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In this chapter, we consider the central issue of minimality of the state-space system representation, as well as equivalences of representations. The question introduces important new basic operators and spaces related to the state-space description. In our time-variant context, what we call the Hankel operator plays the central role, via a minimal composition (i.e., product), of a reachability operator and an observability operator. Corresponding results for LTI systems (a special case) follow readily from the LTV case. In a later starred section and for deeper insights, the theory is extended to infinitely indexed systems, but this entails some extra complications, which are not essential for the main, finite-dimensional treatment offered, and can be skipped by students only interested in finite-dimensional cases.
This note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.
We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product ${{H}_{f}}\,H_{g}^{*}$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products ${{H}_{g}}\,{{T}_{{\bar{f}}}}$ and ${{T}_{f}}\,H_{g}^{*}$, where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic.
In this note we show that two conjectures of George Weiss on admissible and weakly admissible observation operators fail to hold in general for the right shift semigroup in the case that the output space is infinite dimensional.