proceedings of the american mathematical society Volume 111, Number 1. January 1991 MATRICES WITH CIRCULAR SYMMETRY ON THEIR UNITARY ORBITS AND C-NUMERICAL RANGES CHI-KWONG LI AND NAM-KIU TSING (Communicated by Louis J. RatlifT. Jr.) Abstract. We give equivalent characterizations for those n x n complex ma- trices A whose unitary orbits %?(A) and C-numerical ranges WC{A) satisfy ei8&(A) = f/(A) or e'e WC(A) = WC(A) for some real 0 (or for all real 0 ). In particular, we show that they are the block-cyclic or block-shift operators. Some of these results are extended to infinite-dimensional Hubert spaces. 1. INTRODUCTION Let C" be equipped with the standard inner product (•, •) defined by (x, y) = v*x for all x , y eCn. For any A e C"x" , the unitary orbit of A is the set ^(A) = {UAU*:UeCnxn, U*U = I}. If we regard A as a linear operator on C" , then %(A) is the collection of all matrix representations of A with respect to different orthonormal bases of C" . In order to understand the properties of A , it is useful to study its unitary orbit. This is especially true when those "orthonormal bases-free" properties are concerned. For example, to conclude that A, ßeC"x" are not the same linear operator represented in different orthonormal bases, one has to show that W(A)¿,1f(B). Another tool for studying the properties of A e Cnx" is the (classical) nu- merical range of A , which is defined and denoted by W(A) = {x*Ax:xeC", x*x=l}. There is extensive literature on the numerical range, and many nice results have been obtained (e.g., see [3] and its references). In particular, many of these results show the interesting interplay between the geometrical properties of Received by the editors November 22, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 15A60. Key words and phrases. C-numerical range, unitary orbit, linear operator. The research of the first author was supported in part by the National Science Foundation under Grant DMS 89-00922. The research of the second author was supported in part by the National Science Foundation un- der Grant DMC 84-51515 and by the National Science Foundation's Engineering Research Centers Program: NSF CDR 88-03012. ©1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page 19 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use