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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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