COMPARISON OF 2 NORMS OF MATRICES

Any complex n x n matrix A satisfies the inequality parallel to A para llel to(1) less than or equal to n(1/2) parallel to A parallel to(d), where parallel to.parallel to(1) is the trace norm and parallel to.par allel to(d) is the norm defined by parallel to A parallel to(d) = max {[(i=1)Sigma(n) \X-i*AX(i)\(2)](1/2); (X-i) is an element of B}, where B is the set of orthonormal bases in the space of n x 1 matrices. The present work is devoted to the study of matrices A satisfying the ide ntity: parallel to A parallel to(1) = n(1/2)parallel to A parallel to( d). This paper is a first step towards a characterization of matrices satisfying this identity. Actually, a workable characterization of mat rices subject to this condition is obtained only for n = 2. For n = 3, a partial result on nilpotent matrices is presented. Like our previou s study (J. Dazord, Linear Algebra Appl. 254 (1997) 67), this study is a continuation of the work of M. Marcus and M. Sandy (M. Marcus and M . Sandy, Linear and Multilinear Algebra 29 (1991) 283). Also this stud y is related to the work of R. Gabriel on classification of matrices w ith respect to unitary similarity (see R. Gabriel, J. Riene Angew, Mat h. 307/308 (1979) 31; R. Gabriel, Math. Z. 200 (1989) 591), (C) 1998 E lsevier Science Inc. All rights reserved.