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A nonlinear matrix equation connected to interpolation theory. (English) Zbl 1039.15007
The authors study the matrix equation $X=Q+A^*(\widehat{X}-C)^{-1}A$, where $Q$ is an $n\times n$ positive definite matrix, $C$ is $mn\times mn$ positive semidefinite, $A$ is $mn\times n$ (arbitrary) and $\widehat{X}$ is $mn\times mn$, block diagonal, with $m$ diagonal blocks equal to $X$. The authors impose the condition $C<\widehat{Q}$ (i.e. the matrix $\widehat{Q}-C$ 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 {\it 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:
15A24Matrix equations and identities
41A05Interpolation (approximations and expansions)
47A57Operator methods in interpolation, moment and extension problems
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References:
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