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The semigroup of normal singular operators. (English) Zbl 0863.47020

Summary: Given a group \(G\) of invertible operators on a locally convex Hausdorff space \(X\) and a continuous projection \(p\) on \(X\) satisfying \([a,p]= ap-pa\) compact, we set \(S(G,p)=\{ap+bq+t:a,b\in G, t\in\widehat K(X)\}\), where \(q=1-p\) and \(\widehat K(X)\) is the set of all compact operators on \(X\). Then \(S(G,p)\) is seen to be a regular monoid.
In this paper, we consider some algebraic properties of this semigroup. The study of the index of operators in \(S(G,p)\) gives some useful results, such as: \(S(G,p)\) is simple implies \(k(X)\geq 1\) and \(S(G,p)\) is bisimple implies \(k(X)=1\), where \(k(X)\) is the characteristic number of \(X\), as given in Definition 5.1 of [E. Krishnan and K. S. S. Nambooripad, Forum. Math. 5, No. 4, 313-368 (1993; Zbl 0803.47017)].

MSC:

47D03 Groups and semigroups of linear operators
47A53 (Semi-) Fredholm operators; index theories
45P05 Integral operators

Citations:

Zbl 0803.47017
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