ACTA issues

Minimal representations of a contractive operator as a product of two bounded operators

D. Z. Arov, M. A. Kaashoek, D. R. Pik

Acta Sci. Math. (Szeged) 71:1-2(2005), 313-336

Abstract. A particular case of the Kalman--Yakubovich--Popov inequality is studied, namely when the state operator and the feed through operator of the underlying linear time-invariant system are both zero. In this particular case, the system theory problems reduce to problems about representations of a bounded linear Hilbert space operator $K$ as a product $K=CB$, where $B\colon{\cal U}\rightarrow{\cal X}$ and $C\colon{\cal X}\rightarrow{\cal Y}$ are bounded linear Hilbert space operators too. These problems, which are inspired by [2], are of independent interest. We present an example of two pseudo-similar multiplicative representations of $K$, $K=C_1B_1$ and $K=C_2B_2$, such that the first is minimal, i.e, $\mathop{\rm Ker} B_1^*$ and $\mathop{\rm Ker} C_1$ are both equal to $\{0\} $, but the other is not minimal. Given a minimal multiplicative representation $K=CB$ of a contractive operator $K$, we describe the set of all (possibly unbounded) positive selfadjoint operators $H$ on $X$ such that $B_H=H^{1/2}B$ and $C_H =CH^{-1/2}$ define contractive operators which give a minimal multiplicative representation $K=C_HB_H$. The existence of minimal and maximal elements in this partially ordered set of positive operators $H$ is proved. Moreover we show that if $H$ is one of these two extremal elements, then the range of $B$ is a core for $H^{1/2}$. Other properties of this set are derived too.

AMS Subject Classification (1991): 47A45, 93B28, 47A48

Keyword(s): contractions, linear systems, factorization

Received June 8, 2004, and in revised form January 12, 2005. (Registered under 5871/2009.)