Journal of Mathematical Analysis and Applications

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1 J. Math. Anal. Appl ) Contnts lists availabl at ScincDirct Journal of Mathmatical Analysis and Applications Unitary quasi-affin transforms of contractions Frnando Gómz Dpartamnto d Análisis Matmático, Univrsidad d Valladolid, Facultad d Cincias, Prado d la Magdalna, s.n., Valladolid, Spain articl info abstract Articl history: Rcivd 5 Jun 2006 Availabl onlin 8 Octobr 2008 Submittd by J.A. Goldstin Kywords: Hilbrt spac Contraction Sz.-Nagy Foiaş dilation Kérchy unitary asymptot Prigogin xact irrvrsibility Grossmann nstd Hilbrt spac Bakr map Th unitary quasi-affin transforms of contractions ar th cornrston of th xact thory of irrvrsibility introducd by Misra, Prigogin and Courbag. This work shows that th class of contractions with unitary quasi-affin transforms is just C 1 and that vry unitary quasi-affin transform of a contraction is unitarily quivalnt to th rsidual part of its Sz.-Nagy Foiaş dilation. Som connctions with th Grossmann thory of nstd Hilbrt spacs and gnralizd ignvalus ar stablishd. Ths rsults ar applid to th study of th bakr map Elsvir Inc. All rights rsrvd. 1. Introduction W ar intrstd in intrtwining rlations of th form W Λ = ΛU, 1) whr W is a contraction dfind on a Hilbrt spac H, U is a unitary oprator on a Hilbrt spac H and Λ is at first an arbitrary linar oprator from H into H. In particular w ar intrstd in rlations 1) whr Λ is a quasi-affinity, i..λ is a on-to-on and continuous oprator from H onto a dns subspac in H, so that Λ 1 xists on this dns domain, but is not ncssarily continuous; U is thn calld a unitary quasi-affin transform of W. Th unitary quasi-affin transforms of contractions hav bn alrady tratd in th work of Sz.-Nagy and Foiaş [12], and thir adjoints furthr studid undr th nam of unitary asymptots by Kérchy [8]. To b prcis, Sz.-Nagy Foiaş provd that a unitary quasi-affin transform of a contraction W of class C 1 and such that Kr W ={0} is th rsidual part R, R of its minimal isomtric dilation; in such cas, th adjoint of th orthogonal projction onto R rstrictd to H, Λ = P R H), is quasi-affinity that intrtwins R and W s Proposition 6 blow). On of th contributions of this work is a strong convrs of this rsult. Thus w can say that, givn a unitary oprator U and a contraction W with Kr W ={0}), thrxists a quasi-affinity Λ satisfying th intrtwining rlation 1) if and only if iff ) W is of class C 1 and U is unitarily quivalnt to th rsidual part R of th minimal dilation of W s Thorm 10 and its corollaris). This rsult implis in particular that R,Λ) is a Kérchy unitary -asymptot of W. W also connct this thory with that of nstd Hilbrt spacs and gnralizd ignvalus introducd by Grossmann [4]. A nstd Hilbrt spac is a pair of Hilbrt spacs ach of which is in a crtain sns idntifid with a dns subst of This work was compltd with th support of JCyL-projct VA013C05 Castilla y Lón) and MEC-projct FIS Spain). addrss: fgcubill@am.uva.s X/$ s front mattr 2008 Elsvir Inc. All rights rsrvd. doi: /j.jmaa

2 F. Gómz / J. Math. Anal. Appl ) th othr. Ths structurs hav bn usd to study analytic continuation into unphysical shts and to discuss nonnormalizabl stats of quantum-mchanical systms. W will s that, whn Λ is a quasi-affinity, th intrtwining rlation 1) inducs a nstd Hilbrt spac H, H,Λ ) and th gnralizd ignvalus of U ar just th ignvalus of W. On th othr hand, th intrtwining rlation 1) is in th basis of th xact thory of irrvrsibility introducd by Misra, Prigogin and Courbag MPC) in a sris of paprs [9,10] intndd to unify dynamics and thrmodynamics, including a dscription of irrvrsibl bhavior at th microscopic lvl. Until th lat 1970s it had bn a gnral blif that stochastic procsss can aris from dtrministic dynamics only as a rsult of som form of coars-graining or approximations. According to th MPC thory thr is an quivalnc btwn highly unstabl dtrministic and stochastic volutions. Such quivalnc is stablishd through quasi-affinitis and intrtwining rlations lik 1), which convrt unitary groups associatd with dtrministic dynamics to contraction smigroups associatd with stochastic Markov procsss. In contrast with th coars-graining approach thr is no loss of information involvd in such transition. Nowadays th xact MPC thory of irrvrsibility still nds furthr spcification and clarification [5, Chaptr II.1], [1, Sction 5.4]). Obviously, dropping positivity, th abov mntiond rsult Thorm 10 and its corollaris) charactrizs th irrvrsibl dynamics inducd by th xact MPC thory and givs th structur of th rvrsibl volutions admitting such chang of rprsntation and a prototyp for th intrtwining quasi-affinity, on th basis of th Sz.-Nagy Foiaş dilation thory [12]. Hr w apply ths rsults to th study of th bakr map. W will obtain spctral rprsntations for its Frobnius Prron oprator U and th associatd contraction W, and th Grossmann gnralizd ignvalus of U. Th papr is organizd as follows. Sction 2 contains a brif rviw of th trminology and rsults of th Sz.-Nagy Foiaş thory w will us throughout th papr. Sction 3 includs a lifting thorm for oprators Λ satisfying 1) and othr rlatd rsults ncssary in Sction 4 to prov Thorm 10 and its corollaris. Sction 5 is ddicatd to th nstd Hilbrt spacs. Finally, in Sction 6 th study of th bakr map is carrid out. 2. Trminology and prliminaris Our trminology and notations ar thos of [12], xcpt for th Hilbrt spacs of th minimal isomtric and unitary dilations, which ar dnotd hr by K + and K, rspctivly. W rviw thm brifly. Lt H K b two Hilbrt spacs. For oprators A : H H and B : K K w writ A = pr B whn Ah, h ) = Bh, h ) for all h, h H or, quivalntly, Ah = P H Bh for all h H, whr P H dnots th orthogonal projction of K onto H. W call B a dilation of A if A n = pr B n, n = 1, 2,... Two dilations of A, sayb on K and B on K, ar calld isomorphic if thr xists a unitary oprator U : K K such that Uh = h for h H and B = U 1 BU. For vry contraction W on a Hilbrt spac H thr xist an isomtric dilation U + on som Hilbrt spac K + H and aunitarydilationu on som Hilbrt spac K H, which ar morovr minimal in th sns that K + = U+ n H and K = U n H. 0 Ths minimal isomtric and unitary dilations ar dtrmind up to isomorphism, cf. [12, Sction I.4]. In what follows w considr th minimal isomtric dilation U + of W mbddd in its minimal unitary dilation U in th following way: K + U n H, U + U K+. 0 P M will always dnot th orthogonal projction from K + or K onto a closd subspac M. Which spac, K + or K, will b clar by th contxt. Lt V b an isomtry on a Hilbrt spac H. AsubspacL H is calld wandring for V if V n L V m L for vry pair of intgrs m,n 0, m n; sincv is an isomtry it suffics that V n L L for n N. Th orthogonal sum M + L) V n L satisfis VM + L) = 1 V n L = M + L) L. If U is a unitary oprator on H and L is a wandring subspac for U,sincU 1 is also isomtric, U m L U n L for all intgrs m n. Thsubspac ML) = U n L rducs U. ML) dos not dtrmin L, only its dimnsion.

3 86 F.Gómz/J.Math.Anal.Appl )84 96 For a contraction W on th Hilbrt spac H with minimal unitary dilation U on K th subspacs L and L dfind by th ovr bar dnots closur) L U W )H, L U W ) H ar wandring subspacs for U and th spac K can b dcomposd into th orthogonal sum K = U 2 L U L L H L UL U 2 L. ML) and ML ) rduc U and hnc th sam is tru for th subspacs R K M L ), R K ML). W shall call rsidual part and dual rsidual part of U to th unitary oprators R U R and R U R. Now considr th subspac L UL = I UW ) H. Thn L and L ar wandring subspacs for th minimal isomtric dilation U + of W and hnc for U ) such that L L = {0} and K + = H M + L) = R M + L ). R is th subspac of K + which rducs U + and U ) to th unitary part R of U +.Morovr,R ={0} iff n W n = 0, whr dnots limit in strong sns, cf. [12, Sction II.2]. A contraction W on H is calld compltly non-unitary c.n.u.) if for non-zro rducing subspac H 0 for W is W H0 a unitary oprator. To W thr corrsponds a uniqu dcomposition of H into an orthogonal sum of two subspacs rducing W,sayH = H 0 H 1, such that W H0 is unitary and W H1 is c.n.u. In particular, for an isomtry, this canonical dcomposition coincids with th Wold dcomposition, cf. [12, Thorm I.3.2] 3. A lifting thorm Th following rsult is a lifting thorm for oprators Λ intrtwining contractions and unitary oprators. In this cas an xplicit xprssion 2) for th lifting Λ + is givn. Similar xprssions hav bn considrd in th study of Pták gnralization of Toplitz and Hankl oprators, s [11] and rfrncs thrin. Hr w giv a proof basd on that of th classical lifting thorm [12, Thorm II.2.3]. In what follows H and H ar two Hilbrt spacs. Thorm 1. Lt W b a contraction on H with minimal isomtric dilation U + on K +,andltu b a unitary oprator on H.For vry boundd oprator Λ : H H satisfying th intrtwining rlation 1) thr xists a uniqu boundd oprator Λ + : H K + such that U + Λ + = Λ + U, Λ = Λ + and Λ = P H Λ +, which is of th form Λ + = n U n + ΛU n. 2) Proof. Sinc K + = H M + L) th gnral form of an oprator Λ + : H K + satisfying Λ = P H Λ + is Λ + = Λ + B 0 + U + B 1 + U 2 + B 2 +, 3) whr ach B n is an oprator from H into L. From 3) w dduc U + Λ + Λ + U = U+ n B n 1 B n U ), with B 1 = U + Λ ΛU.Bcausof1)whavB 1 = U + W )Λ and thus B 1 is an oprator from H into L. BingU unitary, in ordr that Λ + satisfis U + Λ + = Λ + U, it is thrfor ncssary and sufficint that so that B n = B n 1 U 1 for n = 0, 1,..., B 1 = U + W )Λ, B n = U + W )ΛU n+1) for n = 0, 1,..., and, using 1),

4 F. Gómz / J. Math. Anal. Appl ) Λ + = Λ + U+ n U + W )ΛU n+1) = Λ + U+ n+1 ΛU n+1) U n n+1) + W ΛU = Λ + U n+1 + ΛU n+1) U n + W ΛU n+1) = Λ + = Λ Λ + n U n + ΛU n = n U n + ΛU n, U+ n+1 ΛU n+1) U+ n n ΛU bing th last limit in strong sns on LH) bcaus w ar daling with an orthogonal sum and for th Nth sum and ach h H,sincU + is an isomtric xtnsion of W,whav N 1 2 Λ + U+ n B n )h N 1 = Λh 2 + N 1 B n h 2 = Λh 2 + U + W )ΛU n+1) h 2 N 1 = Λh 2 + ΛU n+1) h 2 ΛU n h 2 ) = Λh 2 Λh 2 + ΛU N h 2 Λ 2 U N h 2 = Λ 2 h 2. Morovr, sinc for all Λ + satisfying Λ = P H Λ + th inquality Λ Λ + holds, w hav Λ = Λ +. Rmark 2. Th oprator Λ + : H K + of Thorm 1 can also b considrd as an oprator from H into th spac K K + whr th minimal unitary dilation U of W is dfind. W will dnot this oprator by Λ + as wll. Obviously Λ + : H K is of th form Λ + = n U n ΛU n 4) and satisfis th conditions UΛ + = Λ + U, Λ = Λ + and Λ = P H Λ +. From now on w shall us ithr manings of Λ + without causing confusion. Daling with intrtwining rlations btwn contractions and unitary oprators, th rlvant parts of th isomtric and unitary dilations ar th rsidual ons. Th following proposition is an asy consqunc of th dcomposition of an oprator that intrtwins two contractions as an oprator matrix with rspct to th Wold dcomposition of thir minimal isomtric dilations givn by Sz.-Nagy and Foiaş in[13]. Proposition 3. Undr th conditions of Thorm 1,thrangofΛ + is containd in th rsidual part R of K + and K. Proof. Rcall that K + = M + L ) R corrsponds to th Wold dcomposition of U +,bingu + R unitary and U + M+ L ) a unilatral shift. On th othr hand, sinc U is unitary, its minimal isomtric and unitary) dilation coincids with itslf and thrfor K + = R = H and U + = U. It is wll known [13, Sction 1] that th oprator Λ + can b rprsntd by a matrix ) A 0 Λ + =, B C with A : M + L ) M +L ), B : M + L ) R and C : R R, such that U + M+ L ) A = AU M + L ), U + R B = BU M + L ), U + R B = BU R. Sinc in this cas M + L ) ={0}, whava = B = 0 and Λ + = C. In rlation to th lifting thorm and th dual rsidual part w can say th following: Proposition 4. Lt W b a contraction on H with minimal unitary dilation U on K,andltU b a unitary oprator on H. For vry boundd oprator Λ : H H satisfying th intrtwining rlation 1) th following limit xists and n U n ΛU n = P R Λ. 5) Proof. For vry h H w hav [12, Proposition II.3.1] n U n W n h = P R h.

5 88 F.Gómz/J.Math.Anal.Appl )84 96 Taking h = Λh, from th intrtwining proprty 1) w obtain n U n ΛU n h = n U n W n Λh = P R Λh, h H. From 5), sinc U is unitary, lim n ΛU n h = P R Λh h H ), and, bing U a dilation of W, n W n ΛU n = P H P R Λ. 4. Unitary quasi-affin transforms of contractions In what follows w will say that th oprator S 1 acting on a Hilbrt spac H 1 is a quasi-affin transform of th oprator S 2 acting on a Hilbrt spac H 2 if thr xists an quasi-affinity Λ : H 1 H 2 such that ΛS 1 = S 2 Λ. Som proprtis of th quasi-affinitis can b found in [12, Proposition II.3.4]: Lmma 5. Lt H 1, H 2 and H 3 b Hilbrt spacs. a) If X : H 1 H 2 and Y : H 2 H 3 b quasi-affinitis, thn Y X : H 1 H 3,X : H 2 H 1 and X =X X) 1/2 : H 1 H 1 ar quasi-affinitis. Morovr, X X 1 xtnds by continuity to a unitary transformation from H 1 to H 2. b) If a unitary oprator U 1 on H 1 is th quasi-affin transform of a unitary oprator U 2 on H 2,thnU 1 and U 2 ar unitarily quivalnt, i.. thr xists a unitary oprator V : H 1 H 2 such that U 2 V = VU 1. Conditions undr which a contraction has as unitary quasi-affin transform th rsidual part of its minimal isomtric dilation ar alrady givn by Sz.-Nagy and Foiaş [12, Proposition II.3.5]: Proposition 6. Lt W b a contraction on H and R, R th rsidual part of its minimal isomtric dilation. If W h 0 and W n h dos not convrg to 0 for ach nonzro h H, thnλ P R H is a quasi-affinity from H to R such that W Λ = ΛR. Proposition 6 will b improvd in Corollary 13. Othr approach looking for unitary quasi-affin transforms of contractions can b found in th proof of [12, Proposition II.5.3]: Proposition 7. Lt W b a boundd oprator on H. Assum that thr xists a slfadjoint oprator A on H such that 0 <Ah, h)<m h 2 0 h H), 6) and lt Λ b th positiv slfadjoint squar root of A, which, by 6), is a quasi-affinity on H.IfAW h, W k) = Ah, k) for all h, k H, thn ΛW Λ 1 xtnds by continuity) to an isomtric oprator U on H satisfying ΛW = UΛ or, taking adjoints, W Λ = ΛU.If, in addition, W h 0 for h 0, thn U is unitary. To prov th main rsults of this work, Thorm 10 and its corollaris, w will nd th following tchnical lmmas. Lmma 8. Lt W b a contraction on H with minimal isomtric and unitary dilations U + and U on K + and K, rspctivly, and R, R th corrsponding rsidual part. For a non-zro h H th following assrtions ar quivalnt: a) h P H R; b) n W n h = 0. Proof. Lt h b a non-zro vctor such that h H and h P H R or, quivalntly, such that h H and h R. Sinc K + = R M + L ), h M + L ) and hnc h has an orthogonal xpansion h = m=0 U m h m, whr h m L and h 2 = m=0 h m 2.Morovr,sincW = pr U 1 and U ν L H for ν > 0, w hav W n h = P H U n h = P H U m n h m = P H U m n h m, m=0 so that n W n h = U m n h m = 0 n m n m n and a) b) is provd. Now, to prov b) a) assum that for a non-zro h H on has n W n h = 0. Thn

6 F. Gómz / J. Math. Anal. Appl ) n 1 U k+1 U W ) W k h = h U n W n h M + L ) and n h U n W n h = h M + L ).Thus,h P H R. An immdiat consqunc of Lmma 8 is th following: Lmma 9. Lt W b a contraction on H and lt R, R b th rsidual part of its minimal isomtric and unitary dilations U + and U on K + and K, rspctivly. Thn th following assrtions ar quivalnt: a) P H R = H; b) W C 1,i.. n W n h 0 for vry non-zro h H. In what follows w com back to th Hilbrt spacs H and H of Sction 3. Th main assrtion of Thorm 10 blow is that if U is a unitary quasi-affin transform of a contraction W and Λ is a quasi-affinity intrtwining both oprators, thn U is also a quasi-affin transform of th rsidual part R of th minimal isomtric dilation of W and th lifting Λ + of Λ is a quasi-affinity intrtwining U and R, providdkrw ={0}. An immdiat Corollary of this fact is that thn U and R ar unitarily quivalnt. Thorm 10. Lt W b a contraction on H with Kr W ={0}, minimal isomtric and unitary dilations U + and U on K + and K, rspctivly, and rsidual part R, R. LtU b a unitary oprator on H which is a quasi-affin transform of W, i.. thr xists a quasi-affinity Λ : H H such that th intrtwining rlation 1) is satisfid, and lt Λ + : H K + or Λ + : H K) th lifting oprator Λ + = n U n + ΛU n givninthorm1.thn a) W C 1,i..W n hdosnotconvrgto0 for ach non-zro h H; b) R M + L) ={0}; c) Λ + is a quasi-affinity from H into R such that RΛ + = Λ + U. Proof. a) Sinc Λ is a quasi-affinity from H to H, whavh = ΛH.FromThorm1,Λ = P H Λ +.ByProposition3, Λ + R. Thus, H = ΛH = P H Λ + H P H R H and thrfor P H R = H. But, by Lmma 9, P H R = H implis that W n h dos not convrg to 0 for ach non-zro h H. b) Suppos thr xists a non-zro k R H.Thnk ML ) and k H, so that k M + L) and hnc k has an orthogonal xpansion k = U n k n, whr k n L and k 2 = k n 2.Sinck 0, thr is at last on non-zro k n ; lt k ν b th first of ths non-zro trms. Thn w hav U ν 1 k = U 1 k ν + U μ k ν+μ+1. μ=0 7) Sinc k R and R rducs U,alsoU ν 1 k blongs to R = K ML ) and, in particular, U ν 1 k L.Morovr,U μ L L for μ 0 and w dduc from 7) that U 1 k ν L and k ν UL.SincH L = UL UH, w conclud that k ν UH. Thus thr xists an h H such that k ν = Uh; consquntly P H k ν = P H Uh = Wh.SincL H, whavp H k ν = 0 and hnc Wh= 0. But l ν 0implish 0, and this is in contradiction with Kr W ={0}. c) By Proposition 3, Λ + H R. W must prov that Λ + is injctiv and Λ + H = R. Th injctivity of Λ + follows from that of Λ. Indd, if thr xist h 1, h 2 H such that Λ + h 1 = Λ +h 2,thnP HΛ + h 1 = P HΛ + h 2, that is Λh 1 = Λh 2 and thus h 1 = h 2. Now suppos that Λ +H R, i.. that thr xists a non-zro k R such that k Λ + H and thn k,λ + h ) = 0 for all h H. Taking into account th xprssion 2) for Λ + and th rlation Λ = P H Λ + s Thorm 1 and Rmark 2) w hav thn k,λ + h ) = lim k, U n ΛU n h ) = lim U n k, P H Λ + U n h ) n n = lim U n Λ n + P HU n k, h ) = 0 for all h H. Butthisisquivalntto n U n Λ + P HU n k = 0,

7 90 F.Gómz/J.Math.Anal.Appl )84 96 which, sinc U is unitary, coincids with Thus, n Λ + P HU n k = 0. lim Λ n + P H U n k, h ) = lim PH U n k, P H Λ + h ) n = lim PH U n k,λh ) = 0 for all h H, n and, sinc Λ is quasi-affinity, ΛH is dns in H and this implis n P HU n k = 0. 8) Now rcall that K + = H M + L) = R M + L ) and that R rducs U to its rsidual part R and thn U n k = R n k R for all k R and n Z. Thrfor 8) implis n U n k R M + L), but,fromb),r M + L) ={0} and w hav n U n k = 0, only possibl if k = 0sincU is unitary. This provs Λ + H = R. Rmark 11. Undr th conditions of Thorm 10 xcpt Kr W ={0}, whavkrw Rang Λ ={0}. Indd, if thr xists a non-zro h Kr W and h = Λh for som h H, from 1), 0 = Wh= W Λh = ΛU h, so that U h Kr Λ and thrfor Λ cannot b a quasi-affinity from H into H. Corollary 12. Evry unitary quasi-affin transform of a contraction W is unitarily quivalnt to th rsidual part of th minimal isomtric dilation of W, providd Kr W ={0}. Proof. This follows from Thorm 10c) and Lmma 5b). Othr consqunc of Thorm 10 is an improvmnt of Proposition 6 in th following sns: Corollary 13. Lt W b a contraction on H with Kr W ={0} and R, R th rsidual part of its minimal isomtric dilation. Thn R is a unitary) quasi-affin transform of W iff W C 1,i..W n hdosnotconvrgto0 for ach non-zro h H. In such cas Λ P R H is a quasi-affinity from H to R such that W Λ = Λ R. Proof. This follows from Thorm 10a) and Proposition 6. W conclud this sction with an obvious consqunc of th prvious facts: Corollary 14. A contraction W on H with Kr W ={0} has unitary quasi-affin transforms iff W C 1,i..W n hdosnotconvrg to 0 for ach non-zro h H. 5. Nstd Hilbrt spacs A nstd Hilbrt spac H 0, H 1, E 01 ) is a structur that consists of two infinit dimnsional sparabl Hilbrt spacs H 0 and H 1, a quasi-affinity E 01 of H 1 into H 0, and th adjoint quasi-affinity E 10 = E 01 of H 0 into H 1. Lt A : H 0 H 0 b a linar oprator and considr in H 1 th oprator j 10 A) E 1 01 AE 01. Th domain of j 10 A) is th subst of E 1 01 DA) consisting of vctors f which ar also such that AE 01 f E 01 H 1.Thmatrix lmnts of j 10 A) ar diffrnt from th matrix lmnts of A, bcaus of th diffrnt dfinition of th scalar product. Th ignvalus of j 10 A ) includ th impropr ignvalus of A, long familiar in quantum mchanics: Dfinition 15. Lt A b an oprator in H 0 such that A is dnsly dfind, so that th adjoint A xists, and j 10 A ) is also dnsly dfind, so that its adjoint j 10 A ) also xists. Th complx numbr z is said to b a gnralizd ignvalu of A if thr xists in H 1 a non-zro vctor f such that j 10 A ) f = zf.

8 F. Gómz / J. Math. Anal. Appl ) This dfinition includs th usual on: Lt Ah = zh h H 0, h 0). Notic that j 10 A ) E 10 A E 1 10 E 10 AE 1 10.So j 10 A ) f = zf,with f = E 10 h. Now lt U b a unitary oprator acting on a Hilbrt spac H, W a contraction on a Hilbrt spac H and Λ aquasiaffinity satisfying th intrtwining rlation 1). Taking adjoints w gt W = Λ ) 1 U Λ. Thus, idntifying in th prvious framwork E 01 : H 1 H 0 with Λ : H H, A : H 0 H 0 with U : H H, j 10 A) E 1 01 AE 01 with W = Λ ) 1 U Λ, w hav th following: Proposition 16. In th nstd Hilbrt spac H, H,Λ ) th gnralizd ignvalus of U ar th ignvalus of W. 6. Th bakr map Th bakr map is on of th first xampls of rvrsibl mixing transformations and was introducd by Hopf [7]. It is dfind on th unit squar [0, 1) 2 as a two-stp opration: 1) squz th unit squar to a 2 1/2-rctangl and 2) cut th rctangl into two 1 1/2-rctangls and pil thm up to rcovr th unit squar: Bx, y) = { 2x, y 2 ), 0 x < 1 2, 2x 1, y+1 2 ), 1 x < 1. 2 It is a typical Kolmogorov systm K-systm) with th Lbsgu masur as an rgodic invariant masur. Th tim volution of th probability dnsitis ρx, y) is govrnd by th Frobnius Prron oprator w drop hr th prim of prvious sctions) Uρx, y) ρ B 1 x, y) ) = { ρ x 2, 2y), 0 y < 1 2, ρ x+1 2, 2y 1), 1 2 y < 1. 9) Th oprator U is unitary on th Hilbrt spac L 2 [0, 1) 2 ) of squar intgrabl functions with rspct to th Lbsgu masur. It is wll known [3] that for K-systms, U has a Lbsgu spctrum: an infinitly dgnrat continuous spctrum on th unit circl plus a point ignvalu at z = Λ-transformation ForthbakrmapaΛ-transformation is constructd as follows [10]. Lt χ 0 b th function { 1, 0 x < 1/2, χ 0 x, y) 1, 1/2 x < 1, and, for ach finit st S ={n 1,...,n r } of intgrs n j n k if j k), st χ S x, y) U n 1 χ 0 x, y)u n 2 χ 0 x, y) U n r χ 0 x, y). Thn th family of functions {χ S } togthr with th unit function 1 form a complt orthonormal st of L 2 [0, 1) 2 ).Not that Uχ S = χ S+1, 10) whr S + 1 ={n 1 + 1,...,n r + 1} if S ={n 1,...,n r }.Now,forachintgrn, dfinthopratore n to b th orthogonal projction oprator onto th subspac spannd by χ S such that n S max{n j S}=n. ThΛ-transformation is dfind by Λ n E n + P 0,

9 92 F.Gómz/J.Math.Anal.Appl )84 96 whr P 0 is th on-dimnsional orthogonal projction onto th subspac of constant functions and { n } <n< is a positiv monotonically dcrasing squnc boundd by 1 such that n+1 / n also dcrass monotonically as n incrass. 1 This lads to W ΛUΛ 1 n+1 = UE n + P 0. 11) n Th oprator W is a contraction such that W 1 = 1 and W n ρ 1) dcrass strictly monotonically to 0 as n Charactristic functions By virtu of 10), U k E n = E n+k U k < n, k < ). Thrfor, ) 2 W n+1 W = E n + P 0, n ) 2 WW n = E n + P 0, n 1 so that th dfct oprators ar D W I H W ) [ ) 2 ] 1/2 1/2 n+1 W = 1 E n, n D W I H WW ) [ ) 2 ] 1/2 1/2 n = 1 E n, n 1 which ar slfadjoint and boundd by 0 and 1, with dfct spacs D W D W H = Kr D W ) = L 2 C, D W D W H = Kr D W ) = L 2 C, if { n } <n< is a strictly dcrasing squnc hr C dnots th spac of constant functions on [0, 1) 2 ). Th charactristic function of W, valud on th st of boundd oprators from D W into D W,isdfindfor in th unit disc D { C: < 1} by Θ W ) [ W + D W I W ) 1 ] D W D W [ = W + k D W W k 1 D W, 12) k=1 ] D W th xpansion bing convrgnt in norm. Sinc for k 0whav W k n = U k E n + P 0, n k from 12) w obtain for D [ Θ W ) = γ n+1 U + β n k+1 α n k U )]E k n = [ 1 + β n k α γ n k+1 U )]UE k n, 13) n+1 1 An xampl of such squnc { n } <n< is givn by 1 n =, τ > n/τ

10 F. Gómz / J. Math. Anal. Appl ) whr, for ach n Z, 1 α n 1 ) 1/2, β n 2 2 n n 2 ) 1/2, 2 n+1 γn n. 14) n 1 n 1 In a similar way, th charactristic function of W, valud on th st of boundd oprators from D W into D W,isgivn for D by Θ W ) [ W ] + D W I W ) 1 D W D W [ = γ n U + α n k+1 β n+k U )]E k n = 6.3. Spctral rprsntations [ 1 + α n k β γ n+k 1 U )]U k E n. 15) n Givn a sparabl Hilbrt spac G, lt L 2 D; G) dnot th st of all masurabl functions v : D G such that 1 2π 2π v iθ ) 2 0 G dθ< modulo sts of masur zro); masurability hr can b intrprtd ithr strongly or wakly, which amounts to th sam du to th sparability of G. Th functions in L 2 D; G) constitut a Hilbrt spac with pointwis dfinition of linar oprations and innr product givn by u, v) 2π 1 2π u iθ ), v iθ )) 0 G dθ u, v L 2 D; G)). Lt H 2 D; G) b th Hardy class of all functions of L 2 D; G) whos kth Fourir cofficints vanish for all ngativ k s. Th lmnts of H 2 D; G) ar just th boundary valus of th G-valud holomorphic functions on D such that 1 2π 2π ur iθ ) 2 0 G dθ 0 r < 1), has a bound indpndnt of r. Elmntary proprtis of vctor and oprator valud functions ar givn in Hill and Phillips [6, Chaptr III]; s also [12, Sction V.1]. For th bakr map w dal with G = D W = D W = L 2 [0, 1) 2 ) C. For almost all iθ D with rspct to th normalizd Lbsgu masur) th following limit xists Θ W iθ ) Θ W ) D, iθ non-tangntially ). Such limits induc a dcomposabl oprator Θ W from L 2 D; D W ) into L 2 D; D W ) dfind by [Θ W v] iθ ) Θ W iθ ) v iθ ) for v L 2 D; D W ). For thos iθ D at which Θ W iθ ) xists, thus a.., st W iθ ) [ I Θ W iθ ) ΘW iθ )] 1/2. W iθ ) is a slfadjoint oprator on D W gnrats by boundd by 0 and 1. As a function of iθ, W iθ ) is strongly masurabl and [ W v] iθ ) W iθ ) v iθ ) for v L 2 D; D W ), a slfadjoint oprator W on L 2 D; D W ) also boundd by 0 and 1. In this cas, from 12) w gt Θ W iθ ) ΘW iθ ) = I 1 Γ 2 iθ ), whr ± lim n ± n and Γ iθ ) β n [ k= ikθ β n+k U k ]E n. Th oprator Γ iθ ) is slfadjoint and such that Γ iθ ) 2 = 2 2 )Γ iθ ).Thus,thoprator P iθ ) 1 2 Γ iθ ) 2 is an orthogonal projction and W iθ ) [ 1 = Γ 2 iθ )] 1/2 = 2 2 )1/2 P iθ ) 1 = 2 Γ iθ ). 2 )1/2

11 94 F. Gómz / J. Math. Anal. Appl ) W hav thn for ach S Thrfor, [ W iθ )] β ns χ S = 2 2 )1/2 W L 2 D; D W ) = span { k= k= ikθ β ns +kχ S+k. } ikθ β ns +kχ S+k : S. 16) Th functional modl for a c.n.u. contraction W on a complx sparabl Hilbrt spac, its dilations and rsidual part can b found in th monograph of Sz.-Nagy and Foiaş [12, Sction VI.2 and Thorm VI.3.1]. For th rsidual part R, R it is givn by ˆR W L 2 D; D W ), ˆRv) iθ v iθ ) v ˆR), or in th light of 16) and th unitary quivalnc btwn R and U Corollary 12), { } ˆR span ikθ β ns +kχ S+k : S, ˆR k= k= ikθ β ns +kχ S+k ) = ik 1)θ β ns +kχ S+k. 17) k= On th othr hand, sinc in this analysis W can b idntifid with its rstriction to L 2 [0, 1) 2 ) C, which is of class C 01, th functional modl of W is of th form Ĥ = H 2 D; D W ) Θ W H 2 D; D W ), [Ŵ u) ] iθ ) = iθ [ u iθ ) u0) ], u Ĥ. 18) To obtain a convnint dscription of th spac Ĥ and th oprator Ŵ lt us considr th Fourir transformation F which carris th spacs L 2 D; D W ) and H 2 D; D W ) onto th spacs of D W -valud squncs l 2 Z; D W ) and l 2 Z + ; D W ), rspctivly, whr Z + {l Z: l 0}. F is givn by F : L 2 D; D W ) l 2 Z; D W ), a S iθ ) [ χ S F a l S ilθ S S l Z + whr a l S = 2π 1 2π ilθ a 0 S iθ ) dθ and )χ S ] = S χ l S F ilθ ) χ S =..., l 1) 0, χ l) S, l+1) ) 0,..., S, l Z. l Z + a l S χ l S, 19) Obviously, {χ l S : S, l Z} is an orthonormal basis for l2 Z; H) and {χ l S : S, l Z± } ar orthonormal bass for l 2 Z ± ; H). By virtu of 15), [ ΘW iθ )] ilθ ) χ S = γns ilθ χ S 1 + α ns β ns +k il+k+1)θ χ S+k ). Thrfor, [ F ΘW iθ ) ) F 1] χs) l = γns χ l S 1 + α n S β ns +kχ l+k+1. S+k W hav thn that an lmnt S j Z + a j χ j S S of l2 Z + ; D W ) is orthogonal to [F Θ W iθ )F 1 ]l 2 Z + ; D W ) iff ) γ ns a l S 1 + α n S β ns +ka l+k+1 = 0, S, l Z +. S+k

12 F. Gómz / J. Math. Anal. Appl ) Taking th quation for S + 1 and l + 1awayfromthquationforS and l w obtain th rcurrnc rlation so that a l+1 S = β n S a l S 1 β, S, l Z+, ns 1 a l S = β n S a 0 S l β, S, l Z+. ns l 20) Thrfor, by virtu of 18), th spctral rprsntation Ŵ : Ĥ Ĥ of W is givn by Ŵ S and, from 20), a S iθ ) χ S ) = S Ĥ = H 2 D; D W ) Θ W H 2 ) D; D W { = a S iθ ) χ S L 2 D; D W ): a S iθ ) = S Summing up: iθ a S iθ ) a 0 S) χs 21) ilθ β n S a 0 S l }. β, S ns l l=0 Proposition 17. Th Frobnius Prron oprator U of th bakr map givn in 9) and th associatd contraction W givn in 11) hav th rspctiv spctral rprsntations 17) and 21). By mans of th Fourir transformation F givn in 19) w can also obtain translation rprsntations from th spctral ons. It is not difficult to show that such translation rprsntations ar isomorphic to th original U and W givn in11) rstrictd to L 2 [0, 1) 2 ) C Grossmann gnralizd ignvalus of U By virtu of Proposition 16, in th nstd Hilbrt spac H, H,Λ ) th gnralizd ignvalus of U ar th ignvalus of W. On th othr hand, th point spctrum of W rstrictd to L 2 [0, 1) 2 C is th st of points D for which Θ T ) is not on-to-on [12, Thorm VI.4.1]. To obtain Kr Θ W ) w must solv th quation [Θ W )] S a S)χ S ) = 0, which amounts to a S ) = γ ns +1α ns +1 k β ns +ka S+k ), S. In particular, subtracting to this quation for S th quation for S + 1multiplidby w gt th rcurrnc rlation so that a S+1 ) = 1 β n S +1 a S ), β ns a S+k ) = k β n S +k β ns a S ), S, k Z. 22) It is immdiat to s that S a S) 2 convrgs whn c + < < c and that S a S) 2 divrgs if c < or < c +, whr c ± = lim k ± k±1 / k. Thrfor: Proposition 18. In th nstd Hilbrt spac H, H,Λ ) th st of Grossmann gnralizd ignvalus of U contains th annulus { C: c + < < 1} and is containd in th annulus { C: c + < 1}. Th corrsponding ignvctors S a S)χ S must satisfy 22). S Antoniou and Tasaki [2] and Tasaki [14] for gnralizd spctral dcompositions including th Pollicot Rull rsonancs.

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