Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices

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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: VW is linear and KV and DW 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 × n Hermitian matrices, D is the n × 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) = XB* XB will be studied for certain cones K of entry-wise nonnegative X's.

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