The Drazin spectrum of tensor product of Banach algebra elements and elementary operators. (English) Zbl 1263.47007
Let \(A\) be a unital Banach algebra. The Drazin spectrum of \(a\in A\) is the set of all complex numbers such that \(a-\lambda\) is not Drazin invertible. If \(B\) is a Banach algebra, then let \(A\overline \otimes B\) denote the completion of \(A\otimes B\) with respect to a uniform crossnorm. For \(a\in A\) and \(b\in B\), the author characterizes the Drazin spectrum of \(a\otimes b\in A\overline \otimes B\). To this end, he characterizes the isolated points of the spectrum of \(a\otimes b\in A\overline \otimes B\). Further, for Banach spaces \(X, Y\) and operators \(S\in L(X)\), \(T\in L(Y)\), he characterizes the Drazin spectrum of the elementary operator \(\tau_{ST}\in L(L(Y,X))\) defined by \(\tau_{ST}(A)=SAT\).
Reviewer: Mohammad Sal Moslehian (Mashhad)
MSC:
| 47A10 | Spectrum, resolvent |
| 46H05 | General theory of topological algebras |
| 47B49 | Transformers, preservers (linear operators on spaces of linear operators) |
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