By Kenneth I. Joy
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3) Lm l ,00 , 00 (R), where (R) Taking into account (ii) and the fact that V(A) one sees that Ae is determined uniquely. 3) is a linear subspace of L7(R). The space X is called the state space, the space ~m the input/output space of the triple a. Suppose e is a realization triple of exponential type 00 and 00 S 00 1 < O. Then e is a realization triple of exponential 45 Symbols analytic in a strip type wI too. To see this, note that (1) and (i) are fulfilled with 00 replaced by wI. When the actual value of is 00 irrelevant, we simply call e a realization triple.
Factorization of Matrix Functions and Singular Integral Operators, OT3 Birkhiiuser, Basel, 1981. , All optimal Hankel-norm approximations of linear multivariable systems and their V°-error bounds, Int. l. Control 39 (1984), 11151193. C. , Convolution Equations and Projection Methods for their Solutions, Amer. Math. Soc. (Providence), 1974. C. , Einfuhrung in die Theorie der eindimensionalen singuliiren Integraloperatoren, Birkhiiuser, Basel, 1979. [H] Helton, J. , A spectral factorization approach to the distributed stable regulator p'roblem: the algebraic Riccati equation, SIAM J.
We W(A)=X:(A) [~p then may _~ J X+(A) factor IS c as ~ c=d * [ p ~ 1 _ q d and a signed spectral factorization, where X+(A)=dW+(A). It remains only to use this connection and the work of Section 2 on Wiener-Hopf factorization to get an analogous result for signed spectral factorization. 1. Suppose that the rational mxm matrix function W(A)=W*(A) has .! signed antispectral factorization Y (A) = Y (00)[1 - - open unit disk D. invertible, ~ m + C - (AI-A - )-I B - 1 * We also assume that Y joo) and Y _(00) = Y _(0) * W(oo) and W(O)=W(oo) * are invertible.