Contractive completions of Hankel partial contractions

A Hankel partial contraction is a Hankel matrix such that not all of its entries are determined, but in which every well-defined submatrix is a contraction. We address the problem of whether a Hankel partial contraction in which the upper left triangle is known can be completed to a contraction. It is known that the 2 × 2 and 3 × 3 cases can be solved, and that 4 × 4 Hankel partial contractions cannot always be completed. We introduce a technique that allows us to exhibit concrete examples of such 4 × 4 matrices, and to analyze in detail the dependence of the solution set on the given data. At the same time, we obtain necessary 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.