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The Laffey-West decomposition. (English) Zbl 0675.47008

Let \(X\) be a real or complex Banach space and let \(L(X)\) denote the set of all bounded linear operators from \(X\) into itself. The following theorem is proved: If \(A\in L(X)\) is an Atkinson operator with index \(i(A)=0\) \((>0;<0)\) then for each \(n\in\mathbb N\) there exists \(V_ n\in L(X)\), invertible (right invertible; left invertible) and there exists \(F_ n\in L(X)\), of finite rank, such that (a) \(A=V_ n+F_ n\) and (b) for each \(j\in \mathbb N\), \(j\leq n\Rightarrow [F_ n,[A^ j,F_ n]]=0\).
This theorem generalises a similar result for Fredholm operators of T. J. Laffey and T. T. West [Proc. R. Ir. Acad. 82A, 129–140 (1982; Zbl 0479.47009)].
Reviewer: R. Gross

MSC:

47A53 (Semi-) Fredholm operators; index theories

Citations:

Zbl 0479.47009
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