Riccati Equation and Positivity of Operator Matrices

• Journal title : Kyungpook mathematical journal
• Volume 49, Issue 4,  2009, pp.595-603
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2009.49.4.595
Title & Authors
Riccati Equation and Positivity of Operator Matrices
Fujii, Jun Ichi; Fujii, Masatoshi; Nakamoto, Ritsuo;

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