On some variational problems in the theory of unitarily invariant norms and Hadamard products

We deal with two recent conjectures of R.-C. Li [Linear Algebra Appl. 278 (1998) 317-326], involving unitarily invariant norms and Hadamard products. In the particular case of the Frobenius norm, the first conjecture is known to be true, whereas the second is still an open problem. In fact, in this paper we show that the Frobenius norm is essentially the only invariant norm which may comply with the two conjectures: more precisely, if a norm satisfies the claim of either conjecture, then it can be controlled from above and from below by the Frobenius norm, uniformly with respect to the dimension. On the other hand, both conjectures remain open in the relevant case of matrices with an upper bound to the rank. As a first partial result in this direction, we prove the first conjecture for matrices of rank 1 and for any unitarily invariant norm. (C) 2001 Elsevier Science Inc. All rights reserved.