Preface This pape r organize s an d expound s th e lecture s whic h th e autho r delivere d a t th e Regional Conferenc e hel d 7—1 1 Jun e 197 1 a t th e Universit y o f Ne w Hampshir e unde r the sponsorshi p o f th e Conferenc e Boar d o f th e Mathematica l Science s an d wit h th e support o f th e Nationa l Scienc e Foundation . The subjec t i s take n fro m th e theor y o f unitar y dilation s o f contractio n operator s and o f th e theor y o f functiona l model s o f suc h operators . Th e fundamenta l resul t i n thi s area, namel y th e existenc e an d uniquenes s o f th e minima l unitar y dilatio n fo r a contrac - tion, wa s prove d b y th e autho r i n 195 3 an d wa s th e startin g poin t o f variou s far-reachin g investigations o f th e structur e an d propertie s o f Hilber t spac e operators . Thi s tren d i s far fro m bein g exhausted . Mos t o f th e investigation s i n thi s directio n wer e mad e b y th e author i n clos e an d lastin g collaboratio n wit h Cipria n Foias ou r boo k o n Harmonic analysis of operators on Hilbert space, quote d i n th e seque l b y [H] , give s a detaile d account o f ou r result s a s wel l a s o f a goo d par t o f th e relevan t result s o f a numbe r o f other mathematicians . In m y lecture s I selecte d som e o f th e chapter s o f th e theor y whic h I expounde d i n some detail , othe r chapter s wer e onl y sketched , an d man y other s no t eve n mentioned . I hav e trie d t o b e quit e detaile d i n th e geometri c (o r purel y operator.theoretic ) aspect s of th e theory , thu s preparin g a natura l an d rapi d introduction , b y "Fourie r representa - tions,' ' o f th e characteristi c function s an d functiona l model s o f operators . I hav e trie d to mak e clea r th e connection s o f th e invarian t su b space proble m wit h th e factorizatio n problem o f th e characteristi c function , bu t I could no t ente r int o suc h application s a s the fairl y complet e spectra l theor y o f wea k contractions . I hav e include d th e element s of th e functiona l calculu s o f contraction s an d o f th e theor y o f operator s o f C Q , an d have indicate d th e mai n result s i n th e Jorda n mode l theor y o f thes e operators , bu t I have no t include d one-paramete r continuou s semigroups , o r th e wa y o f passin g fro m contractions t o (no t necessaril y bounded ) dissipativ e o r accretiv e operators . I hav e given som e attentio n t o th e *'liftin g theorem, " whic h is a mos t usefu l too l i n man y applications, bu t I hav e no t eve n mentione d application s o f th e theor y t o scatterin g North-Holland an d Akademia i Kiado , 1970 . Revise d an d enlarge d editio n o f th e firs t edi - tion, i n French , published b y Masson e t Akademia i Kiad o i n 1967 . A Russian editio n o f th e revised varian t wa s publishe d i n 197 0 by "Mir " i n Moscow . vii
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