JOINT SIMILARITY ORBITS WITH LOCAL CROSS-SECTIONS

L. A. Fialkow and D. A. Herrero have characterized those operators T, acting on a complex Hilbert space H such that the conjugation mapping s:G(H)→S(T) from the linear group of H onto the similarity orbit of T, S(T), has a continuous local cross section defined on some neighborhood of T in 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 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 G(H) is replaced by the unitary group. © 1990 Birkhäuser Verlag.