Maps preserving local spectral subspaces of generalised product of operators
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Maps preserving local spectral subspaces of generalised product of operators H. Benbouziane2
· Y. Bouramdane1 · M. Ech-Chérif El Kettani1
Received: 15 January 2019 / Accepted: 6 September 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract Let B(X ) be the algebra of all bounded linear operators on a complex Banach space X . For A ∈ B(X ) and λ ∈ C, let X A ({λ}) be the local spectral subspace of A associated with {λ}. For an integer k ≥ 2, let (i 1 , . . . , i m ) be a finite sequence with terms chosen from {1, . . . , k}, such that {i 1 , . . . , i m } = {1 . . . k} and at least one of the terms in {i 1 , . . . , i m } appears exactly once. The generalized product of k operators A1 · · · Ak ∈ B(X ) is defined by A 1 ∗ A 2 ∗ · · · ∗ A k = Ai 1 Ai 2 · · · Ai m , and includes the usual product and the triple product. We characterise the form of surjective maps from B(X ) into itself satisfying X φ(A1 )∗···∗φ(Ak ) ({λ}) = X A1 ∗···∗Ak ({λ}), for all A1 , . . . , Ak ∈ B(X ) and λ ∈ C. Keywords Local spectral subspace · Nonlinear preservers problem · Generalised product Mathematics Subject Classification Primary 47A11. Secondary 47A15 · 47B48
1 Introduction Let X be an infinite dimensional complex Banach space and B(X ) be the algebra of all bounded linear operators on X with identity I . The local resolvent of A ∈ B(X ) at point
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H. Benbouziane [email protected] Y. Bouramdane [email protected] M. Ech-Chérif El Kettani [email protected]
1
Department of Mathematics, Faculty of Sciences DharMahraz Fes, University Sidi Mohammed Ben Abdellah, 1796 Atlas Fes, Morocco
2
Departement of Industrial Engineering, National School of Applied Sciences, University Sidi Mohammed Ben Abdellah Fes, Atlas Fes, Morocco
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x ∈ X , denoted by ρ A (x), is the union of all open U ⊂ C for which there exists an analytic function f : U → X such that (A − λ) f (λ) = x for every λ ∈ U . The local spectrum of A at x is defined by σ A (x) = C\ρ A (x). It is a (possibly empty) closed subset of σ (A), the usual spectrum of A. In fact σ A (x) = ∅ for all nonzero vectors x ∈ X precisely when A has the single-valued extension property (SVEP). Recall that an operator A is said to have the SVEP provided that for every open U ⊂ C the equation (A − λ) f (λ) = 0, (λ ∈ U ), has no nontrivial analytic solution f . For any subset ⊂ C, the local spectral subspace, X A (), is defined by X A () = {x ∈ X / σ A (x) ⊂ }. Clearly, if 1 ⊂ 2 ⊂ C, then X A (1 ) ⊂ X A (2 ). For more information about these notions one can see the books [2,14]. The problem of characterizing maps on matrices or operators that preserve certain functions, subsets or relations has attracted the attention of many mathematicians in the last decades; for example see [1,3–11,13,15,16] and their references. Motivated by the first result of the preserver problem in local spectral theory obtained by Bourhim and Ransford in [8], the authors characterised in [10] additive maps on B(X ) preserving the local spec
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