Contractions in Von Neumann Algebras

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File: 580J 281201 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 CORE Codes: 3440 Signs: 2087 . Length: 50 pic 3 pts, 212 mm 0.1. no. analysis article functional of journal 2812 Analysis Functional of Journal l iceefntl eeae rus',wihesl eue oknow to reduces easily which among group groups?'', cyclic a generated of shape finitely the discrete hear one all ``Can Valette: and Robertson, qied Equipe h trigpito hsppri usinrie yP el Harpe, la de P. by raised question a is paper this of point starting The and t pcrmi otie nteui ice yuigaprubto ftekernel the of perturbation a using By circle. unit the in contained K is spectrum its nqaiywihgvsago oto of control good a gives for which result inequality contraction) a (to conjugacy H atclr ercvrsm eut fBazm ae ntento finvariant of notion the on based Beauzamy type.'' of ``functional sub- results of invariant subspace some of recover kinds new we describe particular, to us allows for which spaces kernel asymptotic an space s hskre ogv hr ro fteFgeeKdsntermo h loca- contraction the a on that theorem prove FugledeKadison the to of and proof trace short the for a calculis of give functional tion to of kernel kinds this various use define to us allows class Introduction : hr ro ftepwrieult o h ueia ag n naccurate an and range numerical the for inequality power the of proof short a , eascaet n contraction any to associate We ( T ' nls U Analyse f aoaor d Laboratoire nvrie Claude Universite , egv,frayoperator any for give, we ) C g \ 0012 navnNuanalgebra Neumann von a in ) nteds ler.Fnly eascaet any to associate we Finally, algebra. disc the in otatosi o emn Algebras Neumann Von in Contractions T rmtepstv solutions positive the from , eevdSpebr3,19;rvsdSpebr2,1994 28, September revised 1993; 30, September Received . R . . A ' 4 nls ocinel tProbabilite et Fonctionnelle Analyse . soiea CNRS au associe e lc Jussieu place Bernard 69622 135 0 ilubneCedex Villeurbanne ilsCassier Gilles ( Preliminaries . her Fack Thierry ynI Lyon , 93 (1996) 297338 , T 75230 96Aaei rs,Inc. Press, Academic 1996 fclass of T Metadata, citationandsimilarpapersatcore.ac.uk M ad oegnrly oayoperator any to generally, more (and, and 297 754 T ), eas e eeaie o Neumann von generalized a get also We . X ai Cedex Paris noeao kernel operator an & 43 , f fteoeao equation operator the of C ( nvrie iree ai Curie Marie et Pierre Universite , rT \ olvr du boulevard cigo eaal ibr space Hilbert separable a on acting *) x+ , T France l ihso erdcini n omreserved. form any in reproduction of rights All in g 05 s , ( Copyright rT M C , en qieNo Equipe Jeune 1 France ) T 11 suiayi n nyif only and if unitary is otato naHilbert a in contraction x When . & K Novembre : (0 ( T 96b cdmcPes Inc. 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Length: 45 pic 0 pts, 190 mm ( 298 hswyteivratsbpcsò ucinltp' osdrdby considered type'' functional subspaces. invariant `òf of kinds subspaces new also invariant but Beauzamy, the B. way this yblmpwihdsrbstesltoso 1.W louei oprove to of it X use also solution We positive empty (1). zero non of non solutions the any describes that which map symbol a santwrh atta h kernel the that fact noteworthy a is hc elcsteaypoisof asymptotics the reflects which eeaie o emn nqaiyfroeaoso class of operators for by generated inequality algebra Neumann Neumann von von the generalized in choosen be nqaiyfrtenmrclrne ae nteue of used the on based range, numerical the for inequality both involves and prtrsc that such operator ( h uldaio hoe ntelcto ftetrace. the of location the on theorem FugledeKadison the nrdcdb h is ae uhr(f Cs ] sabpoutof byproduct a as a 3]) by [Cas. generated algebra (cf. dual author the named on results first general the by introduced iehr Term1 oiiease,bsdo h s fa operator an of We use the [Har.-Rob.-Val.]). on (cf. based unitary answer, positive fact a in 1) kernel is (Theorem circle, here unit give the in contained contraction a that eue o otatosto contractions for reduces ( operator o emn inequality Neumann von T f f , , ial,w soit oaypstv solution positive any to associate we Finally, yuigaprubto fteCsirskernel Cassier's the of perturbation a using By g g sasmdt ea be to assumed is r nteds ler,and algebra, disc the in are r h icagba and algebra, disc the are & K f : ( rT T ( & T , K *) f fclass of , eemnsanntiilivratsbpc o T for subspace invariant trivial non a determines : ( ( X rT T x )= + )=( *) & g X X T Xx ( ATA T ( I& rT C & I and nafnt o emn ler,woesetu is spectrum whose algebra, Neumann von finite a in & \ + | ) C 0 ÄT :Ä 2 nuieslbudon bound universal an x& &1 ÄT :Ä ? Xg 1} T | 2 ) cnrcin an -contraction) f ) .W ieas eysotpofo h power the of proof short very a also give We *. &1 sacnrcin oevr eso that show we Moreover, contraction. a is ( ASE N FACK AND CASSIER ( &1 rT e | +( & 0 x +( ) 2 it ? x& T # )+ K I x | * )W s hsaypoi enlt get to kernel asymptotic this use We H.) I & f 2 : # & XT ( ( g e T )Nt htti nqaiyi spatial is inequality this that Note H.) :T ( K & :T e ,adalw st e an get to us allows and ), = it : it *) ( )+ )| *) T X (1), &1 2 rvdsavr hr ro of proof short very a provides ) &1 (K g smttckernel asymptotic & ( e X hs uepitsetu is spectrum point pure whose it re I & & )| X it A ( 2 X T & fteoeao equation operator the of (K , } K K- (0| & X : A re ( pcrloeao.It operator. spectral ) T it x &1 (0| ( ,w e o any for get we ), T T K | : & egtas a also get We . x) |<1) ) : where , ( x T erecover We . : | 2 dt ). |<1), x) ? C asymptotic \ 2 dt which , A ? A san is can (1) File: 580J 281203 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 3382 Signs: 2468 . Length: 45 pic 0 pts, 190 mm esalas osdrthe consider also shall We pcrmof spectrum by and on separable) operators time linear the bounded of (most space Hilbert ler of algebra n by and ler falcmlxvle ucin hc r nltcisd h unit the inside analytic are which functions valued complex all of algebra III. prtrter,a n[eu ] npriua,w hl eoeby denote a shall called be we will particular, In 1]. [Beau. in as theory, operator 0.3. 0.2. ilb eoe by denoted be will rjcinin projection space Hilbert a 2]. on [Cas. ing to refer we notion, this on details more For D on m- iieesof finiteness II. 0. I. = 0.3.1. 0.3.2. 0.3.3. seta u omof norm sup essential M h smttcfntoa aclsadisapplications its and calculus functional asymptotic The contractions. to jugate ..Itouto.02 otn fteppr ..Notations. algebras Neumann 0.3. von paper. in the of contractions Content of Structure 0.2. Introduction. 0.1. Preliminaries. prtr fclass of operators lmnswt pcrmcnandi h lsdui ic ..Cnrcin nfnt von finite for in trace'' Contractions the I.4. `ùnder disc. in unit contained calculus spectrum closed Functional with the algebras in I.3. Neumann contained theorem. spectrum FugledeKadison with elements the to tion I3 h eeaie o emn nqaiy I4 prtr fclass of Operators II.4. inequality. Neumann von generalized The II.3. cone ntecnuayo prtr fcasC class of operators of conjugacy the On I..Tecone The III.4. Notations paper the of Content [z o oedtiso o emn lers erfrt [Dix.]. to refer we algebras, Neumann von on details more For . # S C prtrTheory Operator o emn Algebras Neumann Von pcso nltcFunctions Analytic of Spaces _*( C 1 + = B ( || T T M ( T [e .II2 h kernel The III.2. ). z H M )= n by and , |<1 OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS seuvln oteeitneo atflnra iietrace finite normal faithful a of existence the to equivalent is it .Sc o emn algebra Neumann von a Such ). contraction C hc seuvln oteiett.If identity. the to equivalent is which | where [: + t C T ] ( m # \ T # & I2 plcto otepwrieult o h ueia range. numerical the for inequality power the to Application II.2. . h pnui ic by disc, unit open the Z n h nain usaepolmfor problem subspace invariant the and ) o any For . C W :I Â W 2 A h lrwal lsdielgnrtdby generated ideal closed ultraweakly the | ?] ( . H spoe in proper is T f T ( esaluetesadr emnlg of terminology standard the use shall We . suuly esaldnt by denote shall we usually, As . T any For . steda ler eeae by generated algebra dual the is eksetu _ spectrum weak )= sb eiiina lrwal lsd*-sub- closed ultraweakly an definition by is t onay h armeasure Haar The boundary. its t ueia range numerical its ) K H [(Tx f ealta o emn ler act- algebra Neumann von a that Recall . r , nelement An . # t ( X L .II3 nteoeao equation operator the On III.3. ). T | A ( \ x) S suuly esaldnt by denote shall we usually, As . # T ocontractions to 1 B ] , , S | ( m H 1 & D . *( ,w hl eoeby denote shall we ), ..Tekernel The I.1. . x = ,w hl eoeby denote shall we ), & T T =1 I..Teoperator The III.1. . )of [z # M B ] # B I1 etre enl for kernels Perturbed II.1. . . ( sfnt if finite is T C H ( eie by defined , H || uhthat such ) T H h ler fall of algebra the ) . z |1 K ssprbe the separable, is r T , C t A ( . T \ ] ( ( I dtÂ .I2 Applica- I.2. ). \ D steonly the is t closure, its 1 r con- are >1) h disc the ) S 2 & T _ ? T f & * ( & T T XT n the and on & the ) H = 1 299 the S X a 1 . File: 580J 281204 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2770 Signs: 1895 . Length: 45 pic 0 pts, 190 mm kernel ercl that recall We mutators ilwy(f Cs-a],adalw opoetefloigcomplement following the prove to allows and [Cas.-Fa.]), (cf. (cf. way cial u ro sdfeetadmr iet tue h kernel the uses It 1992, direct. June more by in and obtained Valette different then Alain is was and proof Robertson Our [Bro.]) ans Orle Guyan (cf. the A. Brown during Harpe, of la result de a Pierre on based proof plete lse f functions of) classes in n naHletsaeH space Hilbert a on ing andin tained dis- details): all more Robertson among for group Guyan [Har.-Rob.-Val.] cyclic A. a (cf. of Harpe, groups?'' shape la generated the finitely hear de one crete Pierre ``can by Valette: Alain raised and question a positively and [Hof.] [Gar.], [Du.], to refer we questions, [Rud.]. these on details more For & icadcniuu ntecoe disc closed the on continuous and disc function f 300 r for 0 are u u htaeams vrweetebudr au fsm holomorphic some of value boundary the everywhere almost are that in efrtpoe hoe when 1 Theorem proved first We Theorem answers which theorem, following the is section this of result main The & .Srcueo otatosi iievnNuanAlgebras Neumann von Finite in Contractions of Structure I. S p 9 1 <+ (ii) II): set D, , (i) K hnTi iiepoutof product finite a is T then & . r u , S u n suiayi n nyi sacnrcinwt pcrmcon- spectrum with contraction a is T if only and if unitary is T If t ( & in 1 0 ti elkonthat known well is It <0. T ; ial,ltu eoeby denote us let Finally, . p M 1. sup = nrdcdb h is ae uhr(f Cs ].Acom- A 3]). [Cas. (cf. author named first the by introduced ) H H satp II type a is 0 p ( e ea lmn nafnt o emn algebra Neumann von finite a in element an be T Let p D ( r D <1 adsc onaylmteit o any for exists limit boundary a such (and ) f1 if etn nOeao Algebras Operator on Meeting stesaeo l nltcfunctions analytic all of space the is ) f \ in | 0 2 . L ? c 1 n p | p ( atradi h pcrm_ spectrum the if and factor u ( ASE N FACK AND CASSIER S <+ f ( re )= 1 , it m )| | hs ore coefficients Fourier whose ) 0 p 2 ? 2 dt e ? S & ( + H 1 H esta eight than less int & 1 and D Âp p p f ( ( _ o n ooopi function holomorphic any For . ( S S e ( 1 1 T it h pc fal(equivalence all of space the ) stesaeo l functions all of space the is ) ) ){ 2 dt ? & < u & yuiga operator an using by , c.[Har.-Rob.-Val.]). (cf. ) =sup ( T utpiaiecom- multiplicative z ) # K u D fTi contained is T of r in , | u t ( u ( T D z in )|. nacru- a in ) satisfying H M p ( D act- )). File: 580J 281205 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2913 Signs: 2102 . Length: 45 pic 0 pts, 190 mm lsdui ic o any For disc. unit closed n[ai] u ntersrcinta h pcrmof spectrum the that disc restriction unit the open on but [Vasi.], in nt and unit, satisfied so oiietp when type positive of is andi h lsdui disc unit closed H the space in Hilbert tained a on acting stiil.I at h dao sn systematically using of idea the fact, In trivial). is sunitary is h pcrmof spectrum the enl(e h eaka h n fI13.A xasoa oml for formula expansional An I.1.3). of end the at remark K the (see kernel soitdwt T with associated hr h enlwsue opoeta h map the that prove to used by was kernel the where ] sabpouto eea eut nteda ler eeae ya by generated algebra dual the on results general of byproduct bounded a as 3]) I.1. subject. this on exchanges I.2). interesting section for (see Valette trace the of location the on theorem nqaiy osbc o[a.3.W hl e (cf. Neumann see von shall the We a proof 3]. direct [Cas. a to get to back (e.g. goes space inequality) Hilbert a in tractions r I. h kernel The Theorem oethat Note eaantakPer el ap,A ua oeto,adAlain and Robertson, Guyan A. Harpe, la de Pierre thank again We hskernel This , t ( 1.1 h enlK Kernel The (ii) T (i) 8 led per nRezadNg c.[i.N.,pg 467), page [Rie.-Na.], (cf. Nagy and Riesz in appears already ) eiiino h kernel the of Definition . , ( n . emycos nsc a that way a such in A choose may we K- hr xssapstv netbeeeetAin A element invertible positive a exists There h pcrm_ spectrum The )= T pcrloperator spectral # 2 I . T & K A D K OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS K n r r , re rT e ea lmn nafnt o emn algebra Neumann von finite a in element an be T Let s httebudr eairo h enlwhen kernel the of behavior boundary the that (so ea lmn hs spectrum whose element an be , r t oepaieisdfeec ihtecasclPoisson's classical the with difference its emphasize to , , t r ( ( , & t T ( T t ( T scnandi h lsdui ico radius of disc unit closed the in contained is it lopoie ietpofo h FugledeKadison the of proof direct a provides also ) T a nrdcdb h is ae uhr(f [Cas. (cf. author named first the by introduced was ) T if )=( )( and T n cf ( 0and 0 I r sacnrcin h kernel The contraction. a is .[ T & ih0 with I . . ) & Cas & hntefloigasrin r equivalent are assertions following the Then re sueta h ueia ag fTi con- is T of range numerical the that Assume T fTi otie in contained is T of A re & esalcall shall We . & .3]) it it } T T & ) A r netbeeeet in elements invertible are * K r &1 1and <1 &1 r , +( t & ( T 3. 8 I .Let ). &re ( n )= t K # it T r R _ , T 8 K t A *) ( ( Â * r T S 2 T , eie nteintegers the on defined & t ? &1 1 eaC-ler with C*-algebra a be scnandi the in contained is ) ( the ) . T Z T n K M 9 Moreover & I httekernel the that II) set , osuytecon- the study to ) r scnandi the in contained is , t uhta A that such I ( . T Cassier if loappears also ) n A 0 , r because , ' <1. if kernel s r (ii) &1 Ä 301 TA 1 : M & is File: 580J 281206 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2267 Signs: 1177 . Length: 45 pic 0 pts, 190 mm h lsdui ic ti la rmtedfnto that definition the from clear is It disc. unit closed the iuu ucinof function tinuous t function any for rmteCuh' oml,w get we formula, Cauchy's the From nti ae ehv h olwn inequalities following the have we case, this In oiieon positive n hsi ut()for (5) just is this and K z npriua,tekernel the particular, In lmn na in element opoe() oeta h function the that note (5), prove To 302 o any for ial,w e ysrihfradcalculation, straightforward by get we Finally, 1& = 1+ # r I.1.2. , R t re ( Â T r r 2 it I a eprubdt td prtr nthe in operators study to perturbed be can ) ?Z # K r is rpriso h ase' Kernel Cassier's the of Properties First D , r ih0 with K , esalsti h sequel the in set shall we , t ( r D , T C t ( )= T o any for *-algebra | 0 2 ) ? f X K r in *( 1+ 1& 1ad0| and <1 t r K , 1&| 1+| r r I satisfying t A r ( &| +| & , x z T ( r r t f ( = D # A ( K ) T I r z rT z z z 2 H 2 n 0 and ) dt r *)= re | | | | T ihui,adasm that assume and unit, with , ? ASE N FACK AND CASSIER & t & ota egtb h anc' inequalities Harnack's the by get we that so , )= ( * it = x K x T . T o any for & & z I ) K ) | 2 2 ( 0 T spstv fadol fTi contraction a is T if only and if positive is X 2 r z ? ,& nraiainproperty). (normalization |< )= where , h h f t r ( ( ( ( h z z 1 oeas htw ae o any for have, we that also Note <1. e r T K ( ) ) it 1 Letting <1. z smer) (3) (symmetry). ) ) r r , )= K t (0 (T 1+| 1&| r r r +| &| , (K t ). ( T r z z z z 1 n any and <1) z | | | | ) ( X & & T 2 dt x x =( ? ) & & r x 2 2 e again Let . , Ä , I | & C _ x) 1 ( \ re & T K lse.Frany For classes. eget we , shroi and harmonic is scnandin contained is ) & r , it t T ( T ) t sacon- a is ) &1 # .(4) T R Â ean be 2 ?Z (1) (2) (5) . . File: 580J 281207 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2891 Signs: 2194 . Length: 45 pic 0 pts, 190 mm K oml ntesequel. the in formula n naHletsae nqelna map linear unique a space, Hilbert a on ing [Fug.-Kad.] (cf. trace the of [Ber.]. location and the about theorem FugledeKadison T oiyi otettecs foeaoswt ueia ag niethe inside algebra, range trace Neumann of any numerical von `ùnder commutativity finite with even a of that in operators show contraction of a shall for we case Finally, the disc. unit treat closed to it modify ae ahriefcieayvgeaaoybetween analogy of vague commutativity any non ineffective the rather from makes coming information the of emn algebra Neumann Z .Fc n .d aHre(f F.Hr] oolie41 httenor- the that by proved 4.1) been corollaire has [Fa.-Har.], it (cf. But Harpe normal. la is de trace 249) the P. page that and 3, addition me oreÁ Fack (the in T. [Dix.] ask from follows we trace if central the of uniqueness T eas h pcrlradius spectral the because S arx oeas httekernel the that also Note matrix. a oso' enl ercmedt h obflrae ocompute to reader doubtful the to recommend We K kernel. Poisson's cal I.2. noncommutativity. its of part trivial non a htteiaeof image the that 1 ( r r * oee,w ol iet on u that out point to like would we However, . I.1.3. ealfrtta hr xss o n iievnNuanalgebra Neumann von finite any for exists, there that first Recall ial,nt htw have we that note Finally, h map The sa lutaino h sfleso h ase' enl e sshow us let kernel, Cassier's the of usefulness the of illustration an As , , scnandi h lsdui ic.Ti atimdaeyipisthe implies immediately fact This disc). unit closed the in contained is M t t =0 ( ( plcto oteFuglede the to Application (iii) T T i)Fraypstv element positive any For (iv) ,wihsatisfies: which ), K i)( (ii) i ( (i) )= o arcso h form the of matrices for ) r , O Remark t ( T P T T r TS TS )= T =0. , * t ( Ä = T ) ) OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS I * * hnteoperator the When . + ,where ), T T = =( K * n M o any for : 1 T r ST ilb called be will , T ( t spstv w nyasm eeta h pcrmof spectrum the that here assume only (we positive is ( S r T n with ) ) e * * ne h aoia eta rc fafnt von finite a of trace central canonical the under ) P & r int o any for any for T \ , t ( T T ( T in z ) h ase' kernel Cassier's the *), n )= )of aio Theorem Kadison + Z *I ( n P h eta rc of trace central the : M 1 S T T K r + ( , r ); in ze satisfies S , r N t n ( & T e M T in where , int it { sntùiesl' ehv to have we `ùniversal'': not is ) steuulPisnkre on kernel Poisson usual the is ) T '(..i n r okl h defect the kill to try one if (i.e. '' M in n any and T * T ; Ä n \( M K T T N ehave we , r nr convergence), (norm suiay ehave we unitary, is , .W o' edthis need won't We )1. t * ( sannzr nilpotent zero non a is T T K from loecdsapart a encodes also ) M r in { , t ( h xsec and existence The . ( K T Z M r n h classi- the and ) ( , T M t ( noiscenter its into T and T ); )wl keep will )) * 0and: 0 T .This *. M act- 303 File: 580J 281208 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2491 Signs: 1324 . Length: 45 pic 0 pts, 190 mm 0 egtfo (1) from get we 304 cigo ibr pc H. space Hilbert a on acting observation: this from get immediately t n i spoe,bcue( because proved, is (i) and o n netbeelement of invertible radius spectral any the For because sense, makes This ibr pc H space Hilbert essen- is proposition following The (iii). nature: and in (i) algebraic from tially automatic is mality andi h lsdui disc unit closed the in tained il lmn in element tible osdrnow Consider # oeta h ro f()as hw that shows also (i) of proof the that Note Corollary Proposition Proof R r K Â i)Sm ro sfr() replacing (i), for as proof Same (ii) (ii) <1, (i) 2 : ?Z ( i saoe e o any for set above, As (i) . T fteseta aisof radius spectral the if , ) ehv { have we o n rca tt on { state tracial any For o n t any For * =[( =[ =[ =[ _ X 1. 1. K X X X , M = X X : n ea lmn in element an be T and &1 &1 &1 K ( oiigthat Noticing . &| ( e ea lmn nafnt o emn algebra Neumann von finite a in element an be T Let ( T # I Let : K ( ( ( & ( n + I I R ) I : T =0 r * & & & , Â ÄT :Ä | t 2 )=( =[( 2 ( M :T :T :T ?Z T T | )]( : ))0. * | X *) *) *) eafnt o emn ler cigo a on acting algebra Neumann von finite a be ASE N FACK AND CASSIER I I | 2 XT I n n with r any and : & n D I & & # T &1 &1 &1 | & 2 . M ÄT :Ä :T * )( ÄT :Ä T ÄT :Ä n ( [ X I T + X * X ) *) & +( ) XT &1 ) n get we , &| &1 ( &1 hc sceryapstv n inver- and positive a clearly is which , &1 I ÄT :Ä : & I X = X ] : M & +( # T ) &1 * | : &1 &1 ÄT 2 D X , T ÄT :Ä T ( I M & ssrcl esta n.We one. than less strictly is )+( ( n t any 0 * I &: X * I & ) & I XT &1 &1 by hs pcrm_( spectrum whose , ÄT :T r I :T & ( <1, :T & )( { I ) # *) ( &1 & I *) I T :T R & ] ssrcl esta one. than less strictly is &1 K & :T Q.E.D ) Â * &1 ehv K have we 2 *) ÄT :Ä 1, ?Z 0. I ] *) t . ( * X ) T &1 &1 , &( n n with r any and ) * ] ] * * 0frany for 0 I & .(1) r , T t :T ( T ) ) *) scon- is * 0; M File: 580J 281209 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2589 Signs: 1597 . Length: 45 pic 0 pts, 190 mm scnandin contained is hssosthat shows This o n ope ucinfwihi ooopi nanihoho fthe of neighborhood a on holomorphic is which disc f unit function closed complex any for rca tt on { state tracial h lsdui disc unit closed the otie nteCoe ntDisc Unit Closed the in Contained I.3. K ntesaeo rbblt esrson measures probability of space the in iertransformation linear a pcrm_ spectrum Fix pc H space elements for trace'' also a `ùnder but calculus tractions, functional a of existence esr & measure _( 1, T | Proof I.3.1. Theorem * t ) ( |=| ucinlCalculus Functional * i)( (ii) i)Flosimdaeyfo i.Q.E.D (i). from immediately Follows (ii) T (i) scnandi h lsdui disc unit closed the in contained is # ) * _ i Let (i) . , / xsec faFntoa Calculus Functional a of Existence ( n rca tt on state tracial a { and ( T T h ovxhl ftesetu fT of spectrum the of hull convex The spstv,w have we positive, is T ( Fuglede * T on * 3. n let and ) ) )|= OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS S fT of D D 1 Let _( ehave We . . } uhthat such | Moreover T T . M Kadison 0 D 2 M * ? ea lmn in element an be , e scnandi h lsdui disc unit closed the in contained is ) whenever / it & eafnt o emn ler cigo Hilbert a on acting algebra Neumann von finite a be h rc { trace the T / T eacaatron character a be T `` ( wa lim =weak Ä ne h Trace the under K * ' hoe ntelcto ftetrace the of location the on theorem s 1, , = :T & ehave we t / T ( | T ( + ( T 0 M K 2 _ f ) ? r * ( 1, bI in )= . e Ä T ( o n lmn in T element any For ) t it T ( oso that show to ) scnandin contained is ) 1 K M 2 T dt { ) { ? 1, ( M ) ( S eog otecne ulo the of hull convex the to belongs f } * D K hs pcrmi otie in contained is spectrum whose t 1 ( opoe() tsfie b using (by suffices it (i), prove To . ( . )= T r T , , | '' t hr xssauiu probability unique a exists there ) )) Z ( 0 * 2 T / ? ( o lmnswt Spectrum with Elements for * ( M )) 2 / dt K ? ( scnandi Conv in contained is `` 2 uhthat such ) K 1, dt { . ne h Trace the under ? t 1, stu o nyfrcon- for only not true is ( T t _ ( ( T suethat Assume D. * T * )0adhence and ))0 * M )) scnandin contained is ) 2 hs spectrum whose dt D * ? = = . / / .) ( ( T I o any For ' The .'' )=1. ( * _ ( _ .As ). T 305 ( T D )), ) . File: 580J 281210 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2522 Signs: 1560 . Length: 45 pic 0 pts, 190 mm 306 eafnt o emn ler cigo ibr space Hilbert a on acting on algebra state Neumann tracial von finite a be ypooiin1(ii), 1 proposition By etu eieaRdnmaueon measure Radon a define thus We hieo h eune( point sequence the limit of the choice that shows This S e rm(1) from get ovrigt ,adaylmtpoint limit any and 1, to converging ec oncide. co@ hence ooopi nanihoho ftecoe ntdisc unit closed the of neighborhood a on holomorphic & n ec o n integer any for hence and 2.Tetermi rvd Q.E.D proved. is theorem The (2). pc fpoaiiymaue on measures probability of space ftecoe ntdisc unit closed the function of complex any for have n ec b aigtetrace), the taking (by hence and npriua,fraysqec ( sequence any for particular, In bv qaiy.I olw that follows It equality). above r 1 , Proof I.3.2. T uhthat such ovre when converge, xsec f& of Existence . ikwt h Grothendieck the with Link nqeeso & of Uniqueness | 0 2 ? e int & M i d& ( | T o n element any For . f 0 2 1 o any For . )= ? ( r t Ä K )= D & { f r r { r , ( ( 1 n ( T , t { rT f ) ( | T & f ( Let . n T ( 0 2 ( 1 f oapoaiiymeasure probability a to , n rT ? ASE N FACK AND CASSIER sapstv esr.O h te ad we hand, other the On measure. positive a is )= T ) ( # e T 2 )fraycmlxfunction complex any for )) dt ))= int ovrigt .Cneunl,temeasures the Consequently, 1. to converging Z ? ))= | f r = 0 2 & hc shlmrhco neighborhood a on holomorphic is which & & S & b aigtecmlxcnuaeo the of conjugate complex the taking (by ? ih0 with 2 1 | 1 I 1 ( f 0 suiu n osntdpn nthe on depend not does and unique is | t ' 2 and ( pcrlMaue* Measure Spectral s edwdwt h ektplg) we topology), weak the with (endowed 0 ? and o n integer any for ) 2 r e S ? n it f T ) 1 f ( ) & n hc satisfies which , e ( and K & 1 e it in & ftesqec ( sequence the of 2 it ) r 2 r , ) etopoaiiymaue on measures probability two be d& t 1 set <1, M ( aetesm oet,and moments, same the have d& fra ubr (0 numbers real of T r the , , ( ) T t 2 .(2) ). dt { ( ? t ( .(1) ). I , )=1. Grothendieck & r , & T = & = D r , T & T { ehave We . T n e again Let . & ( 1= because (1)=1 r 0, K hc satisfies which n H , T r , Let . f ) t ( n ' T 1 spectral s hc is which )) r { nthe in dtÂ n <1) ea be 2 M ?. File: 580J 281211 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2619 Signs: 1628 . Length: 45 pic 0 pts, 190 mm rbe ntedomain the in problem xeso fapstv otnosfnto spstv,teformula the positive, is function continuous positive a of extension ntecoe ntdisc unit { closed state the in tracial a with equipped esr on measure computable. esrsascae ih* with associated measures { where a eo oeitrs orlt tt u measure our to it relate to interest some of be may esr * measure by hr \ where o n ope function _( complex any for by olw rmtesbamnct ftefunction the of subharmonicity the from follows amncetninof extension harmonic eie oiieRdnmeasure Radon positive defines C h xsec of existence The . .As T tti n,fraycmlxcniuu function continuous complex any for end, this At Proposition steGohnic' pcrlmeasure spectral Grothendieck's the As _ ); ( (ii) (ii) (i) (i) T f 2 .Ismi rprisaetefloig(f [Bro.]): (cf. following the are properties main Its ). shroi usd h pcrmof spectrum the outside harmonic is z ( ( T T { * & & T ln ( steFgeeKdsndtriatascae ihtetrace the with associated determinant FugledeKadison the is ) T T TS )( * f soitdwith associated = = _ T ( 2 = ( T = z T OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS * | ( # )= z 2. * T uhthat such ) ? 2 ( & D ST * | T ) T D z * _ e ea lmn nafnt o emn algebra Neumann von finite a in element an be T Let ); o any for ( T steBrlmaueon measure Borel the is 2 + )= T . hc sdfndb h formula the by defined is which , zÄ ) hnw have we Then \ D f { | f | z D ( l | (ln ( _ ihbudr au qa to equal value boundary with , ( z to & T ( * T ) T f T _ ehave We . d* )( ) T ( T )( z D n| ln hc shlmrhci egbrodof neighborhood a in holomorphic is which T E , & . T ) f S ..teuiu ouino h Dirichlet's the of solution unique the i.e. , sdfndi Bo]a h nqepositive unique the as [Bro.] in defined is )= sueta h pcrmo scontained is T of spectrum the that Assume ( \ | )= z z T z in and ) ( & ( )(itbtoa derivative), (distibutional |) * ? 1 T | T M ` _ . ) )on | ( T d* ( . d* z ) , f T T E ( S ( { z ( z )& 1 l | (ln ` ) * ), htw hl call shall we that , o any for ) T d* T S h measure the , m 1 sfrfo en xlct it explicit, being from far is T f ( eie nayacEof E arc any on defined (T ( E z z ) Ä ), )|)= f f on ( z & )= S f _ T steharmonic the As . ( 1 T hc sreally is which , eoeby denote , z ) { * # n| ln l | (ln T C h harmonic the ssupported is , f z ( z & )| T d* )on |) f T 307 the ( M S z 1 ) 308 CASSIER AND FACK

Fig. 1. Angle intercepted by an arc E. where .(z, E) is the angle intercepted by the arc E at the point z # D (cf. Fig.1)and m(E) is the normalized Haar measure of E. Proof. (i) For any integer n0, we have

2? n n int z d*T (z)={(T )= e d&T (t). |_(T) |0

It follows that the two positive Radon measures &T and |(*T) have the same moments and hence coincide. (ii) It suffices to prove that we have, for any proper closed arc 1 E=[%1, %2]ofS,

1 &T(E)=*T(_(T)&E)+ .(z, E) d*T (z)&*T(_(T)&D) m(E). ? |_(T)&D

By [Cara.], the harmonic extension f of the characteristic function f=/E of E is given by

1 f (z)= .(z, E)&m(E)(z#D). ?

For any integer n # N with n>2Â(2?&%2+%1), denote by fn the continuous function supported by [%1&1Ân, %2+1Ân], which is equal to 1 on E, and which is both affine on [%1&1Ân, %1] and [%2 , %2+1Ân]. We have

2? it f n(z) d*T (z)= fn(e ) d&T (t) for n large enough, (1) |D |0

File: 580J 281212 . By:BV . Date:12:01:96 . Time:10:16 LOP8M. V8.0. Page 01:01 Codes: 1812 Signs: 857 . Length: 45 pic 0 pts, 190 mm File: 580J 281213 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2879 Signs: 1926 . Length: 45 pic 0 pts, 190 mm n h ih adsd f()cnegsto converges (1) of side hand right the and have pcrm_ spectrum insosta,fraycmlxfunction complex any of for neighborhood that, shows tion S h condition the o n ope ucinfwihi ooopi nanihoho of neighborhood a on holomorphic is which f function complex any for emn algebra Neumann oi ucin erae osm function some to decreases functions monic rca state tracial constant a is there if S (i.e. bounded radially n h ih adsd fti qaiycnegsto converges equality this of side hand right the and nparticular In tutr fBOfntos(f Ko] .36,that 336), p. [Koo.], (cf. functions BMO of structure shroi on harmonic is T n seto i)i rvd Q.E.D proved. is (ii) assertion and derivative iuu ihrsett h armeasure Haar the to respect with tinuous etu have thus We yteLbsu oiae ovrec hoe.O h te ad as hand, other the On theorem. 0 convergence have dominated we Lebesgue the by ibr pc H space Hilbert h =( I.3.3. yuigtemeasure the using By ro f& of Proof tflosfo rpsto that 2 Proposition from follows It Proposition scnandin contained is fteform the of T * T &1 ( h qaiy& Equality The S * ehv,frany for have, We )*. h ( d& ) T T , { f T ) * ' oepeiey ehave: we precisely, More ''. n S Âdm +1 = OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS Ch scnandi h circle the in contained is 1 , 3. D # S & n rca tt on state tracial a { and ). S S T Henceforth, . h _ M 1 nBO hnvrteGohnic' measure Grothendieck's the whenever BMO, in &1 = 1 Let f h operator the , ( ti loes oddc rmPooiin2adthe and 2 Proposition from deduce to easy also is It . { n T n suethat assume and , tsfie oso that show to suffices It . ( [z for K ). T = r M & | , * | T t n z = ( { re |1 esalnwivsiaesm osqecsof consequences some investigate now shall we , T ( ag nuh h eune( sequence the enough, large eafnt o emn ler cigo a on acting algebra Neumann von finite a be & i% f )& T ( f # T &1 r ( D & K z g *))= uhta 0 that such Let . T ) = r * , and 01 | d* f = t ( f ( S T T & ytemxmmmdlsprinciple, modulus maximum the by & { T ))=2 ( )bhvslike behaves *) T ( z S oniewith co@ ncide &1 T f )= S M 1 ( 1 , . T m # ea lmn nafnt von finite a in element an be . ehave we e{ Re & g &1 C o n lmn in T element any For f T on _( hc seulto equal is which D 0sc htfraysector any for that such >0 ( )) r hc shlmrhco a on holomorphic is which % E & T f 1adany and <1 ( 0 T ( S ), X .Tefloigproposi- following The ). z = 1 ) r ihRadonNikodym with , , % d* t & ), ( & T * & T T &1 % f T T ( ( ( f 0 z E )* ftesetu of spectrum the if T n sasltl con- absolutely is + when ) fpstv har- positive of ) when ) tti n,set end, this At . &1 h] t `ne any `ùnder ) # f ( R h on n Â 0,we >0), M n 2 Ä Ä ?Z S whose 1 + * + S , T and 309 1 . is . File: 580J 281214 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2353 Signs: 1316 . Length: 45 pic 0 pts, 190 mm with on state in ehave we u 1 But function tions measure and H, where in 310 ehave We egbrodof neighborhood I.4. eget We n h bv eakgives remark above the and ueyiaiay n tflosfo 1 that (1) from follows it and imaginary, purely hsimdaeyipisthat implies immediately This olw rmterelation the from follows { ( ro fteEult { Equality the of Proof e again Let I.4.1. f S M ( 1 otatosi iievnNuanAlgebras Neumann von Finite in Contractions T hc sdfndby defined is which , { Â f *))= osdrteMbu il ( field Mo bius the Consider . (1& 1 X , ofra naineo h eemnn fT of Determinant the of Invariance Conformal T r M f f & , 2 t T nanihoho ftecoe ntds,ti qaiydirectly equality this disk, unit closed the of neighborhood a on =( re n eoeby denote and , ea lmn in element an be nanihoho of neighborhood a on oecneune ftecondition the of consequences some , { M ( i _ ( f I t ( 1 & & X ( eafnt o emn ler cigo ibr space Hilbert a on acting algebra Neumann von finite a be T % r S ) re , )&1 *))& t 1 )= hr xssb arn' xaso w nltcfunc- analytic two expansion Laurent's by exists there , & it T Â { f (1& { ( ) 1& ( T &1 T f { *)= 2 ( f : &( (( 2 ASE N FACK AND CASSIER re K re ( ( =( & 1 z T f r T i h uldaio eemnn associated determinant FugledeKadison the i M )= , ( ( ( & I t *) * % t T f & ( : T &re & = 1 esalnwivsiae yuigthe using by investigate, now shall We . T & ( *))= = &1 % f t T ) ))= ) D 1 & & )=[2 T ( T & T *)& ))= z it ( &1 : uhthat such )( T )& T 1& ) &1 { { : I .If &1 (K ( # & { )* f f ir f ( re D 2 2 ) . ( 1 f (( r ÄT :Ä &1 T sin( , \ 1 i soitdwt contraction a with associated ( t ( T f % ( 1 z T &1 .As & S ) + *) &1 shlmrhcol na on only holomorphic is t &1 )). t . ) o n holomorphic any For )). ) & &1 . |0 , ))& S % _ ihSetu Contained Spectrum with ), )] 1 ( # T Â { t [1&2 scnandin contained is ) ( Let . 2 _ f ( 2 T ( T ? ). = ))= r . { cos( eatracial a be { ( t f & (T Q.E.D. % &1 ]is )] S )). T 1 , File: 580J 281215 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2873 Signs: 1750 . Length: 45 pic 0 pts, 190 mm pcrm_ spectrum ota egtfo h bv equality above the from get we that so togcnegneof convergence strong egbrodo h lsdui ic ln(( disc, unit closed the of neighborhood if utpyn 2 by (2) Multiplying ibr pc H space Hilbert 2 yPooiin3 Proposition by But n h ro scmlt.Q.E.D chapter. the of beginning the at stated complete. is proof the and to ln(( yuig[i. te rÁm 0 i) .19,w e rm(1) from get we 109), p. (iv), 10, me oreÁ (the [Dix.] using By replaced h uldaio determinant FugledeKadison the a odcniut rprisi the in properties continuity good has trum 2 ( T Proof Proposition esalfrtso that show first shall We I.4.2. yPooiin4 ehv o any for have we 4, Proposition By T ( 2 I T =.A ewr o bet rv ietytecnegneof convergence the directly prove to able not were we As )=1. scmltl o-ntr)teequality the non-unitary) completely is 2 & ( )= _ (( e = ( i% :T I stefunction the As . T ro fTheorem of Proof I & 2 2 2 =,w aemdfe hsagmn.Esnily ehave we Essentially, argument. this modified have we )=1, scnandin contained is ) &1 ( ( ( ( :T T T T T ) ( & : ) &1 I ytema au fln of value mean the by ) *) OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS & scnandi h circle the in contained is 2 :I , 4. { ln aesneb ooopi acls oevr ehave we Moreover, calculus. holomorphic by sense make ) &1 [ln(( (( n rca tt on state tracial a { and :T ) 2 I )= 2 Let & ( 2 &1 2 (( T ( I (( :T 2 I T )=ln T & )( & I (( M & : I & *) :T & I ÄT :Ä to & :I S :T &1 .Ltu o rv hoe ,wihwas which 1, Theorem prove now us Let 1. 2 eafnt o emn ler cigo a on acting algebra Neumann von finite a be :T *) 1 2 ) ,w eueb sn h utpiaiiyof multiplicativity the using by deduce we ), e ÄT :Ä ersial,w ol iet s a least (at use to like would we Heuristically, . ( &1 f )=|exp ( T i% &1 2 &1 = =|exp ( re &1 I z = o n element any for )=1 ( ) )= T ) )=ln((1& &1 i% )]= 2 ( ) &1 : & &1 )= (( = y[i. te rÁm 0 i) .109), p. (ii), 10, me oreÁ (the [Dix.] by ) 2 )= I T )= (& & { re 2 { { r r- )&ln [ln(( [ln(( [ln(( ( 2 ih0 with i% S aibe when variables :T T T 2 M 2 (( when ) 1 : ( : , ( T ). I . )= &1 T n n with : any and 2 I :z I I & o n lmn in T element any For 2 ( & : & & ( I ) ntecrl | circle the on ) ) (1& T & 2 ÄT :Ä &1 &1 :T :T :T )= ( r T :T r ). shlmrhci a in holomorphic is ) ) 1adany and <1 =| &1 &1 *) : &1 re 2 ), &1 & &1 ) ( ). ) : &1 T &1 T i% )) | )]| : T Ä r I )]. oehrwt the with together ) )]| 2 in & Ä ). 1 (( | :T & M 1 : I & & oifrthat infer to , |<1, : *) . hs spec- whose % |= :T &1 # M R r &1 ehave we and ) which , Â 2 2 whose ) &1 ? ( 311 T (2) (1) Z ) : ) File: 580J 281216 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2327 Signs: 1106 . Length: 45 pic 0 pts, 190 mm fteaoeeult scniuu nthe in continuous is letting equality above the of ic h function the Since e snwseta h ih adsd f()i eo ic h function on the Since subharmonic zero. is is (2) of side hand right the that see now us Let 312 n hence and e sso httescn emi h ih adsd f()i eo By zero. is (1) get of we side 109), hand p. right the (iv), in 10, term me oreÁ (the second [Dix.] the using that show us Let nertn hsrlto nthe in relation this Integrating shroi nanihoho ftecoe ntds,adhence and disc, unit closed the of neighborhood a in harmonic is But yFbn' hoe,w deduce we theorem, Fubini's By 0 2 0 2 ? ? ln 2 g ln ( 2 (1& 2 0 re ? 2 (1& | r i% 0 (1& n|1& ln 2 Ä ? ) re ln | d%Â ln re 0 1 2 & 2 ? re & 2 & i% 2 ( ln T & (1& ? T i% re T i% )=|exp 2 )= nthe in & T )= ( i z e ( ) re % | i% Ä d%Â & 0 ln 2 & { & ln ? C [ t i% ln ) 2 ln r- Re 2 egtb h otniyo h envalue mean the of continuity the by get we , T | 2 T { ? ( d%Â variable 2 ( [ln(1& ) T ) 2 0 etu have thus We =0. (ln(1& T z ( d%Â 2 d% ( )= Ä z )= 2 ASE N FACK AND CASSIER re ? & ? = 2 i% g l =,snetefunction the since 1=0, =ln | ? T | ( aibe eget we %-variable, & 0 | 2 0 z lim = = 2 ssbamncon subharmonic is ) ? 0 re )=ln 2 ? re T ? ln r | ln & & _ Ä ) 0 2 i% 2 ? | i% 2 d% 2 1 0 T ? T 2 & ( ln ( ? 2 re & )]|=exp e ))]= | n|1& ln (1& i% 2 0 i% 2 | r- & ? (1& & 0 2 ln aibe ecfrh egtby get we Henceforth, variable. ? T T zT | ln 2 ) 0 ) 2 e re ? d%Â (1& ) 2 d%Â i% { n|1& ln T & (1& [ 2 i ) 2 Re ( ? % re ?. d%Â & .(2) C (ln(1& re i% t h ih adside hand right the , ) 2 T | re & ? 2 d% ) i% & ? d%Â z T i & ( Ä % ) re d& & 2 2 d% ? n|1& ln & t ? T ) . | .(1) i% ( d& t T ). ))], T ( re t ). it z | File: 580J 281217 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2617 Signs: 1697 . Length: 45 pic 0 pts, 190 mm otie in contained rv hoe 1: Theorem prove on algebra Neumann von finite) (non-necessarily otie in contained ihtecaatrzto ftekre fteFgeeKdsnsdetermi- FugledeKadison's 2.5). the (Proposition of [Fa.-Har.] kernel in the given of nant characterization the with unitary. ln h pcrmo h contraction the of spectrum the hence and theory, tral that ae es o oepoint some for sense makes re Ti atsaeeti aryitiie eas h measure the because intuitive, fairly is statement last (This sln(| As u ehv led enthat seen already have we But to ii kernel limit by ti ayt e that see to easy is It & Remark I.4.3. Remark 2 { [ _ L i% (( ( 1 ( i)Ti seto ietyflosfo h eakaoetogether above remark the from follows directly assertion This (ii) T T T 2 i If (i) ] T I 0 )gvnby given 1) (0, & ) = y() eddc that deduce We (2). by )=0 T suiay o ntne let instance, For unitary. is c.[a.,p 0) If 302). p. [Hal.], (cf. . oepeiey egtfo rpsto 3 Proposition from get we precisely, More ).) d%Â Remarks )0 egtln(| get we |)0, K re % { i% 2 .Nt httecondition the that Note 1. .Teaoepofo hoe i infcnl ipiisif simplifies significantly (i) 1 theorem of proof above The 2. ( ( ? T T T K S S 0 ota finally that so =0, *) =lim )= 1 ALGEBRAS NEUMANN VON IN CONTRACTIONS % 1 sauiayeeetin element unitary a is ovrey let Conversely, . ytermr bv,w have we above, remark the By . ( ( &1 Vf T . r ))= Ä )= )( 1 T s & )= { { =( K V (( (( r I , I | ol ezr,afc hc sabsurd. is which fact a zero, be would Id & % T 0 & s ( f T )0 n ec | hence and |)=0, + re e e ( T i% )=( t & i% V ) a ntr,w ol have would we unitary, was # i% T dt T ) T T S &1 &1 I 1 eacnrcinin contraction a be ) V spoel otie in contained properly is & &1 & ) ( scnrcinwoesetu reduces spectrum whose contraction is &1 2 eteVler' nerto operator integration Volterra's the be 0 2 e f _ +( 2 0 ? ( & # ? ( M T o any for ) ln S T i% ln L =.W r o npsto to position in now are We )=1. I 1 T h spectrum the , n satisfies and ) 2 2 # &e 2 0 ) 0 1), (0, ) (1& &1 ( _ e i% ( i% T +( T M T & re =.Ti hw that shows This |=1. 2 o contraction a for ) T &1 ( I i% os' ml ngeneral in imply doesn't ) T & T ) d%Â &1 =,i.e. )=1, s ) e 1). r M d%Â i% & 2 T ? _( ih0 with I uhthat such 0 n hence and =0, 2 *) )=0. ? T & S T &1 = T 1 so course of is ) = eas the because , { ssupported is & l | (ln 2 0 Id ? I ln yspec- by r T _ <1, T Q.E.D 2 |)=0. ( T (1& na in T 313 )is is File: 580J 281218 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2729 Signs: 1738 . Length: 45 pic 0 pts, 190 mm 314 ( ue yNgoa c.[a-o.3) egv eeasihl different slightly a here give We kernel the 3]). uses [Na.-Foi. which (cf. definition NagyFoias by duced netbeoperator invertible il involves tial, lsia tutr hoesfroeaoso class of operators for theorems structure classical yLetting By utpyn hsls qaiyby equality last this Multiplying ( hence and n hence and hr n efcnandpofo h oe nqaiyfrtenumerical the for get II.1. 119). inequality we p. power corollary, 1], the a [Hal. of As (cf. proof theory. range self-contained dilation and unitary short minimal a the using without h ih,w get we right, the ro fteeitne o n operator any for existence, the of proof nvra on,isd the inside bound, universal a class ler eeae by generated that algebra show and operator, conjugacy T I I *T & & I nteCnuayo prtr fClass of Operators of Conjugacy the On II. II.1.1. nti eto,w s etraino h kernel the of perturbation a use we section, this In yuigthe using By e re = etre enl o prtr fCasC Class of Operators for Kernels Perturbed C & i% \ i% I T ( ota finally that so , T \ prtr fCasC Class of Operators &1 { ) 1 e eso ftevnNuanieult,wihi spa- is which inequality, Neumann von the of version new a 1) &1 ( K ) TT &1 +( T r , 2 r % *= I Ä and ( etre kernel perturbed & I T & 1 ))= a ii when limit a has A e I & y[i. te rÁm ,p 8) eddc that deduce we 288), p. 1, oreÁ me (the [Dix.] By . i% e = =(1& = ( T eget we , i% uhthat such T I T ,admyb sdt eoe ucl oto the of most quickly recover to used be may and *, & T { { { *& . (( (( (( T *) e I I I & sunitary. is &1 & & & e r i% ASE N FACK AND CASSIER & 2 T re re re & ) \ i% C ) { prtr fclass of Operators . T A & & & &1 I (( \ K = i% i% i% &1 - I K spstv,w get we positive, is I lse,frtenorm the for classes, T T T +( z & { & ( r I TA ( A , ) ) ) T & K t &1 &1 &1 e re ( I ) & T a ecoe ntevnNeumann von the in chosen be may % & e & ( )+ sacnrcin nadto,w get we addition, In contraction. a is +( )+ i% i% ,w logtfrayoeao of operator any for get also we ), T T T i% e T )0 eas ( because ))=0, T i% { I { *& fclass of ntelf n by and left the on T (( & (( ) &1 *) I I re & & e &1 ( & i% \ r I re re T i% Ä & = T i% i% C C &1 re C K T T 1 + I C \ \ . & ( ) \ % *) i% &1 K \ &1 oContractions to TT \ ( .As T r & T c.[a-o.1]), [Na.-Foi. (cf. ( 1,o positive a of 1), &1 , ) \ A &1 & t &1 =,adhence and )=0, ( *, 0 eeintro- were >0) & T )&1 I ) )&1 } ogtanew a get to ) &1 ) I & & I A ). & &1 re K e & & i% % T i% ( fthe of T T *on )= ) &1 File: 580J 281219 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2660 Signs: 1479 . Length: 45 pic 0 pts, 190 mm adt eo class of be to said condition ntds,adif and disc, unit ssrcl oiieand positive strictly is \ hl call shall ntecoe ntds,i iwo h relation contained the is of view range in numerical whose disc, operators unit closed bounded the all in of set the with o any for ic ehave we Since smallest one operator bounded a eseta h class the that see we of oeta ti biu rmordfnto htthe that definition our from obvious is it that Note K .If z Definition Definition rmterelation the From ikwt h Nagy the with Link II.1.2. T ( T T K scnandi h lsdui disc unit closed the in contained is )= ( z C snnzr n eog osm class some to belongs and non-zero is K ( z T \ etre enl o prtr fCasC Class of Operators for Kernels Perturbed K \ r # \- , 0sc that such >0 ) )&(1& t z D ( & K enlascae with associated kernel (T T x& h ayFisdfnto onie ihours. with coincides definition NagyFoias the , o any for r )+ , )+( .Fraybuddoperator bounded any For 2. .Abuddoperator bounded A 1. OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS t 2 ( \( &2 T I )=( T =2( \ C \ K r ) )=inf &1) \ T C z I \ ( (0< =( I 1& 1 x T oa Definition Foias I on & T & eue otecnrcin.Teclass The contractions. the to reduces # )(1& _ T I I re [\ # H =( H \ 1 re & # \ Re C + it ,0 + C it T zT so class of is |0< \ T Re(e I ( \ ( ÄT zÄ *) & T *) . *) ) \ &1 ota tmkssnet pa fthe of speak to sense makes it that so , r fissetu scnandi h closed the in contained is spectrum its if ) ÄT zÄ )&| &1 ) 1 and <1, &1 \ T I & <+ ) [ ( it &1 h prtrkernel operator the [(2&(2& I Re z Tx & | ti rvdi N.Fi ]that 3] [Na.-Foi. in proved is It . 2 +( C ( (2& o any for T r I | \ 2 & x) D T I fadol ftespectrum the if only and if cigo ibr space Hilbert a on acting &zT *T re \ and + and ) & C t ]( T \ \ T # it \ *) T )) 1& * I then , R z T & T fclass of ))( T Â &1 I # C 2 ]( 2| +2 \ aife h numerical the satisfies re \ 2 D. ? # \ I e sset: us Let . &(2& + I Z & C lse nraewith increase classes & & . r it \ re 2 K z T ] ÄT zÄ |(2& & & z ) C Tx ( &1 ) it T \ \ &1 T ) C , & eie by defined ) ) ( I 2 \ &1 2 \ 0 &1) coincides 1,we 1), . ( z # _ H D 315 ( T ). is ) File: 580J 281220 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2703 Signs: 1601 . Length: 45 pic 0 pts, 190 mm K \-kernel eogt different to belong h olwn rpsto umrzstemi rprisof properties main the summarizes proposition following The 0 \( o any For rt ntesequel the in write n rpryfor property ing II.2. H K 316 nqaiyi rsne sflosb .R amsi Hl ](.119): (p. 1] yields that [Hal. computation in simple This Halmos no 2]). R. is [Hal. there P. matrices, (cf. by two-by-two range follows for numerical as `Èven the presented for is inequality inequality power the quickly z r . T Proof The oilsrt h sfleso the of usefulness the illustrate To Proposition , ( K f t sueta eog otecasC class the to belongs T that Assume T r ( -enl hc ol edfnda h mletpstv etrainof perturbation positive smallest the as defined be could which )-kernel, ( (iii) ii meit rm()adtedfnto of definition the and (i) from Immediate (iii) r T <1 rT plcto otePwrIeult o h ueia Range Numerical the for Inequality Power the to Application (ii) yasaa.W hl s odlte odnt the denote to letter bold a use shall We scalar. a by ) i)UeteHraksieult,a ntepofo h correspond- the of proof the in as inequality, Harnack's the Use (ii) , (i) t c.I.1.2). (cf. ) ( \- T )=(1& i olw meitl rmtecrepnigpoete of properties corresponding the from immediately Follows (i) . . T kernel n n t any and (positivity), )0 K o n ucinfi h icagbaA algebra disc the in f function any For \ nteclass the in (1& r , t ( T \ 5. K r K ) ) and 0 for ) z Â f sacniuu ucino satisfying t of function continuous a is ( r (1+ # , (0) T e eabuddoeao cigo ibr space Hilbert a on acting operator bounded a be T Let K C t R ( ) r T \ , I Â K - sntcnnclyascae ihT with associated canonically not is t 2 c.I12 ( I.1.2, (cf. ) + C ( r lse.Ti a eptrgtb osdrn the considering by right put be may This classes. ?Z T z ) ( \ I | | T r )= 0 0 2 ( 2 1and <1 ; ? o any for )0 ASE N FACK AND CASSIER ? \ K K f 1,w hshave thus we 1), K (e r r z , , t ( K t it ( ( T ) T T r , )if K ) 5 t ) ( )). r T 2 dt , \- ? t ( *)= \ t = T (1+ enl e sso o oprove to how show us let kernel, # \ ) R \I ( 2 dt \ K Â z ? 2 1) r = r ?Z ) ,& Â nraiainproperty); (normalization z (1& re # , t ( it D. o any for n eoeby denote and T ( # K D (symmetry) ) r D r ) ), , o n with r any for I t . ( ehave we T .Q.E.D ). r because , (0 \- K enl and kernel, z K ( r T r <1). , t ): T ( T may ) its File: 580J 281221 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2683 Signs: 1535 . Length: 45 pic 0 pts, 190 mm hoe.W hsgtb rpsto (iii) 5 Proposition by get thus We theorem. w g C htf that When hr h function the where H, ti oee ayt rv,b sn the using say, by if, prove, ingenuity''. to specialized, or force easy is brute however either exponent is requires It case the general but The handling. dimension delicate the not n If result. the n hence and ytkn h don,w obtain we adjoint, the taking By s h atrgthn iei oiie because positive, is side hand right last The have ( Ä 2 hnrltvl aypof xss u vnte eur surprisingly require they even but exists, proofs easy relatively then =2, K ( \ t Proof Proposition T )=(1& r too n f and , 1 )1, % (ii) ( & ( (i) ( (0)=0 oe inequality Power I ( sf ; eget we , T I & i o any For (i) . & ( T # re sasltl otnos esalsea h n fthis of end the at see shall we continuous, absolutely is fTblnst h ls C class the to belongs T If ftenmrclrdu w radius numerical the If hnw then re re A ))=( & i% ( = = & i% and D f f i% OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS ( | | ( )( sT 6. f I 0 0 sT K 2 2 ( ( & ? ? e | f r ree f even or )*) f it , ( [(1& [(1& )) % re ( e eabuddoeao cigo ibr space Hilbert a on acting operator bounded a be T Let T )) z ( &1 &1 ))1. &1 )|1 & f F ) ( =(1& i% T ( ( =(1& % +(1& f I z re re w )0 n hence and ))0, ( & )=(1& ih0 with sT & & ( o n z any for # T re nparticular In i% i% )) H f f n & \ ) ( ( &1 \ re e e ) i% ) it it f I ( i% +( I )) )) re w ( D + f + sT % ( ( &1 &1 ( T & \ ), e 2 T | I # | it i% 0 )) )=Sup ( &re 2 +(1& +(1& 0 ) )) fTi opeeynon-unitary completely is T if 2 \ ? f D n &1 ? ( ? 1), &1 (1& z , . o n oiieitgrn integer positive any for (1& )) and = eget we \- i% 1i oiie yletting By positive. is &1 f &1 f kernel. F ( [ re ( re re sT re hnf then ( K T | r sT * & i% i% , i% s sin is eog oteclass the to belongs ) || )*) , s i% f f f t ( ( ), ( ( f e e uhta 0 that such e T * ( &1 it it it e ( # n h function the and )0 )) )) T )) it &(2& )) W &1 &1 ) &1 A &1 eog oteclass the to belongs ( ( ] &1] T D K K K ) s yWiener's by ) fTsatisfies T of ] , s s t \ , , ( t t T K ) ( ( T T I ) r s , , ) ) 2 t dt 1. ( s 2 ? 2 dt dt T 1 we <1, ? ? , ) ) & 9 . 2 dt such (see ? \I 317 C . \ . File: 580J 281222 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2561 Signs: 1631 . Length: 45 pic 0 pts, 190 mm ucin nteds ler A have algebra we disc the in functions eas ehave we because applies. again proof the hence w opoethat prove To meitl euefo h oiiiyo the of positivity the from deduce immediately is ato i)implies (ii) of part first that w.l.o.g. assumed eto I7 htteaoefntoa aclsetnsto extends calculus functional above the that II.7) section 0 particular In nqaiywihgnrlzsteuulvnNuansieult o con- II for inequality Neumann's von usual tractions. the generalizes which inequality the using n result: ing 318 nqaiybcuei ssailadivle both involves and spatial is it because inequality hr K where ( one prtratn naHletsaeH space Hilbert C a class on acting operator bounded & f ( .3. , rpsto a enpoe yNg n oa c.[a-o.1)by 1]) [Na.-Foi. (cf. Foias and Nagy by proved been has 6 Proposition o n one operator bounded any For Theorem oeta hsieult sfnrta h sa o Neumann's von usual the than finer is inequality this that Note T f g ( r n ) rT i)Tefrtpr f(i olw meitl rm()with (i) from immediately follows (ii) of part first The (ii) <1, h eeaie o Neumann von Generalized The ) & | ffntosi h icagbaA algebra disc the in functions of f 0 2 *) \ : ( ? r rT w , ( | t ehave we x+ \- \ (T ( \[ T *) 1), iain e snwpoe yuigteprubdkre,an kernel, perturbed the using by prove, now us Let dilation. , GnrlzdvnNuansinequality.) Neumann's von (Generalized 4 ) ) f n g x o n otato cigo ibr pc H space Hilbert a on acting T contraction any for ( . steCassier the is ( e + rT & w n eoeby denote and g it ( ) )+ ( T T rT x& n # w ) g ) C ( ( x& T e 2 w it {.Set ){0. 2 )]+(1& ( ' w ASE N FACK AND CASSIER T enlascae ihT with associated kernel s ( T ) | ( S fadol if only and if n 0 D 2 n fclass of ? K o n integer any for )1 o n oiieinteger positive any for ), | r f , ' n x any ( t Inequality s \ ( S e T )[ & = ) it ( f C TÂw )+ D t \-kernel its (0)+ # \ ), n n uhthat such r any and H cigo ibr space Hilbert a on acting g ( n x any ( T . e w sueta eog othe to belongs T that Assume g .A ehave we As ). it \- (0)]| ( )| T kernel 2 T )1. (K # . and n n uhthat such r any and H 2 o n pair any For r . ( , t K ( K T T n r r , 1 n hence and 1, ) , *. t t ( n x ( T T 1 emay we 1, e ea be T Let w | H ) h follow- the ) x x) ( S | , ( ,the )1, 0 x) 2 dt D n pair any ( ? f Q.E.D ,and ), , H,we , \ 2 r g dt ?\ <1, =2, ) of . File: 580J 281223 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2235 Signs: 809 . Length: 45 pic 0 pts, 190 mm hshave: thus where where in5(i) ehave we (iii), 5 tion Proof f ( | g (f rT ( rT F *)=(1& ( G Let . rT = = = = ( ( )=(1& e e it *) | | - _ _ it } } } )= 0 0 | | | )= 2 2 0 0 0 ? ? | 2 2 2 \ - x 0 ? ? ? 2 | | OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS f + & f ? F F | F F F g ( , ( y 0 \ 2 ( | ( ( e ( ( ( g K \ & F g ? e e e e e e it ) ) ( it & & r ( ( & it it )+(1& etofntosi h icalgebra disc the in functions two be f rT , )+(1& e g ) ) t (0) it it K it ( ( ( & (0) | )+ )+ ) T ) 0 r K K ( it 2 , ? x ) )+ t I K r r ( I ,& , y G G + | T | + F t r ( , ( ( y) | ) G T \ ( t t | e e \ ( ( y) | e y 0 ) it it T T 2 ( *) 0 & ) \f Â\ 2 ? e | )| | )| \g Â\ | ? ) ) it it f 2 y) dt x g - )| x x )+ (e ? ( ( 0.Smlry eget we Similarly, (0). 2 | e 0.Frany For (0). | | it K ( 2 y) ( it dt y) y) ) G K ? r ) (because K K , ( K t r e ( 2 , r r 2 2 dt dt dt T , t , it r ? b CauchySchwarz) (by ( ? ? t , t )| ( T ( ) t + + + T ( T 2 x T ) ) *) | ( | | x ) | 0 x 0 0 2 K 2 2 y) 2 K ? dt | ? ? 2 dt | ? r x) G ? r G G , x) , = | t = x t ( ( ( ( ( 2 T e dt e e , T | ? it it it | 2 y dt ) 0 2 ) spositive) is ) 0 ) ) ? 2 ( ? x # ( ( ? G K K K H F | ( x) r r r ( , e , , A e t n 0 and t t it ( ( ( it ( T ) T T ) 2 D dt K ? ) K ) ) .B proposi- By ). x . r x x r , , t t | | | ( ( T y) T y) y) r ) *) 1 we <1, 2 dt 2 2 dt 2 dt dt ? ? ? 2 ? dt , ? } } } 319 , File: 580J 281224 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2745 Signs: 1766 . Length: 45 pic 0 pts, 190 mm e enltgte iharsl n[a..Tepofblwi hre and shorter is below proof The [Pau.]. in result constructive. a more with together kernel bed n[a-o.4.Ntol esa nteHletsaeweeteoperator the where space Hilbert the on stay we only T Not 4]. [Na.-Foi. in 320 ehave We A hnteeeit oiieivril lmn in A element invertible positive a exists there Then II.4. ato stemxmlcoe nain usaeFo aifigthe satisfying H of F subspace invariant closed maximal properties the two is following T of part because inequality, eaal ibr pc H. space Hilbert separable all over supremum the taking By where prove to 10): used proposition be may 2], It Fa. 4. [Cas. theorem (cf. from result follows following immediately the 49) p. 11.4, eaal ibr pc H space Hilbert separable a on h norm the h n ftetermflosb setting by follows theorem the of end The na ale eso fti ae,w o hsrsl yuigtepertur- the Proof using by result this got we paper, this of version earlier an In h olwn ojgc hoe ssrne hntersl fsimilarity of result the than stronger is theorem conjugacy following The h o emn' nqaiyfor inequality Neumann's von The Proposition Theorem uhthat such fclass of [[ \ (ii) i)ln( (ii) . prtr fCasC Class of Operators (i) (i) x d& ]] o any For . = 2 & A + & =Inf F C A A dm T &1 ( A \ , 5 e d+ & & x . & &1 Ai contraction a is TA , ( } +( } x \ T & it & 7. { , TA A )+ 1 cs u eso htapstv netbeoperator invertible positive a that show we but acts, 1) e ea lmn navnNuanalgebra Neumann von a in element an be T Let sasltl otnoswt epc omfrayx any for m to respect with continuous absolutely is A x i inf lim \ , r &1 &1 x Ä &1) x e eacnrcini W a in contraction a be T Let G Âdm sacnrcincnb hsnin chosen be can contraction a is & 1 & # ( & 2 e sbuddb nvra osatdpnigonly depending constant universal a by bounded is H : it \$ Â\ )# | )= set , D \ \ L &1. " ( Then . 0 ASE N FACK AND CASSIER f \ 1 k Let . sueta eog otecasC class the to belongs T that Assume ( : ( =0 >1) q e S & 1 , , u it ; m y )+ k h upr ftecmltl non-unitary completely the of support the x r ojgt oContractions to conjugate are - ), = nteui alof ball unit the in g C K o n x any for ( e \ ru k q it - ( =0 prtr c.[a-o.1,Prop. 1], [Na.-Foi. (cf. operators )+ T \ 1 Q.E.D =1. ) T x 1& k k \ x " # k 2 \ F earpeetto for representation a be d& [ *-algebra & M f ( u [ (0)+ M H uhthat such )| 0 ] egttedesired the get we , nsc a that way a such in . x = g (0)]. M k : =0 q M cigo a on acting T \ cigon acting k ( x \ # k >1). = F , ; x. File: 580J 281225 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2494 Signs: 1220 . Length: 45 pic 0 pts, 190 mm tflosfo h w atieulte that inequalities last two the from follows It uhauiayoperator unitary a such h eiiino h om[ ] esethat that see we prove ]], [[ to } [[ Finally, norm the norm). of definition the [[ emn algebra Neumann h ucinl[ ]tiilystsisteprleormieult;there- Let inequality; norm. parallelogram hilbertian the a satisfies is trivially it fore ]] } [[ functional The representation the considering by hand, other the On of representations such all over infimum the taking By rmteaoetermtgte ihterm1that 1 theorem with together theorem above the from Remark Ux Ux ]]=[[ ]]=[[ .Let 1. & x x x& OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS ]frayuiayoperator unitary any for ]] ]i biu.Q.E.D obvious. is ]] 2 lim lim = lminf lim = lminf lim = M [[ T r r & _ _ _ +(1& k r r : Ä Ä =0 x q ftesetu of spectrum the If . Ä Ä x - & - & ea lmn fclass of element an be 1 1 - 1 1 ]]= & & (T - & & U | | D K D D 0 & [[ 2 2 | - \ omtswt h kernel the with commutes k ? ( ru x& \ D & } ) x & x ( 1lminf lim &1 \ K (x k " Ax k T : =0 ]] q D k ru k k q | : =0 ) : =0 A q 0 =0 ( q & x) - A & x& T u | r k x) e eog to belongs Ä 2 ) u etepstv prtrsc that such operator positive the be u - k q ikt ( d& \ x k k 1 =0 K 1[[ &1 & 2 ( - - ( | o any for \ u K ru )d& x) u K &1 K ) ( r k T , ru ru t - T ( ( ) U ( T T x T & x K ( scnandin contained is ]]. x ) u ) ntecmuatof commutant the in k ) x ru M & x & ) x C T x | . ( A k k x) T k tsfie oso that show to suffices it , \ " & | sacnrcin(o this (for contraction a is & in ) x) 2 ( } } x \ & d& H. d& k A 1 nafnt von finite a in 1) + & K ( x ( &1 2 u dm ru x u = d& ). T eget we , ) ( & T Ix ( 2 ( scnuaeto conjugate is t u ,adequality and ), ) eget we , ) S \ 1 tfollows it , 1 From &1. M But . 321 File: 580J 281226 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2619 Signs: 1833 . Length: 45 pic 0 pts, 190 mm i n)pa oei h netgto fteivratsbpcsfor subspaces invariant any the to of associate investigation shall the we in chapter, role this a play any) (if and enl ecnpoe(f Cs a ] htteqaiseta measures quasi-spectral the that 2]) Fa. [Cas. (cf. prove K can We kernel. eoe h lsia eopsto of decomposition classical the recover cigo eaal ibr space Hilbert separable a on acting eaal ibr space Hilbert separable M measure Ba.4,adso h xsec fivratsbpcsta r o of not are that of subspaces III.1. invariant notion of the existence of the type kind. show interpretation this and functional conceptual 4], of more [Beau. subspace a give invariant we particular, algebra netgt h xsec fivratsbpcsfor subspaces invariant of existence when the investigate (1) of solutions 322 ucinlcluu ysaigo h ibr space Hilbert the on staying by calculus functional beginning the at stated 2 theorem the proves of This operator. unitary some ftecmltl o-ntr at hsas lost eiethe define to allows also This part. continuity absolutely non-unitary the like completely results, related the the of of much and parts singular hs upr contains support whose kernel r Let III.1.1. Remark , + 9 I.TeAypoi ucinlCluu n t Applications Its and Calculus Functional Asymptotic The III. t admc oei fact). in more much (and I ( T satisfying h prtrS Operator The ) T K dtÂ M r + , eacnrcini o emn algebra Neumann von a in contraction a be h oeC Cone The T t Set . 2 ( .Let 2. ? T on , ekycneg,when converge, weakly X C S )in + 1 ( uhthat such T T M )= ea prtro class of operator an be T + C + n h oeC Cone the and ( T ( [X ttrsotta h o-eoelements non-zero the that out turns It H. T S hc svr sflt opeeydsrb the describe completely to useful very is which , T f 1 .Let ). ( a ainlycci etr hsalw to allows This vector. cyclic rationally a has )= & T ASE N FACK AND CASSIER # M )= 0 2 _ [X ? ( T T e | | int X 0 # .Ti us-pcrlmauealw to allows measure quasi-spectral This ). * 2 T ? XT M 0and 0 d+ H f eacnrcini o Neumann von a in contraction a be ( T n eoeby denote and , e | = X r + it ( T T Ä t ) ( osdrdb .Bazm in Beauzamy B. by considered T ))= * in T d+ XT noasltl otnosand continuous absolutely into 1 ) M & T C T ( oauiu quasi-spectral unique a to , = \ + t T n ). ( X] \ o n integer any for aifig()a operator an (1) satisfying *XT 1 na in 1) , H = C i h formula the via , K 1} X] r , t otatos In contractions. M ( . T W t Cassier's its ) *-algebra cigo a on acting n 0 and 0, T X H .In (1) M in File: 580J 281227 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2818 Signs: 1989 . Length: 45 pic 0 pts, 190 mm n that and pc rbe for problem space satisfies: ainaottesqec ( sequence the about mation T ( lmnsin Elements rw o quickly. too grows satisfies and basis, canonical the in diagonalizes ( eaal ibr space Hilbert separable pcsdntapyt hscnrcin eas h eune( sequence the because sub- invariant contraction, non-trivial this of to existence the apply prove don't to spaces 4] [Beau. for in developped cyclic is vector contraction The o ].Nt that Note 4]). Fo. ls C class hf eie ntecnnclbss( basis canonical the on defined shift C ovret ( 0 to converge nasprbeHletspace Hilbert separable a on eaal ibr pc H space Hilbert separable a of osdrdb .Bazm n[eu ]aejs enl fpositive of kernels just are 4] [Beau. in Beauzamy for B. isometrizers by considered T n ( n ealta contraction a that Recall h olwn em umrzssm ai rprisof properties basic some summarizes lemma following The Example. Lemma esalseltrta h nain usae f`fntoa type'' ``functional of subspaces invariant the that later see shall We # T * x T n Z . Ä )= S T (ii) (i) ,where ), T n 0( ,} 0, ( ) [ e n 0 ; 1 R n n ] 1. )= S * h optto of computation The . Ä T C n sapstv otato in contraction positive a is C 1 e sijciei n nyi so ls C class of is T if only and if injective is Let + + e eacnrcini o emn algebra Neumann von a in contraction a be T Let OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS w n + n ( n T Supp( T ( 1if =1 T Ä T hssosteitrs of interest the shows This . T ) C o any for ) C R ilb called be will ) a o fnntiilivratsbpcs ic no since subspaces, invariant non-trivial of lot a has + for eacnrcini o emn algebra Neumann von a in contraction a be 1} T + scnandin contained is T ( -contractions. T n C c.[eu ].Nt oee httemethods the that however Note 3]). [Beau. (cf. H n o any for ) 0and 0 sacne lrwal lsdcn in cone closed ultraweakly convex a is ) ) + n # 0 and 0, ssi obe to said is ( . T T Supp( x T ehave We hn h toglimit strong the Then, H. ) # n & navnNuanalgebra Neumann von a in x H ) (& S w n S 0 x rs.i h eune( sequence the if (resp. T n T C = # oiieioerzr o T for isometrizers positive sotnitrsigt e oeinfor- more get to interesting often is ), o ntne let instance, For . C + H e 4 1 S + (T n S and n fcasC class of & if M T ) ( n ( T ))= n [ # hc eog to belongs which , e 0 ti ayt hc that check to easy is It <0. o any for ) 0 n Z )= ] T C of .If ). [ =0 0 + 4 ] ( l 2| 1 ,} 0, , T 2 n T ( | fadol fTi of is T if only and if (resp. , nteivratsub- invariant the in ) Z e 1} R so class of is n )by ; S T ntecommutant the in T T eteweighted the be ftesequence the of for T M fcasC class of n x ( e S C & ) M cigo a on acting n n T T )= + 0 n c.[Na.- (cf. : C & M ( M cigon acting <0. T 0} n w doesn't x& ). acting then , which n 1} e ) 323 n n )if S 0 +1 T File: 580J 281228 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2562 Signs: 1419 . Length: 45 pic 0 pts, 190 mm ned h operator the Indeed, n S and n hence and get n hence and n hence and ucinlcluu that calculus functional oeta 1 od ihu suigthat assuming without holds (1) that Note S lshefcos eddc rm()adtepoete fthe of properties the and (1) from deduce we factors, Blaschke f e now Let 324 ucin y[a. Crlay65 .8) ota egtfinally inner get of we set that the so in 80), dense p. uniformly 6.5, is (Corollary products Blaschke [Gar.] by all functions of set the But f lshepouti h eklmt(in limit weak the is product Blaschke refrinrfunctions inner for true : : f ( ( ( Proof z T T )=( ) )* (iii) (iv) = (v) T S = S h nynntiilasrini i) oefrtta ehave we that first Note (iv). is assertion non-trivial only The . T : B T ( f f & f I ÄI :Ä ( ( : # S fTi o fcasC class of not is T If S o n ne ucinf function inner any for T S ( T z & T T H B T )* ) )* (T Â sa isometry an is T )= sivril fadol fTi ojgt oa smtyin isometry an to conjugate is T if only and if invertible is T (1& ) S S = ( S *) T D T S f B S ( f T esm n ne ucin o n integer any For function. inner any some be ) T S S Äz :Ä ( ( T S ) T T T T yletting by , ,wee| where ), f R o n iieBack product Blaschke finite any for ( = )= : )= :I o n ne function inner any for ( = T f f ) : & ( = S S f T fteform the of : f T T T ( S : )* ASE N FACK AND CASSIER T ( )=| R T ; =( =| n en otato,w have we contraction, a being ) o n ne function inner any for o n niieBack product Blaschke infinite any for : )* S S 0} <.Ide,w e o any for get we Indeed, |<1. n T S : : I f Ä S , : # & f | | f ( 2 2 : T T ( then ( H T T R ) + S :T f = * ) : f ) T = ( : n S H & T S ( *) ( & T D z S h ees nqaiyi trivially is inequality reverse The . T )= S )=( :T T f S .( ); f ( T : & ( ( D T & T f S *S : ( =1, )o h iiepout fits of products finite the of )) ( : :T T T I )* T & & .( T )) n & sCNU sayinfinite any As C.N.U. is *S =S ÄT :Ä z f n hence and ) ( ÄS :Ä , Â T T (1& ), n fTi C is T if and & T f ) T , n f in T , ÄS :Ä : + Äz :Ä H T fteform the of S ,wee| where ), T T + 1# ( : D # S ). _ B D T . N ( n . S 0 we 0, . U T : ). . |<1. then H M 2 1 ) ) , File: 580J 281229 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2577 Signs: 1437 . Length: 45 pic 0 pts, 190 mm cigo ibr pc H. space Hilbert a on acting h function the g of ogy Letting nqeprilioer U isometry partial unique a o n lsheproduct Blaschke any For uhthat such isometry partial | hence and hence and get n i)i rvd Q.E.D when (p.1) [Beau.-Rom.] proved. is (iv) and hoe c.[a.,term64 .7)acmlxnumber complex a 79) p. 6.4, theorem [Gar.], (cf. theorem o,if Now, isometry lshepouto oecntn fmdlsone modulus of constant some or product Blaschke Proposition Proof h olwn rpsto scoet N.Fi ](.3 n 5 and 65) and 39 (p. 1] [Na.-Foi. to close is proposition following The (ii) (i) oiieisometrizer positive i Let (i) . n B U T f Ä oiieeeetXin X element positive A X ( X * T sa ne ucinin function inner an is in # - + XT B )* # S C M XT = OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS B C n + ( 8. = T B + f (T U ihsm upr as support same with ) n sn 2,w get we (2), using and : ( ( T |= T 0 T X e eaC a be T Let ) f * in ) S .Asm ovreythat conversely Assume ). sijciei n nyi U if only and if injective is saBack rdc.B 3,w get we (3), By product. Blaschke a is n - - f eapstv lmn in element positive a be = ( : S M ( - T X XU T f ) f yplrdcmoiin hr xssaunique a exists there decomposition, polar By . XT : = . = : ( ihspoteult supp( to equal support with X ( = B T T S S = - sueta a es range dense has T that Assume )* U )* - )* B T U etu e,b writing by get, thus we ( S X n T n T - XT 1} - T ) = , g f o n lsheproduct Blaschke any for = X tgvsajsiiainfrteterminol- the for justification a gives It . M : ( X cnrcini o emn algebra Neumann von a in -contraction XT ( T - # T = = S )* si C in is M H f ) XU : U - n = n ( X f g ihsm upr sX as support same with ( - XT T T ( ( n uhthat such and * D T )) * X U =S o n integer any for + ) ,teeeit yFrostman's by exists there ), XT . ; n - ( f T : X = f M X ( T ) (T T , # = fadol fteeexists there if only and if X fteeeit partial a exists there If . X ) ) , C n , - sunitary is + X- B ( T U = - .W have We ). - f XT X : B . : g X n = .(3) . # 0. = where )=supp( D X , - , uhthat such uhthat such XT g ,we sa is 325 X M ), File: 580J 281230 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2809 Signs: 1590 . Length: 45 pic 0 pts, 190 mm kernel X rmtedefinition: the from egbrodo h lsdui ic ehave we disc, unit closed the of neighborhood supp( III.2. h olwn rpriso h smttckernel asymptotic the of properties following The that of T 326 K e sseta h ia upr of support final the that see us Let iewt h lsr fterneof range the of closure the with cide eaal ibr space Hilbert separable ml that imply r a es ag,s htI ( Im that so range, dense a has fteei ocnuin esalsml write simply shall we confusion, no is there If Let Definition Let , h operator the t U ( ii o any For (iii) i)If (ii) T K i)Frany For (ii) X i o any For (i) h smttcKre K Kernel Asymptotic The X ) r K T , steiett.Cnesl,if Conversely, identity. the is =dand )=Id t h smttckre soitdwt T with associated kernel asymptotic the ( # r Xf , X mlilctv omo h kernel) the of form (multiplicative eacnrcini o emn algebra Neumann von a in contraction a be t C ( U )# T K + ( X rT , r ( suiay oti n,nt httefnlspotof support final the that note end, this To unitary. is M , K X .Frany For 3. T t sijcie h nta upr of support initial the injective, is ( )= sapstv lmn in element positive a is ) r ,where ), For . T , aypoi omo h kernel) the of form (asymptotic t X , ( T X | X r 0 sijcie Q.E.D injective. is 2 , r )= and ih0 with ? # K X X f ih0 with C )=(1& r ( = , (0 H. X e t + T ( it ( S T ( ) I X T sacnrcini o emn algebra Neumann von a in contraction a is ASE N FACK AND CASSIER T & =toglim )=strong K r esalset shall we , ,any ), f # 1and <1 r r re ( , C 1adany and <1 r rT t r , ( & r 2 t - + T ( )( n 1adayfunction any and <1 it *) X , ( Ä T XT I T X r ) & U + - ) X ,w hl alkre soitdwith associated kernel call shall we ), ) ih0 with &1 re = steiett o,afc hc will which fact a too, identity the is XT 2 sdnein dense is ) dt ? +( it | T U C t 0 T 2 u Im( But . K # ? * + I = * t & r f R &1 ( n , ( # U T t Â r e ( K re 2 . ) R 1adany and <1 T & satisfying ) T ?Z r X K , it , Â it )= U X t 2 K T ) ( ( r ? , ). T I H K = - r *) t & , ( K suiay ehave we unitary, is Z ) t T r ( n h ia support final the and X &1 U , T r ehave we , T re , t , ( sdnein dense is ) t T X , n T ( & X , . S X )= X , f it M T & r immediate are ) X T n hl call shall and ) steidentity. the is nltco a on analytic t ) ) K X # &1 2 cigo a on acting dt r ? R , . t . ( Â X 2 U ?Z .Note ). H on- co@ the , and M . File: 580J 281231 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2786 Signs: 1506 . Length: 45 pic 0 pts, 190 mm us-pcrlmeasures quasi-spectral lim egbrodo h lsdui disc unit closed the of neighborhood x, npriua,if particular, In o n ar( pair any for if and, polynomials, positive in values with ihrsett h armeasure Haar the to respect with esr oooy when topology, measure eacnrcini o emn algebra Neumann von a in contraction a be T eaal ibr space Hilbert separable H, rmtepstvt of positivity the From Lemma Let Proof y n n X and # Ä ii o any For (iii) i)(smttcvnNuaninequality): Neumann von (Asymptotic (iv) i)Frany For (iv) & (ii) H; (i) [ + T f aepofa o hoe .Q.E.D 4. Theorem for as proof Same . ( T eacnrcini o emn algebra Neumann von a in contraction a be + & .(eeaie o emn nqaiyfrisometrizers). for inequality Neumann von (Generalized 2. d+ + # K [ *) T T T C n f r X x , ( + ( f X + , t , OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS M rT f ( , T X P T ( + T g =eklim )=weak ( n , T y T hc aife supp( satisfies which , x & ffntosin functions of ) d+ , *) ( , ) sCNU,temeasures the C.N.U., is X Xg ). X x x f , X X X X o n ope otnosfunction continuous complex any for ) & Q ) hnfraypair any for Then , ( ( # | f + T in T & ( 0 2 T C ) X & )] y ? Xg ) ( X C + & H x K | t + , r x & )= f + ( + g ( r Ä T (e X & , ( rT K ( T (0 and n T t , T T ( n any and ) , Ä & r ) P r T x ,t oiieqaiseta measure quasi-spectral positive a to 1, = )] , y ,w have we ), , it + ( - t ( sC.N.U. is , x ( )+ e t X r K T X )= x it & 1ad0 and <1 H ) m & ,w get we ), ,and ), r X , T , # g 2 Q t n x any & f ( ( * C on and T ( e ( ( n e e + + it S , 2 0 + it it )| x T ? 1 X ( S ) ) T ); ( T ) 2 | # 1 g ) d+ # # ( f + ; f (K .I ses opoeta the that prove to easy is It ). , ( H dtÂ f ( X n n r any and H e g e supp( X ) , it ehave we , ) + & x , 2 T r ) M K x , , T ? it y , d+ faayi ucin na on functions analytic of t n , ( y )+ r and , x r boueycontinuous absolutely are , T ( t cigo ibr space Hilbert a on acting , t t + X ovrei h weak the in converge 2 ( , x o n ar( pair any for ) T , T g X x ( ) , .Mroe,w have we Moreover, ). ) e y ? + ( it x ). T t ,0 )| K ) , | x 2 (+ x) r , f M , d+ x t on ( . T r T 2 X dt <1, cigo a on acting ( ? , ) S f x . , 1 ) x n any and , x ( P ehave we t ), , | Q y) )of 327 Let + = X File: 580J 281232 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2695 Signs: 1691 . Length: 45 pic 0 pts, 190 mm where L eaal ibr pc H. space Hilbert separable 328 eegengiil,i ntr.Ide,teasltl otnospart continuous absolutely the Indeed, unitary. is negligible, Lebesgue ti o ifcl ocekta h map the that check to difficult not is It euaysfntoa ersnainfor representation functional Beauzamy's ftecoe ntds,where disc, unit closed the of o any for ec qa ozr ypooiin9 hswudipythat imply would This 9. and proposition measure, by Lebesgue zero the to to respect equal with hence continuous absolutely be would of atwihi bud eae eutwsotie yA tmn using Atzmon, A. 26-27). by p. obtained remark was 1], result [Atz. related (cf. A methods absurd. other is which fact when T have ntecmlmn of complement the in soitdwith associated kernel L n hence and 2 Remark Proposition h olwn rpsto hw h pcrlntr fthe of nature spectral the shows proposition following The Proof Remark ( sCNU,ti oa ucinlcluu oniewith coincide calculus functional local this C.N.U., is T ( S S 1 szr,bcuei o h smttcqaiseta esr of measure quasi-spectral asymptotic the not if because zero, is 1 , r , a + K Ä m o any For . n x x r , , ,adw aefrany for have we and ), sthe is # x 1 t S ( ,gvn salclfntoa aclswihextends which calculus functional local a us giving ), .Any 1. .Fraytiooercpolynomial trigonometric any For 2. & H T + 1 o n otnosfunction continuous any For . , & T n n ar( pair any and X ( _ + 9. n P )=(1& + ( X hFuircefcetof coefficient Fourier th X ) T ( x ) e eacnrcini W a in contraction a be T Let f and t # = + ) C # _ x X supp( & 1} R ( = r ( N T x -contraction Â 2 f 2 : )( n S )in # )= ?Z o n X any For : 0 n I 1 ASE N FACK AND CASSIER H + <0 & & f X . | , a S Supp( uhthat such _ re g a ). & 1 ( faayi ucin naneighborhood a on functions analytic of ) n etu have thus we , n it f T T T T # + ) f * *) # * ) L X | f n , n # x &1 ( | supp( T S , e C S ( x hs eihrlsetu is spectrum peripheral whose , it S T T + C X P ) x e 1 f x stesaa pcrlmeasure spectral scalar the is , d+ it ( 1 ( + ( Ä T + f m n I cnrcin c.[eu ].If 2]). [Beau. (cf. -contractions + Â & on X ), # X n ) _( + 0 : 0 ( Z ). T re ehave we t T : S )=0, n ). ( & a ,tenr ii fthe of limit norm the ), P 1 *-algebra n it P N ) ihspotcontained support with S T = x a T ) n &1 T nqeyetnsto extends uniquely S n T & x xssadi zero is and exists T , N f n Ä M x n T + . < + ac cigo a on acting X X N 's is ( a f n C Q.E.D ) z n x 0} ,we T T ,a on ac ac File: 580J 281233 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 3215 Signs: 2211 . Length: 45 pic 0 pts, 190 mm prxmto,issfie opoeta ehave we that prove of to spectrum suffices the its outside approximation, poles with function rational re rsrigsreto rmL from surjection preserving order in + for atfl(i.e. faithful on to ....that w.l.o.g. salna ujcinfo L from surjection linear a is TisC III.3. prxmto,uigtepeiu oml n h us-pcrlmeasure, quasi-spectral that the get and formula we previous the using approximation, hsi biu o n element any for obvious is This that eas ehave we because C meitl mle httersrcino h map the of restriction the that implies immediately oyoiladthe and polynomial ini h erho nain usae for subspaces invariant of search the in tion when equation operator the of solutions the of description olw rmlma1(v,b sn lopoet i)o the of (ii) property also using by (iv), 1 lemma from follows eaal ibr pc H space Hilbert separable Then T + yuigteaypoi pcrlmeasure spectral asymptotic the using By Proof Theorem , L C x ( S y , T eodStep Second is step First ( 1 , + x = ti la httemap the that clear is It . T nteOeao qainT Equation Operator the On ( ,i aoie by majorized is ), T . h map the T N S uhthat such ) , o any for R is fal oethat note all, of First . 1 y . a ainlycci etr(emyawy sueti condi- this assume always may (we vector cyclic rationally a has U , , ( y + T ., T sasltl otnoswt epc to respect with continuous absolutely is ) + S 6 ehv C have we x + ) . T : T with , + T OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS h clrmaue+ measure scalar The ( 0 eoeby Denote {0. , y f e eacnrcini W a in contraction a be T Let rsre h re.T rv h hoe,w a assume may we theorem, the prove To order. the preserves : y , T = implies )=0 y ujciiyo h a f map the of Surjectivity # *+ sasltl otnoswt epc otemeasure the to respect with continuous absolutely is : o h measure the So, H. & R X i T d+ + r nteoe ntds n outside and disc unit open the in are ( ( & z ( S f 1 h element The 1. T )= T ) , T T . y )= n aife poet ii fthe of (iii) (property satisfies and , , = sueta a ainlycci vector cyclic rationally a has T that Assume y P ( + ( t ( C + z f )=| S Y f x =0). ) T + T , f 1 Ä Â x , Ä ( ( ( ainlycci etrfor vector cyclic rationally a * , y + z *XT x f R f XT + ( & + T T ( + o n ope otnosfunction continuous complex any for ) S fteform the of ( T T , T T e ) 1 x : + ( = , , )) it ( = ( 1 noC onto x f + T )| f )( f , X slna,adta t restriction its that and linear, is ) o n ne ucinfi H in f function inner any for sbscfr+ for basic is )# T 2 X ) x z Ä , d+ ) & x + + Y C . T + T + T : ( = , ( noC onto , T T 2 x T .Mr rcsl,w have we precisely, More ). T x )}}}( , ( , o any for ) 2 1 x emygv complete a give may we , ), x *-algebra f y (Re( ( .Let ). = t sbscfor basic is hc nue faithful a induces which ) f z Ä + R T T + & X ( nfc yuniform by fact In . ( T ehv oprove to have We . + T T )+ , : T x ). ) k , x X ,where ), ( x S where , nparticular In M f f o any for T _ ea element an be # )to eog to belongs ) ( L T + cigo a on acting + T X X . .Btthis But ). + C , , ( x T x S , + , y P R 1 y This . ( , y s By 's. 's) T + ( # sa is sa is 329 D T )is , H. if ), ). . File: 580J 281234 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2450 Signs: 1298 . Length: 45 pic 0 pts, 190 mm yrtoa ylct of cyclicity rational By pcrmof spectrum hoe stu opee Q.E.D III.4. complete. thus is theorem o any for on pcrm nti eto,w hl soit oaynnzr element non-zero any to associate shall we section, X this In spectrum. point oiie. positive d+ f Let prtr.I olw that follows It operators. finally ntesm a,w e Im( get we way, same the In hspoe h ujciiyo h map the of surjectivity the proves This 330 esalrcvrti a h nain usae `ffntoa type'' functional `òf subspaces invariant the way this recover shall We h bv ro ple to applied proof above the # Let ial,if Finally, Y in H , x , C h oeC Cone The . x T ( = + D y eaCNU otato fclass of contraction C.N.U. a be ( # lma1,adhence and 1), (lemma ) .d+ T in eafnto in function a be n hence and H, uhthat such ) (YR T L T ehave we , T , x ( ( sCNU,w have we C.N.U., is Re , S T x + 1 o n ainlfunction rational any For . ) , X ( ( T + x X = T ) )= | x Re uhthat such ) _ R n h nain usae rbe o T for Problem Subspaces Invariant the and (Yy Y eddc that deduce we , p ( + ( T ( = T T X ASE N FACK AND CASSIER ) ){ ( + )+ | X x) X f T y) 1 )= ,where ), ( < isedof (instead = = = = = i . = Im( eas both because ) + | | | | (+ o-rva nain usaefor subspace invariant non-trivial a X C L 0 0 0 0 (+ T 2 2 2 2 ? ? ? ? = X + ( T . d+ | | T f f ( ( R R )= S ( + S 2 T ( Ä . ( T ,where ), C e ( ( Y T 1 . )= e e ) = it , , 1} + + ( it it ) R ) + R . )| )| T T ( y d+ S Y T n eoeby denote and , T ( C ). 2 2 ( ( T f f ) | ,sosteeitneo a of existence the shows ), ,0 ), ( + 1 f . d+ x T f T y) ) 1 =2 , , .Nt ht if that, Note ). ( ) ( R R + R x e Y Y f f ( ( T it o n ne function inner any for T ( , 2 | if . ihplsotiethe outside poles with x T ) ) and ) R , sin is x 2 x d+ &1. x ) h ro fthe of proof The )). ( ). , ( ( . t R T t ) T ( ) 1 uhthat such 1, + T ) , x T ) L x) , x x ( ( . ( t . _ t ( ) r positive are ) ) S p ( X 1 T , + # t pure its ) T C ,and ), + ( T T ), . File: 580J 281235 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 3003 Signs: 2233 . Length: 45 pic 0 pts, 190 mm inltp o r xcl h enl feeet nC in elements of kernels the exactly are T for type tional clrproduct scalar extension + a ylcvco x. vector cyclic a has oe subset Borel hyperinvariants . suethat assume nasprbeHletsaeH space Hilbert separable a on type'': to ``functional Beauzamy's meaning of of operational subspace an adaptation invariant gives easy original of theorem an the next notion is The the than it framework. general although our more to 2-4], ideas slightly [Beau. is in type'' given ``functional one of subspace of seulto equal is a class us of give contraction type. also C.N.U. functional will of this aren't but which subspaces 2-4]), invariant [Beau. of (cf. kind Beauzamy new B. by considered sauiayoeao in operator T unitary a is L any fteform the of oevr as Moreover, epc oteHa measure Haar the to respect lmn X element eko that know we eapstv ucinin function positive a be o oefunction some for 0 Proof Theorem III.4.1. sa is sapstv ucinin function positive a is on ( S is Step First t 1 # S , fortiori a 1 + B .As usae f`fntoa ye'aedfndi Ba.24 sthose as 2-4] [Beau. in defined are type'' ``functional of Subspaces . 0 o n integer any For . .Ltu write us Let ). f eak nIvratSbpcso ucinlType Functional of Subspaces Invariant on Remarks T ( # + x 7 C T of B . | T T T , + OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS : . x y , f[,2 [0, of T sCNU,temeasure the C.N.U., is ( suiay h measure the unitary, is =lim )= ylcvco for vector cyclic a x e eacnrcini o emn algebra Neumann von a in contraction a be T Let Existence a ylcvector cyclic a has T L because oteHletcompletion Hilbert the to ) f n uhthat such ( in Ä S Then + 1 L ? , + with ] B + , L T (T 0 ( , ( x L H = S C o n oiiefnto f function positive any for dniiswith identifies ) E h B The 1 n sin is 1 ( ( c.[eu-o.) h ylcvector cyclic The [Beau.-Rom.]). (cf. ) 1} = , .dm n ( 1 h one function bounded the 1, S S x . + f 1 m 1 H cigo eaal ibr space Hilbert separable a on acting sueta sC is T that Assume ( + , 0 , | T .I at hspeetto ftenotion the of presentation this fact, In ). + 0 m T H . & ( on ) 0 hr h aoioy derivative RadonNikodym the where , T S Beauzamy ). n .Teeeit constant a exists There ). u Ker( y) hc eemnsaseta measure spectral a determines which , 1 and & | ttrsotta h canonical the that out turns It x. S & + = + 1 B )= T n hence and , 0 T f , =,sc ht0 that such )=0, (S oniewith coincide x ( , T (Xu extends x T )) ' sasltl otnoswith continuous absolutely is nain usae ffunc- of subspaces invariant s x H L | | of y) )frayu any for &) ( S . N T H 1 , .As . U + H ( # ihrsett the to respect with x T ., + , , L x y T fcasC class of ( x , f , S + x # x n ( )= 1 scci for cyclic is , = ( H S Let . net into injects ) x T f , 1 , C ( ) f/ , & ). n hence and , L e 0ada and >0 + # it B 0 hyare They M ) Let H. +1 ), ( 1} S and H, T acting 1 fan of C x , , Ân ea be + and 331 for for 0 T is ). f , File: 580J 281236 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2047 Signs: 837 . Length: 45 pic 0 pts, 190 mm n ec yletting by hence and T f etu eieasqec ( sequence a define thus We titypstv,s htteeeit function a exists there that so positive, strictly 332 epc to respect because o n ar( pair any For o any for n =| e snwpoeTerm7 oefrtthat first Note 7. Theorem prove now us Let o any for h n | 2 y S ..Set m-a.e. # ( 1 X + ( & f H f 0 ( # etu get thus We . ( T n ec togycnegst oeelement some to converges strongly hence and , B T C u ( ) ) , f + u & is u ( & - ( T | fpit in points of ) T + | & ) XTy (X & 0 )+ .Ide,w aefrany for have we Indeed, ). u )+ ngiil,and -negligible, n X | Ä n 1 n & n x n 1 & = )= ( + 2 | ( T = u T y) h ASE N FACK AND CASSIER u (X X n (T ( | = = = n T | & ) & f )= n )* H u * | | (S )= 1 = 0 0 = = XTy 2 2 ehave we , | S ? ? | (X &) T | | T \\ 0 feeet in elements of | f 2 h h 0 0 n ? 2 2 h + ( n ? ? n n | n e ( | ( f T ( h u \ \ e T y) + it , T n it u ) f/ f | ) ( , )| )( + n 1 d+ e o any for &) = & y B 2 + it + sasltl otnoswith continuous absolutely is n 1 )| | d+ T (Xy ( + h n , T 2 1 y n n 1). y T ( , d+ E ) ( + y e , T # u ( y =Ker( it o any for h ( | t , T ) E ) C e y ) | n y) , ( u it y) d+ u & + t , ) , in + ) & =0, ( & d+ ( T T t , # ) H u X hc satisfies which ) T , H , & sivratby invariant is ) u ( n . , t ( 1, & ) D ( t X uhthat such ) ) f # C + ( T ). File: 580J 281237 . By:BV . Date:27:02:96 . Time:11:48 LOP8M. V8.0. Page 01:01 Codes: 2508 Signs: 1499 . Length: 45 pic 0 pts, 190 mm ucinin function pcso ucinltype functional of spaces that n hence and H sw have we As hr exists there rvsthat proves hence and nteohrhn,w nwb tp1ta hr exists there that 1 Step by know we hand, other the On omtn of commutant e sso that show us Let eget we athyperinvariant fact htteeeit oiiefunction positive a exists there that enl feeet nC in elements of kernels hyperinvariant). that any for hence and where (Xu ( | orhStep Fourth hr Step Third eodStep Second D E 0 2 ? =Ker( uhthat such ) g X f =| | & ( = &) e # it n hence and H, u X f - L ) X = E | Ker( # d+ f 2 X = XTy n ec Ker( hence and , emyasm ....that w.l.o.g. assume may we , (+ OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS T ( ( : sivratby invariant is ). R T S X f yse ,teeeit eune( sequence a exists there 1, step By . : , enl feeet C elements of Kernels : 1 ( * T u E f Beauzamy , T Let . , XR Beauzamy 0 tflosthat follows It =0. ( + X & =Ker( in ) ( f X 0 t .Cnesl,if Conversely, ). u ) .A ehave we As ). )=( srn lim =strong srn lim =strong C u | + & | Let . X )= Ker( ( + f &) f T X ( ( ( ein be T ' T uhthat such ) T .If ). nain usae ffntoa yeaein are type functional of subspaces invariant s = (Xu ' ) nain usae ffntoa yeare type functional of subspaces invariant s .Let ). ) f u | X n n u T ( 0 because =0, X 0 2 Ä T Ä u C )=Ker( ? | | ))=Ker( w hl hwblwthat below show shall (we + + # & ein be f + &) )= ( E ( f e E T ehave we , it h h u in = ) + (X and ) n n Ker( # d+ ( ( ( XTy C H T L X T T o any for f + T )* f )* u g ) & )= , ( ( ( u | R r Beauzamy are f 0 n hence and =0, T T S S H , R Ker( &) X & spstv.B h is step, first the By positive. is 1 )), T .W nwb hoe 6 Theorem by know We ). ( # H * , (Xu t ,w ae( have we ), sdnein dense is h S + B o any for ) & n T T ( ( f H T Rh ( Ker( u uhthat such ) T | , h ), o any for ea lmn nthe in element an be ) & &) ) where )), n n ( ) # T n f 0frany for =0 1 H. ). ( X T ' nain sub- invariant s f H ffntosin functions of )b tp2. step by )) # f hsproves This . ( Ty C u E T X u , + ) f , & = # si fact in is ( u & # ssome is T E # | + H such ) This . & & H, T . )=0 # ( 333 f H ). , File: 580J 281238 . By:BV . Date:27:02:96 . Time:11:49 LOP8M. V8.0. Page 01:01 Codes: 2673 Signs: 1720 . Length: 45 pic 0 pts, 190 mm h lsdvco subspace vector closed the o-eoegnau fXadaynnzr etr!i uhta X! that such H in ! vector non-zero any and X of : eigenvalue non-zero theory spectral by subspaces invariant many has it pc ffntoa yefrT for type functional of space for snon-empty is AinB rmterelation the From sa isometry an is T for subspace invariant E non-trivial to a is H of 334 n hence and ylcvco x. vector cyclic problem subspaces invariant algebra C osqety ehave we Consequently, Ry h eain ewe h cone the between relations the Ker( ol eueu o te purposes. other for useful be could + Theorem oeprecisely More fteeeit o-eoeeetXi C in X element non-zero a exists there If Remark III.4.2. Ker( # ( T T : (3) (2) (1) X , ! n any and ) ( ,adhence and ), scnuaet nisometry an to conjugate is E M : X either ihrXi invertible, is X either either , e ido nain Subspaces Invariant of Kind New A h relation The . ! cigo eaal ibr pc H space Hilbert separable a on acting ) o any for ) , 8. satisfying . hnThsanntiilivratsubspace invariant non-trivial a has T then R Ker( e eacnrcino ls C class of contraction a be T Let Ker( , * R ehv h olwn alternative: following the have we XR R ntecmuatof commutant the in *T y X X XRy = srn lim =strong nKer( in )= & ){ * A & & E n R R & T [ 0(s h oiiiyof positivity the (use =0 [ (XRy : & & } , 0 R n 0 ) ASE N FACK AND CASSIER R ! 2 2 R & n snon-invertible is X and ] ] =span *XR ; htThsdnerange dense has T that * X A X lim strong , = C S n scnuaet ntr in unitary a to conjugate is T and n . &1 ,adKer( and ), n E and Ä T + T | R : & + ( Ry) * T hr xsside nivril operator invertible an indeed exists there n & n R [T n h nain usae problem subspaces invariant the and ) R - Ä & h *RT =Ker( = R & n + n : ( 2 ! & T &1 (R X 2 T | X )* + n S hc od o any for holds which , h n so needn neet It interest. independent of is , & . * shprnain.Q.E.D hyperinvariant. is ) ( n T 0 R X T X ( XRy oevrtersrcino T of restriction the Moreover , T h ettermshows theorem next The . & * & ) ) R S )* , . ] hs uepitspectrum point pure whose sa yeivratsub- hyperinvariant an is & ; uhta A that such T Assume 2 S | Rh T y) T 1} , * . h n sC is n nti case this In ( 0frany for =0 n X T navnNeumann von a in T . ( T .W hshave thus We ). n ) ( eddc that deduce we , . ) N suulyi the in usually as . ( U T ., | M n a a has and E , , : o any for , othat so ! ) = X y A :! &1 in in , File: 580J 281239 . By:BV . Date:27:02:96 . Time:11:49 LOP8M. V8.0. Page 01:01 Codes: 2543 Signs: 1311 . Length: 45 pic 0 pts, 190 mm rv 1,nt is httetopstv measure positive two the that first note (1), prove etrin vector rpsto that 8 Proposition inltype. tional E tflosta ehv,fraypstv polynomial positive any for have, we that follows It (0 ie eas ehave we because cide, esalpoeta ehv,frany for have, we that prove shall we ti la that clear is It As M operator rmi o-mt.W aet itnus he ijitcases: disjoint three distinguish to have We non-empty. is trum =Ker( eodcase Second (K Proof hr case Third . is case First X r r 1 n 0 and <1) sasmdt ennivril,ti ilide ml that imply indeed will this non-invertible, be to assumed is , t ( Let . T X X H , sanntiilhprnain usaefor subspace hyperinvariant non-trivial a is ) X a yhptei o-eoeigenvalue non-zero a hypothesis by has Ker( : uhthat such Ker( : ) ! : OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS E X | sinvertible is X sannzr nain usaefor subspace invariant non-zero a is eannzr lmn in element non-zero a be !) X t T & X 2 = = = = ){ P = )= ( X! :(K :( & & ( (X T X [ ? ( ) 0 :(! (X! [ ned ehave we Indeed, . I = [( &1 E (K ( ! & ] 0 I r & nti ae eko yTerm7that 7 Theorem by know we case, this In . =span ] I & , :! Â & 2 2 & re t , y ( U r = = | , T | n set and , & re & t u snon-invertible is X but !) .As re T ( !) ) 1 : : | 1 T , it & 0 X & ! T 2 , & | ? it - X 0 it X - ) 2 | T [T 1 T | ? &1 !) P 1 : ) ) Â XP | ) T 2 &1 y ( P ! & n &1 e ! ! where - # ( it | a es ag,i olw from follows it range, dense has e ! | | +( ( )| E !) it Xy T X!) n | , 2 )| 0 C ) !) = d+ I 2 ! & & + d+ & U .(1) :(K + T + ( ] 2 , re T T , . X ! :( , ( , , hs uepitspec- point pure whose ) it X ! ! ( T r ( + , , I ( t ! sauiayeeetin element unitary a is t X I *) & : ) ( ( P T & , t T Let . nti ae the case, this In . ! ) &1 , osethat see To . re , ) T re ! ! it hc so func- of is which , & and it T | T ! *) I !) *) ] eanon-zero a be &1 :+ ! &1 o any for | T E X! !) , ! { ! , | E ! | H !) !) on- co@ { .To 335 H r , File: 580J 281240 . By:BV . Date:27:02:96 . Time:11:49 LOP8M. V8.0. Page 01:01 Codes: 2978 Signs: 1989 . Length: 45 pic 0 pts, 190 mm ! soitdwt h o-eoeigenvalue non-zero the with associated by ial,w aefrany for have we Finally, operator that (1) bounded from & follows positive it Moreover, a [[ exists on There ]] } [[ norm new operator the invertible consider an end, of this To existence isometry. the show to remains it theorem, hence and 336 se[a-o.2 n Ba.2) oecmlt eutaottemulti- the about result complete of more A powers any 2]). of [Beau. plicity and 2] [Na.-Foi. (see n hence and eopstv operator positive zero type. functional of ucinlcluu,ta hr xss o n ...contraction C.N.U. any for exists, there that for calculus, subspace invariant functional non-trivial a of existence the insure to fices subspace fte`June eTe redsOe aer' edi uiy hr hspprwsfirst was paper this where Luminy, session 1993 in the held of Ope rateurs'' organizers the des to orie The thanks de our presented. express ``Journees also the We of topics. these on exchanges integer B ibr pc uhthat such space Hilbert A = ( etakBradBazm,Per el ap,adAanVltefrinteresting for Valette Alain and Harpe, la de Pierre Beauzamy, Bernard thank We Remark Remark Remark E y &1 T e ]] satisfying ) n &1 & 2 1 weew suefrinstance for assume we (where hspoe that proves This . = N 0 E & (A Â = A & .Frtecontraction the For 1. .Tesm ro saoesosta h xsec fanon- a of existence the that shows above as proof same The 2. .I svr ayt rv,wt h epo h asymptotic the of help the with prove, to easy very is It 3. ATA : - , ( y ! N T 2 : & ssandb the by spanned is ecfrh ehave we Henceforth, . y 0 1 | ( &1 T E | & y) A uhthat such )1 ) Â y - & A & 2 } &1 : =[[ = =[[ & o any for X & A [[ S - & sa isometry. an is T & &1 T in y - A E y Xy 0(and {0 a enotie yA tmn(f Az 2]). [Atz. (cf. Atzmon A. by obtained been has A TA in & && Acknowledgment ASE N FACK AND CASSIER XA : ]]= H , &1 1 ! & E &1 A aifigKer( satisfying y snthprnain,adhnecno be cannot hence and hyperinvariant, not is &1 o any for &1 ]] Â e y & y - k ]] T - y & # swith 's 2 : & = N T Xy E 2 fortiori a n uhthat such and , 2 A = eceryhave clearly We . a occi etrfrany for vector cyclic no has (A - osdrdi I..,w nwthat know we III.1.2, in considered n & & (because sivril in invertible is y : - 0 sa iesaeof eigenstate an is <0) 2 # &1 k A : XTA E ( &1 y =4 oahee h ro fthe of proof the achieves To . & X n o any for # X y n ec sntinvariant not is hence and , ){ E & &2| | &1 T . ). A [ * n y A &1 0 | XT ee h invariant the Here, . & ] 2 y) C A and = in 1} B ( = E T ( cnrcin,an -contraction), X E & | eie by defined T ) E n satisfies and ) y& E & * ) A XT 2 A uhthat such , & &1 X T N T - Q.E.D X = . san is A & na in suf- S X N T in & 0 , . File: 580J 281241 . By:BV . Date:27:02:96 . Time:11:49 LOP8M. V8.0. Page 01:01 Codes: 4190 Signs: 3339 . Length: 45 pic 0 pts, 190 mm [Beau.-Rom.] 5] [Beau. Ba.4] [Beau. Ba.3] [Beau. [Fa.-Harp.] [Ber.] Ba.2] [Beau. 1] [Beau. Az 2] [Atz. 1] [Atz. [AK.-Glaz.] [Foi.-Kov.] [Fug.-Kad.] Cs 1] [Cas. [Cara.] [Bro.] Cs 5] [Cas. 4] [Cas. 3] [Cas. 2] [Cas. [Du.] [Dix.] [Cas.-Fa.] induecnrcind classe de contraction d'une Rome tion M. and Beauzamy B. Beauzamy B. eBanach, de Beauzamy B. .Beauzamy B. nc Cie^ evnNuanfne continues, finies Harpe Neumann la von de de P. and Fack T. 1970. emn ler ffnt type, finite of algebra Neumann Berberian K. S. .Beauzamy B. 1988. Amsterdam, North-Holland, Beauzamy B. .Atzmon New A. Ungar, Transl. Theory English Atzmon (1966); A. Glazman Moscow, 1961. M. Nauka, York, I. spaces,'' and Hilbert Akhiezer I. N. Theory Math finies, Kovacs Neumann I. and Foias C. 4973. (1980), .FgeeadR .Kadison V. R. and Fuglede B. lrfil ec@ cdn a u 'le r duale, bre l'algeÁ sur pas ncident co@ ne ultrafaible 1954. Cassier York, Pitman G. New 135, Chelsea, pp. II, and Eds.), Harlow Effros, Longman, Caratheodory 123, G. C. Vol. E. Math., and Notes Araki Research (H. Algebras'' Operators Brown L. qain prtrTheory Operator Equations etcniu 'le rsduales, d'algeÁ bres continus ment Cassier G. Hilbert, de l'espace Cassier 1991. G. 2, No. LAFP print, Pre Cassier G. 95 Cassier G. 325333. (1986), .L Duren 1969. L. Paris, P. GauthierVillars, Neumann),'' von Dixmier J. Fack Acad T. and Cassier G. 773776. OTATOSI O EMN ALGEBRAS NEUMANN VON IN CONTRACTIONS 18) 1732. (1989), . . . Sci 55 54 11 4 . 18) 287288. (1980), 15) 520530. (1952), 18) 619627. (1982), isi' hoe ntetp Icase, II type the in theorem Lidskii's , è le rsdoe aer aslepc ibrin(le rsde bres (AlgeÁ Hilbertien l'espace dans rateurs d'ope algeÁ bres ``Les , Paris 18) 340. (1984), le rsdae nfre u 'saed Hilbert, de l'espace sur uniformes duales bres AlgeÁ , hmshletesfilmn otnse hmsultrafaible- champs et continus faiblement hilbertiens Champs , neepedoe aerpu eullstplge abeet faible topologies les lequel pour rateur d'ope exemple Un , hmsdagÁbe ulse le rsdae nfre sur uniformes duales algeÁ bres et duales bres d'algeÁ Champs , caMath Acta utlna apnsadetmtso multiplicity, of estimates and mappings Multilinear , Ensembles , nteeitneo yeivratsubspaces, hyperinvariant of existence the On , `hoyof ``Theory , otatossn etusccius rvt communication. private cycliques, vecteurs sans Contractions , `nrdcint prtrTer n nain Subspaces,'' Invariant and Theory Operator to `Ìntroduction , egtdbltrlsitwt occi vectors, cyclic no with shift bilateral weighted A , osepcsivrat etp ocine aslsespaces les dans fonctionnel type de invariants Sous-espaces , ocin 'n otato eclasse de contraction d'une Fonctions , , Acta e i I Se rie rc n h ovxhl ftesetu navon a in spectrum the of hill convex the and Trace , `hoyo ucin faCmlxVral,'Vl.I Vols. 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