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A nonlinear matrix equation connected to interpolation theory. (English) Zbl 1039.15007

The authors study the matrix equation $X=Q+{A}^{*}{\left(\stackrel{^}{X}-C\right)}^{-1}A$, where $Q$ is an $n×n$ positive definite matrix, $C$ is $mn×mn$ positive semidefinite, $A$ is $mn×n$ (arbitrary) and $\stackrel{^}{X}$ is $mn×mn$, block diagonal, with $m$ diagonal blocks equal to $X$. The authors impose the condition $C<\stackrel{^}{Q}$ (i.e. the matrix $\stackrel{^}{Q}-C$ is positive definite) and prove the existence and uniqueness of the solution in a certain class of positive definite matrices.

These solutions are important in a problem from optimal interpolation theory, see L. A. Sakhnovich [Interpolation theory and its Applications. (Mathematics and Its applications. (Dordrecht). 428 Dordrecht: Kluwer Academic Publishers.) (1997; Zbl 0894.41001), Chapter 7], where existence and uniqueness of the solutions is conjectured.

##### MSC:
 15A24 Matrix equations and identities 41A05 Interpolation (approximations and expansions) 47A57 Operator methods in interpolation, moment and extension problems