Let A be a complex Banach algebra with unit. For an integer n≥0\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 0$$\end{document} and ϵ>0\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon >0$$\end{document}, the (n,ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$(n,\epsilon )$$\end{document}-pseudospectrum of a∈A\documentclass[12pt]{minimal}
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\begin{document}$$a\in A$$\end{document} is defined by Λn,ϵ(A,a):=λ∈C:(λ-a)is not invertible inAor‖(λ-a)-2n‖1/2n≥1ϵ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varLambda _{n,\epsilon } (A,a):=\left\{ \lambda \in \mathbb {C}: (\lambda -a) \text { is not invertible in } A \text { or } \Vert (\lambda -a)^{-2^{n}}\Vert ^{1/2^n} \ge \frac{1}{\epsilon }\right\} . \end{aligned}$$\end{document}Let p∈A\documentclass[12pt]{minimal}
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\begin{document}$$p\in A$$\end{document} be a nontrivial idempotent. Then pAp={pbp:b∈A}\documentclass[12pt]{minimal}
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\begin{document}$$pAp=\{pbp:b\in A\}$$\end{document} is a Banach subalgebra of A with unit p, known as a reduced Banach algebra. Suppose ap=pa\documentclass[12pt]{minimal}
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\begin{document}$$ap=pa$$\end{document}. We study the relationship of Λn,ϵ(A,a)\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda _{n,\epsilon }(A,a)$$\end{document} and Λn,ϵ(pAp,pa)\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda _{n,\epsilon }(pAp,pa)$$\end{document}. We extend this by considering first a finite family, and then an at most countable family of idempotents satisfying some conditions. We establish that under suitable assumptions, the (n,ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$(n,\epsilon )$$\end{document}-pseudospectrum of a can be decomposed into the union of the (n,ϵ)\documentclass[12pt]{minimal}
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\begin{document}$$(n,\epsilon )$$\end{document}-pseudospectra of some elements in reduced Banach algebras.
