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Theorems of the alternative for cones and Lyapunov regularity of matrices. (English) Zbl 0902.15011
Summary: Standard facts about separating linear functionals will be used to determine how two cones $$C$$ and $$D$$ and their duals $$C^*$$ and $$D^*$$ may overlap. When $$T:V\rightarrow W$$ is linear and $$K\subset V$$ and $$D\subset W$$ are cones, these results will be applied to $$C=T(K)$$ and $$D$$, giving a unified treatment of several theorems of the alternate which explain when $$C$$ contains an interior point of $$D$$. The case when $$V=W$$ is the space $$H$$ of $$n\times n$$ Hermitian matrices, $$D$$ is the $$n\times n$$ positive semidefinite matrices, and $$T(X) = AX + X^*A$$ yields new and known results about the existence of block diagonal $$X$$’s satisfying the Lyapunov condition: $$T(X)$$ is an interior point of $$D$$. For the same $$V,W$$ and $$D,T(X)=X-B^*XB$$ will be studied for certain cones $$K$$ of entry-wise nonnegative $$X$$’s.

##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 15A24 Matrix equations and identities 15B57 Hermitian, skew-Hermitian, and related matrices
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