Strictly contractive and positive completions for block matrices (Q919438)

This paper deals with the following two completion problems for block matrices: (1) Find all matrices of norm less than one having prespecified entries in a lower triangular region. (2) Find all positive definite matrices having prespecified entries on a number of central diagonals. The first problem was considered and solved probably first by \textit{W. B. Arveson} [J. Functional Anal. 20, 208-233 (1975; Zbl 0309.46053)] in an infinite dimensional operator theoretic setting while the second problem was studied by \textit{H. Dym} and \textit{I. Gohberg} [Linear Algebra Appl. 36, 1-24 (1981; Zbl 0461.15002)]. \textit{I. Gohberg} and the reviewer [ibid. 74, 87-150 (1986; Zbl 0596.15011)] developed a general approach using shift invariant subspaces for the study of such problems and found a linear fractional parametrization for the set of all solutions of such problems when solutions exist. The present author develops a more elementary approach based on the study of elementary linear fractional maps and a finite matrix analogue of the Schur algorithm to produce the linear fractional description for the set of such solutions. This approach has overlap with recent work of \textit{E. F. Deprettere} and \textit{P. Dewilde} [NATO ASI Ser., Ser. F34, 211-238 (1987; Zbl 0632.65020)] on time-varying analogues of the Schur algorithm. Also included is a discussion of the case where solutions are required to be Toeplitz matrices and of the maximum entropy principle in this context.