Integral Equations and Operator Theory Vol. 13 (1990) 0378-620X/90/010001-4851.50+0.20/0 (c) 1990 Birkh~user Verlag, Basel JOINT SIMILARITY ORBITS WITH LOCAL CROSS SECTIONS Esteban Andruchow, Lawrence A. Fialkow I), Domingo A. Herrero I) , Marta B. Pecuch-Herrero I) and Demetrio Stojanoff 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-I). 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 sup- port 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. i. INTRODUCTION Let H denote a complex, separable, infinite dimensional Hilbert space, and let [(H) be the algebra of all (bounded linea~ operators acting on H. If G(H) denotes the group of invertible elements in i(H), and S(T) = {WTW-I: W c G(H)} is the similarity orbit of T ~ L(H), then G(H) acts naturally on S(T) via conjugation; moreover, if A'(T) = {A c L(H) : AT = TA} is the commutant of T, then G(H) n A' (T) coincides with G[A' (T)], the group of invertible elements in A' (T), and we have the follow- ing commutative diagram (G(H) and G[A'(T)] are topological groups i) The research of these three authors was partially supported by Grants of the National Science Foundation.