## Positive solutions to $$X=A-BX^{-1}B^*$$.(English)Zbl 0702.15009

The authors study the positive (semidefinite) solutions to the matrix equation $$X=A-BX^{-1}B^*$$ under the assumption that $$A\geq 0$$. It is shown that positive solutions exist if and only if a certain block tridiagonal operator is positive, in which case the solution is given by the generalized Schur complement of that operator. The Schur complement is considered to act on a proper subspace of a finite or infinite dimensional Hilbert space with inner product.
Reviewer: M.de la Sen

### MSC:

 15A24 Matrix equations and identities 15B48 Positive matrices and their generalizations; cones of matrices
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### References:

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