CONTRACTIVE COMPLETIONS OF HANKEL PARTIAL CONTRACTIONS

A Hankel partial contraction is a Hankel matrix such that not all of i ts entries are determined, but in which every well-defined submatrix i s a contraction. We address the problem of whether a Hankel partial co ntraction in which the upper left triangle is known can be completed t o a contraction. It is known that the 2 x 2 and 3 x 3 cases can be sol ved, and that 4 x 4 Hankel partial contractions cannot always be compl eted. We introduce a technique that allows us to exhibit concrete exam ples of such 4 x 4 matrices, and to analyze in detail the dependence o f the solution set on the given data. At the same time, we obtain nece ssary and sufficient conditions on the given cross-diagonals in order for the matrix to be completed. We also study the problem of extending a contractive Hankel block of size n to one Of size n + 1. (C) 1996 A cademic Press, Inc.