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Nonlinear fixed points preservers

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Abstract

Let \({\mathcal {B}}(X)\) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. For \(A\in {\mathcal {B}}(X)\), let F(A) be the set of all fixed points of A. For an integer \(k\ge 2\), let \((i_1,\dots ,i_m)\) be a finite sequence with terms chosen from \(\{1,\dots ,k\}\) and assume that at least one of the terms in \((i_1,\dots ,i_m)\) appears exactly once. The generalized product of k operators \(A_1,\dots ,A_k \in {\mathcal {B}}(X)\) is defined by

$$\begin{aligned} A_1*A_2*\cdots *A_k=A_{i_{1}}A_{i_{2}}\dots A_{i_{m}} \end{aligned}$$

and includes the usual product and the triple product. In this paper we characterize the form of surjective maps from \({\mathcal {B}}(X)\) into itself satisfying

$$\begin{aligned} \dim F(\phi (A_{1})*\dots *\phi (A_{k}))=\dim F(A_{1}*\cdots *A_{k}) \end{aligned}$$

for all \(A_1,\dots ,A_k \in {\mathcal {B}}(X)\).

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Bouramdane, Y., Ech-Cherif El Kettani, M. & Lahssaini, A. Nonlinear fixed points preservers. Rend. Circ. Mat. Palermo, II. Ser 70, 1269–1276 (2021). https://doi.org/10.1007/s12215-020-00558-7

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  • DOI: https://doi.org/10.1007/s12215-020-00558-7

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