Abstract
Research in the area of infinite dimensional system theory has a relatively short history. This is not surprising inasmuch as system theory as such has started to flourish only in the last two decades. Now the passage from finite dimensional linear theory to the infinite dimensional case obviously necessitated the replacement of algebraic machinery by an analytic one. Since structure theory is so intimately related with system theory then almost without exception functional analysis and especially the theory of operators and semigroups in Banach and Hilbert spaces, as well as the theory of distributions, became the source for methods and ideas to be used in a theory of infinite dimensional systems. Here there is an immediate difficulty. Structure theory for the general bounded operator in a Hilbert space, not to say Banach spaces, is an area in which not much is known. Thus research has been concentrated in special classes of operators, compact, selfadjoint, normal, spectral etc. In the case of normal operators the whole story is known given by the spectral theorem and this can be applied to system theoretic problems [6]. There is another class of operators which have been studied extensively in recent years and for which a structure theory has been developed which has so much resemblance to the finite dimensional case that it is natural to develop system theory in that context. We refer here to the class of C0 contractions studied in detail by Sz.-Nagy and Foias [39] consisting of those contraction operators T for which ∅(T) = 0 for some nonzero bounded analytic function in the unit disc.
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Fuhrmann, P.A. (1976). Algebraic Ideas in Infinite Dimensional System Theory. In: Marchesini, G., Mitter, S.K. (eds) Mathematical Systems Theory. Lecture Notes in Economics and Mathematical Systems, vol 131. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48895-5_16
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