Geodesic
MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS
Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 709-718

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DOI

Let X and Y be infinite-dimensional complex Banach spaces, and ${\mathcal B}$(X) (resp. ${\mathcal B}$(Y)) be the algebra of all bounded linear operators on X (resp. on Y). For an operator T ∈ ${\mathcal B}$(X) and a vector x ∈ X, let σT(x) denote the local spectrum of T at x. For two nonzero vectors x0 ∈X and y0 ∈ Y, we show that a map φ from ${\mathcal B}$(X) onto ${\mathcal B}$(Y) satisfies$\sigma_{\varphi(T)\varphi(S)}(y_0)~=~\sigma_{TS}(x_0),~(T,~S\in{\mathcal B}(X)),$if and only if there exists a bijective bounded linear mapping A from X into Y such that Ax0 = y0 and either φ(T) = ATA−1 or φ(T) = -ATA−1 for all T ∈ ${\mathcal B}$(X). 
BOURHIM, ABDELLATIF; MASHREGHI, JAVAD. MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS. Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 709-718. doi: 10.1017/S0017089514000585
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     author = {BOURHIM, ABDELLATIF and MASHREGHI, JAVAD},
     title = {MAPS {PRESERVING} {THE} {LOCAL} {SPECTRUM} {OF} {PRODUCT} {OF} {OPERATORS}},
     journal = {Glasgow mathematical journal},
     pages = {709--718},
     year = {2015},
     volume = {57},
     number = {3},
     doi = {10.1017/S0017089514000585},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000585/}
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