Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Vào khoảng cách giữa quỹ đạo giống nhau. | J. OPERATOR THEORY 10 1983 65 75 Copyright by INCREST 1983 ON THE DISTANCE BETWEEN SIMILARITY ORBITS DOMINGO A. HERRERO 1. INTRODUCTION Consider the finite dimensional vector space C n 1 with its usual inner product and Hilbert space norm and let M C denote the Banach algebra of all n X n complex matrices under the norm IIz4 max y4 v xe C x 1 . If A BeM C then a straightforward computation shows that 1 t B ll n n for all A B 6 M C such that A is similar to A and B is similar to B where t 7 denotes the trace of R e M C . The main result of this note says that if A is a cyclic operator this is equivalent to saying that the minimal monic polynomial of A coincides with d4 Ấ determinant AỴ and B is not a multiple of the identity then the above lower bound cannot be improved. More precisely if for T e M C f T WTW-1 We M C is invertible denotes the similarity orbit of T then we have the following Theorem 1. If A Be M C n 2 A is cyclic and B is not a multiple of the identity then dist Gi y B f inf M - B ll Á e 9 A B e 9 B -- . n The case when A ẰI for some complex Ằ will be treated separately Theorem 8 below . An example will illustrate about the difficulties of the general case. 5-1105 66 DOMINGO A. HERRERO Several consequences can be derived from Theorem 1. Among others we have Proposition 2. If jVeM C is a normal operator such that 1 ea N the spectrum of N and a N . 6 c Re z 0 then dist M Ổ e M C Q 0 . 2pi If A is a nilpotent equivalently ff 1 - 0 ơ ổ 0 1 and rank-ổ 1 then the result of Theorem 1 follows from Proposition of 4 see also 2 Example . If N is positive hermitian then the result of Proposition 2 is Proposition of the same reference. For future purposes it will be convenient to introduce the notation T R to indicate that T are similar operators. The main result of this note was developed during a short visit to the University of California at San Diego. The author is deeply indebted to Professors L. c. Chadwick and J. w. Helton for several helpful
