Let X be an infinite-dimensional complex Banach space and B(X)\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {B}}(X)$$\end{document} be the algebra of all bounded linear operators on X. Let n be an integer, such that n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document}, and let (i1,i2,⋯,im)\documentclass[12pt]{minimal}
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\begin{document}$$(i_1,i_2,\dots ,i_m )$$\end{document} be a sequence with terms belong to {1,2,⋯,n}\documentclass[12pt]{minimal}
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\begin{document}$$\{ 1,2,\dots ,n\}$$\end{document}, such that at least one of the terms in (i1,i2,⋯,im)\documentclass[12pt]{minimal}
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\begin{document}$$(i_1,i_2,\dots ,i_m )$$\end{document} arises exactly once. The generalized product of n operators T1,T2,⋯,Tn∈B(X)\documentclass[12pt]{minimal}
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\begin{document}$$T_1,T_2,\dots , T_n \in {\mathscr {B}}(X)$$\end{document} is given by T1∗T2∗⋯∗Tn=Ti1Ti2⋯Tim\documentclass[12pt]{minimal}
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\begin{document}$$T_1*T_2*\cdots *T_n = T_{i_1}T_{i_2}\cdots T_{i_m}$$\end{document}. For a fixed λ0∈C\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0\in {\mathbb {C}}$$\end{document}, we describe maps ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} on B(X)\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {B}}(X)$$\end{document} for which F2m-2(X)⊂ψ(B(X))\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}_{2m-2}(X)\subset \psi ({\mathscr {B}}(X))$$\end{document} and Xψ(T1)∗⋯∗ψ(Tn)({λ0})=XT1∗⋯∗Tn({λ0})for allT1,⋯,Tn∈B(X),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} X_{\psi (T_1)*\cdots * \psi (T_n)}(\{ \lambda _0\})=X_{T_1*\cdots *T_n}(\{ \lambda _0\}) \text { for all }T_1,\dots ,T_n \in {\mathscr {B}}(X), \end{aligned}$$\end{document}where F2m-2(X)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}_{2m-2}(X)$$\end{document} and XT({λ0})\documentclass[12pt]{minimal}
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\begin{document}$$X_T(\{ \lambda _0 \})$$\end{document} denote, respectively, the set of all operators of rank at most 2m-2\documentclass[12pt]{minimal}
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\begin{document}$$2m-2$$\end{document} and the local spectral subspace of T∈B(X)\documentclass[12pt]{minimal}
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\begin{document}$$T\in {\mathscr {B}}(X)$$\end{document} associated with {λ0}\documentclass[12pt]{minimal}
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\begin{document}$$\{\lambda _0\}$$\end{document}.
