Schur products and matrix completions (Q1119991)

A partially defined matrix is a specification of a set J in \(n\times n\) of matrix entries, a completion is a specification of the remaining entries. It is partially positive if and only if it is symmetric and every specified principle submatrix is positive semidefinite. The authors give a necessary and sufficient condition that a partially positive matrix have a positive semidefinite completion, that a certain mapping sending \((a_{ij})\) to \((a_{ij}t_{ij})\) sends positive semidefinite matrices to positive semidefinite matrices. They give applications and generalizations to matrices of operators.