NOTATION GUID E XI &H£ xn (AS) V T i T(f,T) ClDl(A) X(S),X+(s),X_(S) jrX,(* n) Lp-forward-referenee flag Backward periodi c flag M The operato r T ai T^ + T fl2 Tfl*2. Closed uni t bal l o f H™ xn . Chapter 6 Given r/,Y G L°°, a u a2 G J?00, / integer , /OOfindL' . { ( J ) : a BL°° , 0 inne r o f degree / s.t . 3GUG2 G H°° wit h a = ^G i + FG 2 , ^/c = aiG i + a2(?2} - {aG Y + /#°° ': ||a| | 1} . {i4 L(H , U):TA = AT}, H , H Hilber t spaces T G L(H), f G L(H) contractions . The se t o f all almos t contractiv e intertwining dilation s o f A = {Aoo G L(K, K): UAooU. PA = AP and J - A^AQO ha s a t mos t / negativ e eigenvalues}. Her e {/ , U denot e th e minima l isometric dilation s o f T, T, t o K, K, P: K -+ H, P: K - + H ar e orthogona l projections. A matrix-valued function , an d it s uppe r triangular an d strictl y lowe r triangula r parts. Chapter 7 Set o f all sequences {Mj}^_ 1 o f subspace s of L\ suc h tha t zM.\ c M n , an d fo r eac h y, Mj+i C Mj an d eac h Mj i s full rang e and simpl y invarian t (FRSI ) wit h respec t to multiplicatio n b y z. A flag Mf D Mf D o , suc h tha t Mi = ^ + e^fJ£ , t = 1,... , n, an d Ck = Fi D - - D Fn D 0 a, conventional flag of n subspace s o f C k . A collection {M^ : 1 j n} o f subspace s of L 2 N suc h tha t zMf D M* and , fo r eac h j \ M* + 1 D M* an d M * i s FRSI wit h respect t o backwar d shif t multiplicatio n b y z~\ [Mi,M2,.. .,Mn), th e se t of L|xn-functions suc h tha t th e jt h colum n i s an elemen t o f the subspac e Mj C L\. 48 53 57 57 57 65 65 67 73 74 74 75
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