BELASZ.-NAGY the ba r denotin g closure . Le t D^ , 5^. , b ^ hav e th e correspondin g meanin g fo r T . The obviou s equatio n T(l - T*T) = T - TT*T = ( /- TT*)T implie s Tp(D 2 ) = p(D|) T for an y polynomia l /?(A) , an d henc e fo r an y continuou s functio n o n 0 A 1 , i n particu - lar fo r th e functio n A . Thi s give s th e equation s TD=D.T, DT* = T*D^ r^cSL f i c S . (1.2.1) whence w e deduc e (1.2.2) Next w e prov e Theorem 1 . Every contraction operator T on a Hilbert space § has a unitary di~ lation U on a Hilbert space ft. One can require that this unitary dilation be minimal in the sense (1.2.3) V U n § = ft U is then determined by T uniquely {that is, up to an isometric isomorphism leaving the vectors of § invariant). Proof. For m th e Hilber t spac e ft oi vector s (1.2.4) * = /••-, h_ v h_ v g j , h v b v - ^ with component s h Q e § , h n e 2) , h_ e 3 * fo r « 1 , an d nor m 1/2 IWI = Z H* OO, and embe d § i n ft b y identifyin g h e S p wit h y •• , 0 , ]T] , 0 , y - Th e orthogona l projection P ^ i s the n give n b y P ^ = \* •, 0 , l&J , 0 , \ = & Q . Defin e o n ft th e operators V, U' b y (1.2.5) W : and ( ' ' ^ _ 3 ' ^ _ 2 ' D * * ^ + Th Q , - T ib_ 1 + Dh 0 , i r 4 2 , •- ^ (1.2.6) ( A = / . -, 4 _ r D ^ Q - T i j . T h Q + Dh x , & 2 , ^3, ) these definition s ar e correc t becaus e o f (1.2.2) . Immediat e computation s base d o n th e relations (1.2.1 ) yiel d tha t U is isometri c an d UU' = 1$ henc e w e conclud e tha t V is unitary an d U' = U~ = U . From (1.2.5 ) i t readil y follow s tha t Unh = / . . . , 0 , Wnh\, DT n ~lh, ... , Dh, 0, - \
Previous Page Next Page