Linear Systems and Operators in Hilbert Space | Date: 15 April 2011, 11:08
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PREFACE
Great progress has been made in the last few years in the direction of establishing
a system theory in the context of infinite dimensional spaces. Although this direc-
tion of research has by no means been exhausted it seems that the available theory
has reached a level of maturity where a more systematic description would be in
order. This would be of help to other workers in the field.
My aim in this book is to reach different sets of readers-the mathematically
oriented researcher in system theory on the one hand and the pure mathematician
working in operator theory on the other. I think that the power, beauty, and ele-
gance of that part of operator theory touched upon in this book are such that the
interested system scientist who is ready to invest some, maybe even considerable,
time and effort in its study will be rewarded with a significantly increased set
of methods for tackling multivariable systems and a deeper understanding of the
finite dimensional theory. The operator theorist might find that system theory
provides a rich ground of interesting problems to the mathematician which might
be otherwise overlooked. Mathematics has always benefited from the transplant-
ing of ideas and motivations from other fields. It seems to me that system theory
besides being intellectually exciting is today one of the richest sources of ideas
for the mathematician as well as a major area of application of mathematical
knowledge.
I have tried to present the fairly diverse material of the book in a unified
way as far as possible, stressing the various analogies. In this sense the concept
of module is fundamental and the key results deal with module homomorphisms,
coprimeness, and spectral structure.
The book is divided into three uneven chapters. The first one is devoted to
algebraic system theory and serves also as a general introduction to the subject.
The various possible descriptions of linear time invariant systems are described.
Thus transfer functions, polynomial system matrices, state space equations, and
modules are all touched upon.
In the second chapter the necessary operator and function theoretic back-
ground is established. The material includes a short survey of Hilbert space
theory through the spectral theorem. We use here the classical approach based
on integral representations of certain classes of analytic functions. This approach
is taken to stress the close connection between representation theory and realiza-
tion theory. We continue with a sketch of multiplicity theory for normal operators.
Next we study contractions, their unitary dilations, and contractive semigroups.
The Cayley transform is extensively used to facilitate the translation of results from
the discrete to the continuous case. A special section is devoted to an outline of
the theory of the Hardy spaces in the disc and in a half plane. Shift and trans-
lation invariant subspaces are characterized. Next we describe the main results
concerning shift operators as models including the functional calculus, spectral
analysis, and the theory of Jordan models.
In the last chapter we study the mathematical theory of linear systems with
a state space that is a Hilbert space. Emphasis is on modeling with shift operators
and translation semigroups. The operator theoretic results developed in the second
chapter are brought to bear on questions of reachability, observability, spectral
minimality, and realization theory all in discrete and continuous time. Iso-
morphism results are derived and the limitations of the state space isomorphism
theorem are delineated. A special section is devoted to symmetric systems.
Many of the ideas and results as well as the general structure of the book
have been conceived during my two years' stay with Roger Brockett at Harvard.
Without his help, influence, and encouragement this book would not have been
written. It is a pleasure to acknowledge here my deep gratitude to him. I would
also like to recall the many stimulating exchanges over the past few years with my
colleagues J. S. Baras, P. Dewilde, A. Feintuch, J. W. Helton, R. Hermann, S. K.
Mitter, and J. C. Willems. For her excellent typing of the entire manuscript I
want to thank Mrs Y. Ahuvia. I gratefully acknowledge the support of the Israel
Academy of Sciences-the Israel Commission for Basic Research for its support
throughout the writing of this book.
Most of all I want to thank my wife Nilly for her love, moral support and
encouragement.
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