Theorems of the alternative for cones and Lyapunov regularity of matrices
Czechoslovak Mathematical Journal, Tome 47 (1997) no. 3, pp. 487-499
Voir la notice de l'article provenant de la source Czech Digital Mathematics Library
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.
@article{CMJ_1997__47_3_a8,
author = {Cain, Bryan and Hershkowitz, Daniel and Schneider, Hans},
title = {Theorems of the alternative for cones and {Lyapunov} regularity of matrices},
journal = {Czechoslovak Mathematical Journal},
pages = {487--499},
publisher = {mathdoc},
volume = {47},
number = {3},
year = {1997},
mrnumber = {1461427},
zbl = {0902.15011},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_1997__47_3_a8/}
}
TY - JOUR AU - Cain, Bryan AU - Hershkowitz, Daniel AU - Schneider, Hans TI - Theorems of the alternative for cones and Lyapunov regularity of matrices JO - Czechoslovak Mathematical Journal PY - 1997 SP - 487 EP - 499 VL - 47 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/CMJ_1997__47_3_a8/ LA - en ID - CMJ_1997__47_3_a8 ER -
%0 Journal Article %A Cain, Bryan %A Hershkowitz, Daniel %A Schneider, Hans %T Theorems of the alternative for cones and Lyapunov regularity of matrices %J Czechoslovak Mathematical Journal %D 1997 %P 487-499 %V 47 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/CMJ_1997__47_3_a8/ %G en %F CMJ_1997__47_3_a8
Cain, Bryan; Hershkowitz, Daniel; Schneider, Hans. Theorems of the alternative for cones and Lyapunov regularity of matrices. Czechoslovak Mathematical Journal, Tome 47 (1997) no. 3, pp. 487-499. http://geodesic.mathdoc.fr/item/CMJ_1997__47_3_a8/