A review of the Nagy-Foias type functional model and its applications

Dmitry Yakubovich (Madrid)

Wednesday 6th May, 2009 15:00-16:00 325


In this talk, we give a review of the construction of a linearly similar model, sug- gested a few years ago by the speaker, and of a related model by Alexei Tikhonov. These constructions generalize the Nagy{Foia»s one, but have much more °exibility. The model is constructed not only in a disc or a semiplane, as in the original Nagy{ Foia»s setting, but rather in a wide class of domains in the complex plane. Even if the domain is ¯xed, the model is far from unique. Nor is unique the (generalized) characteristic function, which determines the model. The main ingredient to choose are two auxiliary operators, which are the analogues of the Nagy-Foias defect oper- ators. We will explain the construction and its relation with certain notions from the linear control theory in the Hilbert space setting, such as exact controllability and exact observability. (No previous knowledge of these things is assumed.) Several examples of unbounded operators, to which this construction applies, will be given. In particular, we will explain why any generator of C0 group of operators on a Hilbert space admits a Nagy-Foia»s type model in a vertical strip. Another example concerns generators of analytic semigroups on a Hilbert space. This part is a recent joint work with Jose Gale and Pedro Miana. It is related with the work by Allen McIntosh and others on H-infinity calculus. We prove that a sectorial operator admits an H1-functional calculus if and only if it has a functional model of Nagy-Foia»s type. Concrete formulas for generalized characteristic functions of such operators are given. More generally, this approach applies to any sectorial operator by passing to a slightly different norm. We show that there is only a logarithmic gap between these two norms. We will also discuss an application of our model to the pole placement in the infinite dimensional setting and differences between the speaker's and Tikhonov's settings.

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