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Riccati Equation and Positivity of Operator Matrices

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

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

  1. W. N. Anderson, Jr. and G. E. Trapp, Operator means and electrical networks, Proc. 1980 IEEE International Symposium on Circuits and systems, 1980, 523-527.
  2. T. Ando, Topics on Operator Inequalities, Lecture Note, Sapporo, 1978.
  3. R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17(1966), 413-415. https://doi.org/10.1090/S0002-9939-1966-0203464-1
  4. J. I. Fujii and M. Fujii, Some remarks on operator means, Math. Japon., 24(1979), 335-339.
  5. S. Izumino and M. Nakamura, Wigner's weakly positive operators, Sci. Math. Japon., 65(2007), 61-67.
  6. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246(1980), 205-224. https://doi.org/10.1007/BF01371042
  7. R. Nakamoto, On the operator equation THT = K, Math. Japon., 24(1973), 251-252.
  8. G. K. Pedersen and M. Takesaki, The operator equation THT = K, Proc. Amer. Math. Soc., 36(1972), 311-312.
  9. L. M. Schmitt, The Radon-Nikodym theorem for $L^p$-sapces of $W^*$-algebras, Publ. RIMS, Kyoto Univ., 22(1986), 1025-1034. https://doi.org/10.2977/prims/1195177060
  10. Ju. L. Smul'jan, An operator Hellinger integral, Mat. Sb., 91(1959), 381-430.
  11. Ju. L. Smul'jan, A Hellinger operator integral, Amer. Math. Soc. Translations, Ser. 2, 22(1962), 289-338.
  12. G. E. Trapp, Hermitian semidefinite matrix means and related matrix inequalities - An introduction, Linear Multilinear Alg., 16(1984), 113-123. https://doi.org/10.1080/03081088408817613
  13. E. P. Wigner, On weakly positive operators, Canadian J. Math., 15(1965), 313-317.