On a new positive extension problem for block Toeplitz matrices (Q1375098)

Positive extension problems for block Toeplitz matrices are studied. The given entries form a stalactite type pattern. More precisely, let \(A= [S_{j-k}]^N_{j,k=1}\), where only part of the entries of the \(n\times n\) matrices \(S_j\) are given subject to the equality \(S_j= S_{-j}^*\) and having the property that if an entry is given in a certain column of some \(S_j\) \((j=0,\dots, N)\) then all entries up to and including the diagonal entry in the same column of \(A\) are given as well. The problem is then to determine the non-given entries of \(A\) (treated as independent complex variables) in such a way that the matrix \(A\) becomes positive definite. This problem, including a formula for all solutions (provided they exist), is solved by reducing it to a one-sided tangential interpolation problem for Carathéodory functions with symmetries. In turn, a fundamental matrix inequality is used to solve the interpolation problem. The solutions exist if and only if the information matrix associated with the problem is positive semidefinite. If it is moreover positive definite, all the solutions are parametrized by a linear fractional transformation the coefficients of which are given in a certain rational matrix function. The rational matrix function is explicitly given in a realized form, it has unitary (with respect to two signature matrices) values on the unit circle and expansive values on the unit disc.