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Joint similarity orbits with local cross sections. (English) Zbl 0721.47024

Summary: L. A. Fialkow and D. A. Herrero have characterized those operators T, acting on a complex Hilbert space \({\mathcal H}\) such that the conjugation mapping s: \({\mathcal G}({\mathcal H})\to {\mathcal S}(T)\) from the linear group of \({\mathcal H}\) onto the similarity orbit of T, \({\mathcal S}(T)\), has a continuous local cross section defined on some neighborhood of T in \({\mathcal S}(T)\) \((s(W)=WTW^{-1})\). In this article the authors raise a conjecture on the answer to the analogous problem for the case when T is replaced by an m-tuple of operators and \({\mathcal S}(T)\) is replaced by the joint similarity orbit of this m-tuple. They offer several partial results to support this conjecture. The results include a complete solution for the analogous problem for the case when the similarity orbit is replaced by the joint unitary orbit and \({\mathcal G}({\mathcal H})\) is replaced by the unitary group.

MSC:

47A65 Structure theory of linear operators
47D03 Groups and semigroups of linear operators
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