Fractional Hadamard powers of positive semidefinite matrices

We consider the class Y-n of all real positive semidefinite n x n matrices, and the subclass Y-n(+) of all A is an element of Y-n with non-negative entries. For a positive, non-integer number a and some A is an element of Y-n(+), when will the fractional Hadamard power A(lozengealpha) again belong to Y-n(+)? It is known that, for a specific a, this holds for all A is an element of Y-n(+) if and only if alpha > n - 2. Now let A is an element of Y-n(+) be of the form A = T + V, where T is an element of Y-n(+) has rank 1 and V is an element of Y-n has rank p greater than or equal to 1. If the Hadamard quotient of T and V is Hadamard independent ('in general position') and V has 'sufficently small' entries, then a complete answer is given, -depending on n, p, and alpha. Special attention is given to the case that p = 1. (C) 2003 Elsevier Inc. All rights reserved.