it is not incumbent upon you to complete the work, nor are you free to shirk it Rabbi Tarfon, from Sayings ofthe Fathers as he wrote was sometimes a t hundert ng locomotive, all cars attached except caboose, cracking along the clicking tracks into a country whose topography he suspected but did not know tili he got there. The Tenants, B. Malamud Preface These notes evolved from a set of lectures given under the auspices of the CBMS at the Case Institute of Technology in September 1984 . The original objective of these lectures was to present an introduction to the theory and applications of J inne r matrices. My own interest i n « / inner matrices was kindled by Patrick Dewilde, whom I first me t when he and I were both guests of Tom Kailath at Stanford in the summer of 1978 . It may encourage beginners in the subject to reveal that I spent much ofthat summer being mystified by the manuscript of [DVK] which, i n th e course of time , move d fro m th e realm o f the mysterious to the less exalted role of a special case of subsequent Joint collaborations with Patrick. But the beginning was hard. Actually I had begun to correspond with Patrick several months earlier due to a sequence of Chance encounters which would intrigue students of synchronicity (a term which honesty compels me to reveal I learned from Nero Wolfe, not Carl Gustav Jung), but that is a tale best saved for another day. The applications which I lectured on at the CBMS Conference stemmed largely from a number of collaborations with Patrick Dewilde, Israel Gohberg and two of my then students, Daniel Alpay and Andrei Iacob. I also discussed connections wit h th e wor k o f Kailat h an d Lev-Ar i o n lattic e filtering for nonstationary sequences ([L], [LK]). More than three years have passed since the lectures were delivered and I first started t o writ e the m u p o n a part-tim e basi s while continuin g o n and off to investigate related issues with Patrick, Israel, Daniel and Andrei. Earlier investigations with Daniel [AD1], [AD2], which predated the lectures, made extensive use of de Branges' theory of reproducing kernel Hubert spaces ßf{U) fo r J inne r matrices U and of the de Branges Rovnyak theory of re- producing kernel ßf(S) space s for p x q contractive analytic matrix valued functions S. A s my own understanding of these Spaces increased (partiall y because of the aforementioned collaboratio n and perhaps even more so be- cause of th e nee d t o reexplai n part s o f i t t o divers e audiences , includin g C.V.K. Prabhakara Rao, who, by forcing me to answer scores of questions in his Student days at Delft, pushe d me to a sharper comprehension), it grad- ually became clear that the y were also ideal tools for treatin g a large dass vii

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