Kyungpook Mathematical Journal

  • 제49권4호
  • /
  • Pages.595-603
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  • 2009
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  • 1225-6951(pISSN)
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  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
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Riccati Equation and Positivity of Operator Matrices

  • 투고 : 2009.04.05
  • 심사 : 2009.05.04
  • 발행 : 2009.12.31
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초록

We show that for an algebraic Riccati equation $X^*B^{-1}X-T^*X-X^*T=C$, its solutions are given by X = W + BT for some solution W of $X^*B^{-1}X$ = $C+T^*BT$. To generalize this, we give an equivalent condition for $\(\array{B&W\\W*&A}\)\;{\geq}\;0$ for given positive operators B and A, by which it can be regarded as Riccati inequality $X^*B^{-1}X{\leq}A$. As an application, the harmonic mean B ! C is explicitly written even if B and C are noninvertible.

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참고문헌

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