Nonlinear fixed points preservers
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Nonlinear fixed points preservers Y. Bouramdane1 · M. Ech‑Cherif El Kettani1 · A. Lahssaini1 Received: 29 February 2020 / Accepted: 31 August 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract 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 (i1 , … , im ) be a finite sequence with terms chosen from {1, … , k} and assume that at least one of the terms in (i1 , … , im ) appears exactly once. The generalized product of k operators A1 , … , Ak ∈ B(X) is defined by
A1 ∗ A2 ∗ ⋯ ∗ Ak = Ai1 Ai2 … Aim 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(A1 ) ∗ ⋯ ∗ 𝜙(Ak )) = dim F(A1 ∗ ⋯ ∗ Ak ) for all A1 , … , Ak ∈ B(X). Keywords Fixed points · Preserver map · Generalized product Mathematics Subject Classification Primary 47B49 · Secondary 47B48 · 47A10 · 46H05
1 Introduction Throughout this paper, X denotes an infinite-dimensional complex or real Banach spaces and B(X) the algebra of all bounded linear operators acting on X. The dual space of X will be denoted by X ∗ and the Banach space adjoint of an operator A ∈ B(X) will be denoted by A∗ . For a vector x ∈ X and a linear functional f in the dual space X ∗ , x ⊗ f stands for the operator of rank at most one defined by (x ⊗ f )y ∶= f (y)x, (y ∈ X). The rank one operator x ⊗ f is idempotent if and only if f (x) = 1 . Denote by F1 (X) and P1 (X) the set of all * Y. Bouramdane [email protected] M. Ech‑Cherif El Kettani [email protected] A. Lahssaini [email protected] 1
LaSMA Laboratory, Department of Mathematics, Faculty of Sciences Dhar El Mehraz Fez, University Sidi Mohamed Ben Abdellah, 1796 Atlas Fez, Morocco
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rank one operators and the set of rank one idempotent operators in B(X) respectively. A vector x ∈ X is said to be a fixed point of an operator A ∈ B(X) if Ax = x . The set of all fixed points of an operator A will be denoted by F(A). Note that if we consider the rank one operator x ⊗ f for x ∈ X and f ∈ X ∗ then
x ⊗ f ∈ P1 (X) ⟺ F(x ⊗ f ) = ⟨x⟩
(1.1)
x ⊗ f ∉ P1 (X) ⟺ F(x ⊗ f ) = {0},
(1.2)
where ⟨x⟩ is the linear subspace spanned by x. The study of maps on operator algebras preserving certain properties is a topic which attracts the attention of many authors see [1–11] and the references therein. In [8], Taghavi and Hosseinzadeh proved that if X is a complex Banach space with dim X ≥ 3 and if a surjective map 𝜙 ∶ B(X) → B(X) satisfies dim F(𝜙(A)𝜙(B)) = dim F(AB) for all A, B ∈ B(X) , then there exists an invertible operator S ∈ B(X) such that 𝜙(A) = ±SAS−1 for all A ∈ B(X) . In [9], the authors studied the surjective maps 𝜙 ∶ B(X) → B(X) which satisfy F(𝜙(A) + 𝜙(B)) = F(A + B) for all A, B ∈ B(X) , they conclude that 𝜙(A) = UA + R for all A ∈ B(X) where U = I − 2𝜙(0) and R = 𝜙(0) . They c
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