ , A k ∈ B ( X ) .