Abstract
L. A. Fialkow and D. A. Herrero have characterized those operators T, acting on a complex Hilbert spaceH such that the conjugation mapping s:G(H)→S(T) from the linear group ofH onto the similarity orbit of T,S(T), has a continuous local cross section defined on some neighborhood of T inS(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 andS(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 andG(H) is replaced by the unitary group.
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The research of these three authors was partially supported by Grants of the National Science Foundation.
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Andruchow, E., Fialkow, L.A., Herrero, D.A. et al. Joint similarity orbits with local cross sections. Integr equ oper theory 13, 1–48 (1990). https://doi.org/10.1007/BF01195291
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DOI: https://doi.org/10.1007/BF01195291