Abstract

Any complex n × n matrix A satisfies the inequality ‖ A ‖ 1 ≤ n 1 2 ‖ A ‖ d where ∥.∥ 1 is the trace norm and ∥.∥ d is the norm defined by ‖A‖d = max ∑ i=1 n |X i +AX i| 2 1 2 i( i)∑ β , where B is the set of orthonormal bases in the space of n × 1 matrices. The present work is devoted to the study of matrices A satisfying the identity: ‖A‖ 1 = n 1 2 ‖ A ‖ d This paper is a first step towards a characterization of matrices satisfying this identity. Actually, a workable characterization of matrices subject to this condition is obtained only for n = 2. For n = 3, a partial result on nilpotent matrices is presented. Like our previous 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 study is related to the work of R. Gabriel on classification of matrices with respect to unitary similarity (see R. Gabriel, J. Riene Angew, Math. 307/308 (1979) 31; R. Gabriel, Math. Z. 200 (1989) 591).

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