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5 - State Equivalence, State Reduction
from Part I - Lectures on Basics with Examples
Patrick Dewilde, Technische Universität München, Klaus Diepold, Technische Universität München, Alle-Jan Van der Veen, Technische Universiteit Delft, The Netherlands
Book:

Time-Variant and Quasi-separable Systems

Published online:

24 October 2024

Print publication:

31 October 2024, pp 73-86
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Summary

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.

Characterizations of Hankel operators in the essential commutant of quasicontinuous Toeplitz operators
Part of
Yi Yan
Journal:

Canadian Mathematical Bulletin / Volume 65 / Issue 1 / March 2022

Published online by Cambridge University Press:

09 February 2021, pp. 44-51

Print publication:

March 2022
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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.

Bounded Hankel Products on the Bergman Space of the Polydisk
Part of
- Special classes of linear operators
Yufeng Lu, Shuxia Shang
Journal:

Canadian Journal of Mathematics / Volume 61 / Issue 1 / 01 February 2009

Published online by Cambridge University Press:

20 November 2018, pp. 190-204

Print publication:

01 February 2009
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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.

ADMISSIBLE AND WEAKLY ADMISSIBLE OBSERVATION OPERATORS FOR THE RIGHT SHIFT SEMIGROUP
Birgit Jacob, Jonathan R. Partington, Sandra Pott
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 45 / Issue 2 / June 2002

Published online by Cambridge University Press:

17 June 2002, pp. 353-362
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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.
AMS 2000 Mathematics subject classification: Primary 47D06; 32A37; 93B28

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