Fractional Hadamard powers of positive semidefinite matrices

We consider the class of all real positive semidefinite n×n matrices, and the subclass of all with non-negative entries. For a positive, non-integer number α and some , when will the fractional Hadamard power A α again belong to ? It is known that, for a specific α, this holds for all if and only if α>n−2. Now let be of the form A=T+V, where has rank 1 and has rank p 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 α. Special attention is given to the case that p=1.