Operator inequalities, functional models and ergodicity

a Xi :1908.05032 3 [ma h.FA] 2 Jul 2020 OPERATOR INEQUALITIES, FUNCTIONAL MODELS AND ERGODICITY LUCIANO ABADIAS, GLENIER BELLO, AND DMITRY YAKUBOVICH Abs ac . We discuss when an ope a o , subjec o a a he gene al inequali y in he edi a y o m, admi s a uni a ily equi alen unc ional model o Agle ype in he ep oducing ke nel Hilbe space associa ed o he inequali y. To he con a y o he p e ious wo k, he ke nel need no be o Ne anlinna-Pick ype. We de i e some consequences conce ning he e godic beha io o he ope a o . 1. In oduc ion 1.1. Mo i a ion. Le α( ) be a unc ion ep esen able by he powe se ies P∞ n=0 αn nin D:= {| |<1}, whe e he coe icien s αna e eal numbe s, and le T∈L(H) be a bounded linea ope a o on a Hilbe 1space H. Pu (1.1) α(T∗, T) := ∞ X n=0 αnT∗nTn, whe e he se ies is assumed o con e ge in he s ong ope a o opology SOT in L(H). When αis a polynomial, he se ies abo e is jus a ini e sum, and he e is no con e gence p oblem. In pa icula , when α( ) = 1 − , he igh hand side o (1.1) is I−T∗T, so T∈L(H) is a con ac ion i and only i (1 − )(T∗, T)≥0. In he 1960’s Sz.-Nagy and Foias de eloped a beau i ul spec al heo y o con ac ions (see [62]) based on he cons uc ion o hei unc ional model. In his landma k pape [5], Agle showed ha i Thas spec um σ(T) con ained in he uni disc Dand α(T∗, T)≥0, hen i is na u al o model Tby pa s o B⊗IE, whe e Bis a sui able weigh ed backwa d shi and IEis he iden i y ope a o on some auxilia y Hilbe space E. (By a pa o an ope a o we mean i s es ic ion o an in a ian subspace.) Mo e gene ally, when σ(T)⊂D, i has been ound in a ious pa icula cases ha ins ead o B⊗IE one should conside ope a o s o he o m (B⊗IE)⊕S, whe e Sis an isome y o a uni a y ope a o . This ep esen a ion is called a coanaly ic model. As Agle p o ed in [6], i holds, in pa icula , o m-hype con ac ions, i.e., ope a o s T∈L(H) such ha (1 − )j(T∗, T)≥0 o j= 1,...,m. Agle ’s heo em was gene alized in [46] by M¨ulle and Vasilescu o uples Da e: July 6, 2020. Key wo ds and ph ases. dila ion; unc ional model; ope a o inequali y; e godic p ope ies. 1All Hilbe spaces will be assumed o be sepa able 1 2L. Abadias, G. Bello, and D. Yakubo ich o ope a o s. The i s esul s on Agle model echniques a e exposed in he book [7] by Agle and McCa hy. In [48], Olo sson ob ained ope a o o mulas o wande ing subspaces, ele an in he models o m-hype con ac ions. His esul s we e gene alized by Eschmeie in [31] o uples o commu ing ope a o s, and by Ball and Bolo niko in [10] o wha hey call β-hype con ac ions. M¨ulle s udied he case whe e α=pis a polynomial in [45]. He conside s he class C(p) o ope a o s T∈L(H) such ha p(T∗, T)≥0. He p o es ha any con ac ion T∈ C(p) has a coanaly ic model whene e p(1) = 0,1/p( ) is analy ic in D, and 1/p( ¯wz) is a ep oducing ke nel. This las condi ion is equi alen o he ac ha all Taylo coe icien s o 1/p( ) a he o igin a e posi i e. M¨ulle also conside s some ope a o inequali ies o Twi h in ini ely many e ms, wi h he same p ope y o posi i i y. This pe mi s him o show ha any ope a o T is uni a ily equi alen o a pa o a backwa d weigh ed shi wi h he same spec al adius (see [45, Co olla y 2.3]). In [50], Olo sson deals wi h he case whe e αis no a polynomial. His assump ions a e ha αis analy ic on D, does no anish on D, and 1/α has posi i e Taylo coe icien s a he o igin. Unde his se ing, he s udies con ac ions Ton Hsuch ha α( T∗, T )≥0 o e e y ∈[0,1). Wi h mo e assump ions, he ob ains he coanaly ic model o his class o ope a o s. In [11], he las wo au ho s conside ed unc ions αin he Wiene algeb a AWo analy ic unc ions in he uni disc wi h summable sequence o Taylo coe icien s, subjec o ce ain condi ions. I was assumed ha he se ies PαnT∗nTncon e ges in no m. The ope a o s s udied he e u n ou o be simila o con ac ions (see [11, Theo em I]). This will no longe be ue in he se ing o he p esen pape (see Example 7.3). In [11], an explici model in he spi i o Sz.-Nagy and Foias model was cons uc ed o he class o ope a o s conside ed he e. The oles o he de ec ope a o and he de ec space we e played by (1.2) D:= (α(T∗, T))1/2,D:= DH, whe e he non-nega i e squa e oo is aken. 1.2. Ou se ing. He e he ope a o Dand he space D, de ined by (1.2) whene e α(T∗, T)≥ 0, will also play an impo an ole. Recall ha now we conside he con e gence o (1.1) in SOT. As i will be seen om Example 7.3, his is he app op ia e con e gence in his con ex . Ou assump ions a e he ollowing. Hypo heses 1.1. Suppose αis a unc ion in AWwhich does no anish on D. We pu k( ) = 1/α( ) = ∞ X n=0 kn n ∈D, wi h α0=k0= 1, and assume ha kn>0 o e e y n≥1. Ope a o Inequali ies, Func ional Models and E godici y 3 Unde Hypo heses 1.1, we deno e by Hk he weigh ed Hilbe space o powe se ies ( ) = P∞ n=0 n nwi h ini e no m k kHk:= ∞ X n=0 | n|2kn1/2 . Le Bkbe he backwa d shi on Hk, de ined by (1.3) Bk ( ) = ( )− (0) . De ini ion 1.2. Fix a unc ion αsa is ying Hypo heses 1.1, and le Tbe an ope a o in L(H). We say ha Tis α-modelable i Tis uni a ily equi alen o a pa o an ope a o o he o m (Bk⊗IE)⊕S, whe e Sis an isome y. We ema k ha Bk⊗IEac s on he Hilbe space Hk⊗ E, which can be iden i ied wi h he weigh ed Hilbe space o E- alued powe se ies ( ) = P∞ n=0 n nwi h no m gi en by k kHk⊗E =∞ X n=0 k nk2 Ekn1/2 . I ac s acco ding o he same o mula (1.3). I is na u al o pose he ollowing ques ion. Ques ion 1.3. Gi en a unc ion αsa is ying Hypo heses 1.1, gi e a good su icien condi ion o an ope a o T∈L(H) o be α-modelable. One o he s onges esul s in his di ec ion is con ained in he ecen pape s by Bickel, Ha z and McCa hy [17] and by Clouˆa e and Ha z [21]. I is s a ed o sphe ically symme ic uples o ope a o s. Fo he case o a single ope a o , hei esul can be o mula ed as ollows. Theo em 1.4 ([21, Theo em 1.3]).Le αbe a unc ion wi h α0= 1 and αn≤0 o all n≥1. Suppose ha k= 1/α has adius o con e gence 1,kn>0 o e e y n≥0and (1.4) lim n→∞ kn kn+1 = 1. Then Bkis bounded, and a Hilbe space ope a o Tis α-modelable i and only i α(T∗, T )≥0. I is easy o see ha he hypo heses o Theo em 1.4 imply Hypo heses 1.1. This heo em conce ns he Ne anlinna-Pick case, ha is, when α0= 1 and αn≤0 o n≥1. Al e na i ely, we say ha kis a Ne anlinna-Pick ke nel. In he ecen wo k [22], Clouˆa e, Ha z and Schillo es ablish a Beu ling–Lax–Halmos heo em o ep oducing ke nel Hilbe spaces in he Ne anlinna-Pick con ex . We e e he eade o [26, 49, 56, 57] o mo e esul s in he Ne anlinna-Pick case. In he ecen p ep in [32], Eschmeie and To h ex end p e ious esul s by Eschmeie [31] o all comple e Ne anlinna-Pick ke nels, in he con ex o ope a o uples. 1.3. Main esul s. The ollowing esul gi es a new answe o Ques ion 1.3. 4L. Abadias, G. Bello, and D. Yakubo ich Theo em 1.5. Assume Hypo heses 1.1. I k∈AW, and i s Taylo coe icien s {kn}sa is y k1/n n→1,sup kn/kn+1 <∞and (1.5) lim m→∞ sup n≥2mX m≤j≤n/2 kjkn−j kn = 0, hen Bkis bounded, and he ope a o T∈L(H)is a pa o Bk⊗IE( o some Hilbe space E) i and only i bo h P|αn|T∗nTnand PknT∗nTncon e ge in SOT and α(T∗, T)≥0. Mo eo e , in his case one can ake E=D. As i will be seen la e , he SOT-con e gence o P|αn|T∗nTnimplies he he SOT- con e gence o PαnT∗nTn. No ice ha in Theo em 1.5, he isome ic pa Sis unnecessa y (see Theo em 1.12 (ii) below o mo e in o ma ion). This heo em shows ha Tis α-modelable in many cases when kis no a Ne anlinna-Pick ke nel, and so Theo em 1.4 does no apply. No much abou hese ke nels has been known p e iously. Gi en an in ege N≥2, he e a e examples o unc ions ksa is ying he hypo heses o Theo em 1.5 wi h wha e e p esc ibed signs o he coe icien s α2,...,αN(see Example 5.1). No e ha α1=−k1is always nega i e. Rema k 1.6. Suppose ha k∈AWand he sequence kn kn+1 1 + 1 n+ 1a is inc easing o some a > 1. Then (1.5) holds. This is close o [60, P oposi ion 34]. Indeed, pu k∗ j:= (j+ 1)−a, and de ine ρj:= kj/k∗ j. Then ou condi ion educes o he condi ion ρn+2/ρn+1 ≥ρn+1/ρn, o all n, which implies ha ρjρn−j/ρn≤C, o 0 ≤j≤n. Since kjkn−j kn =ρjρn−j ρn k∗ jk∗ n−j k∗ n and {k∗ n}sa is ies (1.5), i ollows ha {kn}also sa is ies (1.5). Hence, o su icien ly egula sequences {kn}, he condi ion (1.5) is a he close o he condi ion Pkn<∞. I can be added ha , in ac , in Theo em 1.5 {kn}need no be egula ; mo eo e , he quo ien s kn/kn+1 need no con e ge (see Rema k 5.2). The echniques employed in he p oo a e di e en om [21]. We use, basically, a combi- na ion o M¨ulle ’s a gumen s in [45] and Banach algeb as echniques. The abo e heo ems open he ques ion o desc ibing in a ian subspaces o Bk⊗IEand o cons uc ing a unc ional model o ope a o s unde he s udy, which ce ainly would be in e es ing. We do no add ess his ques ion in his pape . Gi en an ope a o C:H→ E, whe e Eis an auxilia y Hilbe space, we de ine (1.6) VCx(z) = C(IH−zT)−1x, x ∈H, z ∈D. Ope a o Inequali ies, Func ional Models and E godici y 5 The nex esul shows ha whene e Tis α-modelable, he ope a o VD:H→ Hk⊗Dis a con ac ion, and we can gi e an explici model o T( ha is, gi e explici ly E,Sand he ans o m which sends he ini ial space in o he model space). Fi s we need o s a e one mo e echnical hypo hesis, whose meaning will be clea la e . Hypo heses 1.7. Le αbe a unc ion sa is ying Hypo hesis 1.1. Pu (1.7) β( ) = X n≥0 βn n,whe e βn=|αn|, and γ( ) = β( )k( ). We assume ha kn/kn+1 ≤C′and γn≤C′′kn o all n≥0. The condi ion kn/kn+1 ≤C′is equi alen o boundedness o Bk. I i holds, hen he second condi ion is sa is ied whene e he e is some Nsuch ha ei he αn≥0 o n≥N, o αn≤0 o n≥N. Theo em 1.8 (Explici model).Assume Hypo heses 1.1 and 1.7. Le Tbe α-modelable. Then α(T∗, T)≥0,VDis a con ac ion, and hence we can de ine W= (IH−V∗ DVD)1/2,W=WH. Mo eo e , S:W → W, gi en by SW x := WTx, is an isome y and he ope a o (VD, W) : H→(Hk⊗D)⊕W,(VD, W )h= (VDh, Wh) p o ides a model o T, in he sense ha (VD, W)is isome ic and ((Bk⊗ID)⊕S)·(VD, W) = (VD, W)·T. Rema k 1.9. Suppose αsa is ies he abo e wo hypo heses, and suppose ha Tis an α- modelable ope a o , which is gi en al eady by i s model wi hou he isome ic pa . Tha is, he e is an in a ian subspace Lo an ope a o Bk⊗IE, ac ing on Hk⊗ E, such ha Tis he es ic ion o his ope a o o L. Then D=E(iden i ied wi h he cons an unc ions in Hk⊗E), and VDis he iden i y ope a o on L. This ollows om Co olla y 2.13 below. Simila ly, in he gene al case, i Tis a pa o an ope a o (Bk⊗IE)⊕S, whe e Sis an isome y, he e is a uni a y ope a o usuch ha he ans o m (VD, uW ) is jus he iden i y. I i is known ha Tis α-modelable, one can ask abou he uniqueness o he model. Fo answe ing his ques ion, we need he ollowing de ini ions. De ini ion 1.10. Le Lbe an in a ian subspace o (Bk⊗IE)⊕S, whe e S:W → W is an isome y. We will say ha he co esponding model ope a o (Bk⊗IE)⊕S|L is minimal i he ollowing wo condi ions hold. (i) Lis no con ained in (Hk⊗E′)⊕W o any E′$E. 6L. Abadias, G. Bello, and D. Yakubo ich (ii) Lis no con ained in (Hk⊗E)⊕W′ o any W′$Win a ian by S. In Rema k 3.5 we show ha he explici model ob ained in Theo em 1.8 is indeed minimal. No e ha unde Hypo heses 1.1, αis de ined on he closed uni disc Dand does no anish on he in e al [0,1). Since α(0) = α0= 1, we ob ain ha α(1) ≥0. We dis inguish he ollowing wo cases. This dis inc ion appea s al eady in [21, Subsec ion 2.3] o he Ne anlinna-Pick case. De ini ion 1.11. Suppose ha αmee s Hypo heses 1.1. We will say ha αis o c i ical ype (o , al e na i ely, ha we ha e he c i ical case) i α(1) = 0. I α(1) >0, we will say ha α is o subc i ical ype (o , al e na i ely, ha we ha e he subc i ical case). Theo em 1.12 (Uniqueness o he minimal model).Suppose ha αmee s Hypo heses 1.1 and 1.7. Le Tbe an α-modelable ope a o . (i) In he c i ical case, he minimal model o Tis unique. Mo e p ecisely, he pai o ans o ms (VD, W0), whe e W0= (I−V∗ DVD) : H→ W0and W0:= Ran(I−V∗ DVD), gi es ise o a minimal model, and any minimal model is p o ided by (VC, W), whe e C= D,W=wW0:H→ W and , w a e uni a y isomo phisms. (ii) In he subc i ical case, he minimal model o Tis no unique, in gene al. Howe e , he e always exis s a minimal model gi en by V=VD, in he sense ha VD:H→ Hk⊗IDis an isome y such ha (Bk⊗ID)VD=VDT. No e ha in his case he isome y Sis absen . We ema k ha he e a e o he wo ks ha gi e answe s o he abo e Ques ion 1.3. In pa icula , Po [52] ga e a model o ope a o s sa is ying wo inequali ies (1 −p)(T∗, T)≥0 and (1 −p)m(T∗, T)≥0, whe e pis a polynomial wi h nonnega i e coe icien s, m≥1 and p(0) = 0 ( his class is a gene aliza ion o m-hype con ac ions). In ac , she ea s uples o ope a o s. In [10], Ball and Bolo niko conside a unc ion α( ) in he Wiene algeb a such ha k= 1/α has posi i e coe icien s sa is ying 0 < ε ≤kn/kn+1 ≤1 o all n(so ha Bk is a con ac ion). They show ha an ope a o Tis α-modelable, wi h absen isome ic pa , i and only i i α(T∗, T)≥0 as well as in ini ely many addi ional inequali ies hold (Tis β- hype con ac i e, whe e βn= 1/kn), and Tis wha hey call β-s able. See [10], Theo em 4.3. In [10], Theo em 7.2, Ball and Bolo niko gi e a model o Tin e ms o hei gene aliza ion o he cha ac e is ic unc ion, which is an in ini e amily o ope a o - alued unc ions. Whe eas hese au ho s ea bo h subc i ical and c i ical cases, Theo em 1.5 only conce ns he subc i ical case (because o he condi ion k∈AW). 1.4. Consequences o he model. I an ope a o Tis α-modelable, i is na u al o s udy wha consequences can be de i ed om he model. He e we ob ain wo ypes o consequences: (1) when he de ec ope a o Dhas ini e ank ( ha is, dim D<∞), and (2) e godic consequences when α( ) = (1 − )awi h 0 < a < 1. Ope a o Inequali ies, Func ional Models and E godici y 7 We will use he space Rk=H˜ k, whe e ˜ kn= 1/kn. I is easy o see ha i is he ep o- ducing ke nel Hilbe space, co esponding o he posi i e de ini e ke nel k(z, w) := k( ¯wz). The pai ing h , gi=P n¯gn( ∈ Hk, g ∈ Rk) makes Rkna u ally dual o Hκ. In his in e p e a ion, he adjoin ope a o o Bkis he ope a o g(z)7→ zg(z), ac ing on Rk. I αis o subc i ical ype, we ha e he ollowing esul ela ed o he Ca leson condi ion. Theo em 1.13. Le Tbe an ope a o simila o a pa o Bk⊗ID, ac ing on he space Hk⊗D, whe e Rkis a Banach algeb a and Dis ini e dimensional. Suppose ha lim n→∞in j≥0 kj kn+j1/n = lim n→∞k1/n n= 1, and also ha (1.8) ∞ X n=N kn≤CN−ε∀N≥0, o some posi i e cons an s Cand εwhich do no depend on N. Suppose ha he spec um σ(T)does no co e D. Pu E:= (σ(T)∩D)∩T and le {lν}deno e he leng hs o he ini e complemen a y in e als o E(in T). Then he Lebesgue measu e o Eis 0, and he Ca leson condi ion holds: X ν lνlog 2π lν <∞. Some o he a gumen s employed in he p oo o his heo em a e ela ed wi h he so-called index o an in a ian subspace o Rk⊗E; see Sec ion 6 o mo e de ails. In he c i ical case, an impo an amily o unc ions αa e hose o he o m α( ) := (1− )a, o a > 0. No e ha hey sa is y Hypo heses 1.1. When a=mis a posi i e in ege , i is said ha T∈L(H) is an m-con ac ion i (1 − )m(T∗, T)≥0, and ha Tis an m-isome y i (1 − )m(T∗, T) = 0. The pape s [13, 15, 16, 35, 55] (among o he s) s udy m-isome ies. The pape [43] is dedica ed o a p o ound s udy o 2-isome ies. In [36], Gu ea s a mo e gene al class o (m, p)-isome ies on Banach spaces, and in [37], he discusses m-isome ic uples o ope a o s on a Hilbe space. In [20], Cha an and Sholapu ka s udy ano he in e es ing class o ope a o s: Tis a join comple e hype expansion o o de mi (1 − )n(T∗, T)≤0 o e e y in ege n≥m. Tha wo k, in ac , is de o ed o uples o commu ing ope a o s. He e we in oduce he case when he exponen ais no an in ege . The de ini ions o a-con ac ion and a-isome ies a e he na u al ones: we say ha Tis an a-con ac ion i (1 − )a(T∗, T)≥0, and Tis an a-isome y i (1 − )a(T∗, T) = 0. No e ha α( ) := (1 − )ais o Ne anlinna-Pick ype when 0 < a < 1. In his case, wi h he help o he model gi en by Theo em 1.4, we will ge he ollowing wo e godic esul s. 8L. Abadias, G. Bello, and D. Yakubo ich Theo em 1.14. I Tis an a-con ac ion, wi h 0< a < 1, hen Tis quad a ically (C, b)- bounded o any b > 1−a. Tha Tis quad a ically (C, b)-bounded (whe e he le e C s ands o Ces`a o) means ha he e exis s a cons an c > 0 such ha sup n≥0 1 kb+1(n) n X j=0 kb(n−j)kTjxk2≤ckxk2(∀x∈H), whe e he numbe s k−s(n), called Ces`a o numbe s, a e de ined by (1 − )s=: ∞ X n=0 k−s(n) n. As we will show (see Example 7.3), o any a∈(0,1), he class o a-con ac ions on His s ic ly la ge han he class o con ac ions. I is ob ious ha any con ac ion is quad a ically (C, b)-bounded (due o he equali y Pn j=0 kb(n−j) = kb+1(n) o any b > 0). The meaning o he abo e ac is ha some e godic p ope ies o con ac ions s ill hold ue o a-con ac ions. Theo em 1.15. Le Tbe an a-con ac ion wi h 0< a < 1and le b > 1−a. Then he ollowing s a emen s a e equi alen . (i) The isome y Sdoes no appea in he (1 − )a-model o T. (ii) Fo e e y x∈H, (1.9) ∃lim n→∞ 1 kb+1(n) n X j=0 kb(n−j)kTjxk2= 0. (iii) Fo e e y x∈H, lim in n→∞ kTnxk= 0. Rema k 1.16. Fo any a∈(0,1), he e a e a-con ac ions which a e no con ac ions. This ollows om Theo em 7.2 below. The same holds o a > 1. Indeed, i m < a ≤m+ 1, whe e mis an in ege , hen i is easy o ge (see ou o hcoming pape [1]) ha any (m+1)-isome y Tis also an a-isome y, which means ha (1 − )a(T∗, T) = 0. The e a e (m+ 1)-isome ies ha a e no con ac ions, and each o hem is an example o his ype. 1.5. Con en s. The pape is o ganized as ollows. In Sec ion 2 we in oduce wo amilies o ope a o s in L(H) depending on a ixed unc ion α( ) = Pn≥0αn n: Admw αand Cw α. Essen ially, Admw αis he amily o ope a o s T o which we can de ine α(T∗, T), and i s sub amily Cw αconsis s o hose T o which α(T∗, T )≥0. We use he supe sc ip no a ion “w” in Admw αand Cw α o make i easie o compa e he esul s om [11] and om he p esen pape . No ice ha in [11], only he con e gence o We ob ain some in e es ing p ope ies o hese amilies and cha ac e ize he membe ship o backwa d and o wa d weigh ed shi s o hem. In Sec ion 3, we p o e Theo ems 1.8 and 1.12. Ope a o Inequali ies, Func ional Models and E godici y 9 The p oo o Theo em 1.5 is gi en in Sec ion 4. In Sec ion 5 we s udy he scope o Theo em 1.5. The e we p esen examples sa is ying he hypo hesis o Theo em 1.5, whe e Theo em 1.4 does no apply. In Sec ion 6 we p o e Theo em 1.13. The p oo s o Theo ems 1.14 and 1.15 a e gi en in Sec ion 7. In ou o hcoming pape [1], we will s udy models up o simila i y (ins ead o uni a y equi alence). The e we will conside unc ions α ha may ha e ze oes in D. We will p o e ha unde ce ain hypo heses, any ope a o in Cw αis simila o an a-con ac ion i α( ) “beha es like” (1 − )ain a neighbo hood o 1. We will also s udy a-con ac ions in mo e de ail. 2. P elimina ies on classes de ined by ope a o inequali ies In his sec ion we in oduce he ope a o classes Admw αand Cw αassocia ed o a unc ion α( ) = Pn≥0αn n, wi h αn∈R. A e s udying hem, we analyze why Hypo heses 1.1 a e na u al. Finally, a he end o he sec ion we discuss he membe ship o weigh ed shi s in he classes Admw αand Cw α. 2.1. The classes Admw αand Cw α.Be o e en e ing in o he de ini ions and basic p ope ies o hese classes, le us men ion he ollowing well known esul ha will be used epea edly. Lemma 2.1 (see [38, P oblem 120]).I an inc easing sequence {An}o sel adjoin Hilbe space ope a o s sa is ies An≤CI o all n, whe e Cis a cons an , hen {An}con e ges in he s ong ope a o opology. De ini ion 2.2. Gi en a unc ion α( ) = Pn≥0αn nwi h αn∈R, we pu (2.1) Admw α:= T∈L(H) : ∞ X n=0 |αn|kTnxk2<∞ o e e y x∈H. No e ha his class o ope a o s is no a ec ed i we change he signs o some coe icien s αn’s. I Xand Ya e wo quan i ies ( ypically non-nega i e), hen X.Y(o Y&X) will mean ha X≤CY o some absolu e cons an C > 0. I he cons an Cdepends on some pa ame e p, hen we w i e X.pY. We w i e X≍Ywhen bo h X.Yand Y.X. P oposi ion 2.3. The ollowing s a emen s a e equi alen . (i) T∈Admw α. (ii) P∞ n=0 |αn|kTnxk2.kxk2 o e e y x∈H. (iii) The se ies P∞ n=0 |αn|T∗nTncon e ges in he s ong ope a o opology in L(H). 16 L. Abadias, G. Bello, and D. Yakubo ich o e e y x∈Hand e e y non-nega i e in ege n. Fix a posi i e in ege N. Then N X n=0 knkDTnxk2= N X n=0 kn ∞ X m=0 αmTm+nx2 =∞ X j=0  X n+m=j, n≤N knαm Tjx2=: ∞ X j=0 τjTjx2. Since αk = 1 we ge τ0= 1 and τ1=···=τN= 0. Mo eo e , τN+i=k0αN+i+···+kNαi<0 o e e y i≥1, because all he αj’s abo e a e nega i e o ze o and he kj’s a e posi i e. The e o e N X n=0 knkDT nxk2≤ kxk2 o e e y Nand hence he se ies PknkDTnxk2con e ges o e e y x∈H. This gi es kVDxk2=∞ X n=0 knkDT nxk2≤ kxk2, as we wan ed o p o e.  The ollowing ac is simple and well-known . P oposi ion 3.2. Le T∈L(H)wi h σ(T)⊂D, and le Ebe a Hilbe space. A bounded ans o m V:H→ Hκ⊗E sa is ies (3.1) V T = (Bκ⊗IE)V i and only i he e is a bounded linea ope a o C:H→ E such ha V=VC(see (1.6)). P oo . I is well-known (and s aigh o wa d) ha any bounded ans o m VCsa is ies (3.1). Con e sely, suppose ha V T = (Bκ⊗IE)V. De ine an(x) by V x(z) := ∞ X n=0 an(x)zn, x ∈H. Then ∞ X n=0 an(Tx)zn=V Tx = (Bκ⊗IE)V x =∞ X n=0 an+1(x)zn. The e o e an+1(x) = an(Tx). The s a emen ollows, pu ing C:= a0, which has o be a bounded linea ope a o .  P oposi ion 3.3. Le C:H→ E be a bounded ope a o and le T∈ Cw α. Then he e exis s a bounded ope a o W:H→ W such ha he ope a o (VC, W )is isome ic and ans o ms Ope a o Inequali ies, Func ional Models and E godici y 17 Tin o a pa o he ope a o (Bk⊗IE)⊕S, whe e S∈L(W)is an isome y, i and only i he ollowing condi ions hold. (i) VC:H→ Hk⊗E is a con ac ion. (ii) Fo e e y x∈H, kxk2−kVCxk2=kTxk2−kVCTxk2. P oo . Le us suppose i s he exis ence o such ope a o W. Since (VC, W ) is an isome y, (i) holds. No ice ha (ii) is equi alen o p o ing ha kWxk2=kWTxk2 o e e y x∈H. Bu his is also immedia e since SWx =W T x and Sis an isome y. Con e sely, suppose now ha (i) and (ii) a e ue. By (i), we can pu W:= (I−V∗ CVC)1/2 and W:= Ran W. Using (ii) we ha e (3.2) kW xk2=kxk2−kVCxk2=kTxk2−kVCTxk2=kWT xk2. We de ine S(Wx) := W T x, o e e y x∈H. No e ha Sis well de ined, since kSW xk=kWxkby (3.2). Since WH is dense in W,Scan be ex ended o an isome y on W. By he de ini ion o W, we know ha (VC, W) is an isome y and i is immedia e ha (Bk⊗ID)VC=VCTand SW =WT. This comple es he con e se implica ion.  P oposi ion 3.4. Le T∈ Cw α. Assume ha C:H→ E and W:H→ W a e any bounded ope a o s such ha (VC, W )is isome ic on (Hk⊗E)⊕W and ans o ms Tin o a pa o (Bk⊗IE)⊕S, whe e S∈B(W)is an isome y. Then Cand Da e ela ed by (3.3) kDxk2=kCxk2+α(1)kWxk2,∀x∈H. P oo . Since (VC, W) is isome ic, we ha e (3.4) kxk2=kVCxk2+kW xk2=∞ X n=0 knkCTnxk2+kWxk2, o e e y x∈H. Subs i u ing xby Tjxabo e and mul iplying by αj, we ob ain ha αjTjx2=∞ X n=0 αjknCTn+jx2+αjWTjx2 =∞ X n=0 αjknCTn+jx2+αjkWxk2, 18 L. Abadias, G. Bello, and D. Yakubo ich whe e we ha e used ha kWxk2=kWTxk2. The e o e kDxk2=∞ X j=0 αjTjx2=∞ X j=0 ∞ X n=0 αjknCTj+nx2+ ∞ X j=0 αj kWxk2 (⋆) =∞ X m=0  X j+n=m αjkn kCTmxk2+α(1) kWxk2. Since αk = 1, he only non- anishing summand in he las se ies abo e is o m= 0 and we ob ain (3.3). Finally, no e ha he ea angemen in (⋆) is co ec as ∞ X j=0 ∞ X n=0 |αj|knCTn+jx2≤∞ X j=0 |αj|Tjx2<∞, whe e we ha e used (3.4) and ha T∈ Cw α. Recall he de ini ion o he minimal model (De ini ion 1.10). Rema k 3.5. Suppose ha Tis α-modelable. Then Tis uni a ily equi alen o (Bk⊗IE)⊕ S|L, whe e L=Ran (VC, W). This model is minimal i and only i (a) Ran C=E; and (b) Ran W=W. Indeed, in his case, i is easy o see ha (a) is equi alen o (i), and (b) is equi alen o (ii) in De ini ion 1.10. P oo o Theo em 1.12. Suppose ha he hypo heses a e sa is ied. Fi s we no ice ha α(T∗, T)≥0, as i ollows om Co olla y 2.13 and P oposi ion 2.6. The e o e Dis well- de ined. (i) In he c i ical case (i.e., α(1) = 0), (3.3) gi es kDxk=kCxk ∀x∈H, so he e exis s a uni a y ope a o such ha C= D. This implies he s a emen . (ii) Suppose we a e in he subc i ical case (i.e., α(1) >0). Fi s , we ema k ha he model is no unique in gene al. Fo ins ance, ake T=Uany uni a y ope a o . Using P oposi ion 2.3 and ha α∈AW, we ob ain ha T∈Admw α. Since ∞ X n=0 αnkTnxk2= ∞ X n=0 αn!kxk2∀x∈H, we ge ha α(T∗, T) = α(1)I≥0. Ob iously, T=Uis a minimal model o T(whe e E= 0, and W=H). Mo eo e , i k= 1/α i s (1.5), hen Theo em 1.5 (which is p o ed in he nex sec ion, bu i s p oo is comple ely independen ) gi es ano he model o T. (See Example 5.1 and Rema k 5.2.) Ope a o Inequali ies, Func ional Models and E godici y 19 Now suppose ha Tis any α-modelable ope a o and (VC, W ) p o ides i s model. Le us see ha he e exis s a minimal model o Twi h V=VDand Wabsen . Changing xby Tnx in (3.3) we ob ain kDT nxk2=kCT nxk2+α(1) kWxk2, whe e we ha e used ha kWT xk=kW xk. The e o e kVDxk2=∞ X n=0 knkDT nxk2=∞ X n=0 knkCTnxk2+k(1)α(1) kW xk2 =kVCxk2+kWxk2=kxk2, so VD:H→ Hk⊗E is an isome y and he e o e p o ides a model o T. The space Lis jus Ran VDin Hk⊗D(which is closed). This model is minimal, because Ran Dis dense in D. (See Rema k 3.5.) This gi es all s a emen s o (ii).  P oo o Theo em 1.8. I is an immedia e consequence o Theo em 1.12 and P oposi ion 3.3 (i) ha VDis a con ac ion. Finally, o p o ing ha (VD, W) gi es a model, we jus need o use he same a gumen employed in he ecip ocal implica ion o P oposi ion 3.3.  Resul s close o Theo ems 1.8 and 1.12 appea in Schillo’s PhD hesis [58]. He deals wi h he gene ali y o uples o commu ing ope a o s, bu o he case o one ope a o , he hypo heses needed he e a e mo e es ic i e han ou s. Fo example, in [58, Theo em 5.16], he uniqueness o he coex ension is p o ed when T is wha he calls a s ong k-con ac ion. Fo one single ope a o Tand using ou no a ions, hese a e ope a o s such ha α(T∗, T)≥0, he limi Σ(T) := IH−lim N→∞ N X n=0 knT∗nα(T∗, T)Tn exis s (in SOT), Σ(T)≥0, and Σ(T) = T∗Σ(T)T. In [58, Co olla y 5.17], he gi es an explici model in ol ing he de ec space DT. His assump ions a e somewha echnical (see [58, Assump ion 5.8]). He also assumes he exis ence o α(B∗ k, Bk), o which [58, P oposi ion 2.10] says ha a su icien condi ion is ha he coe icien s {αn}o he unc ion αha e e en ually he same sign. Recall ha ou Theo em 1.12 (ii) says ha o he subc i ical case he model is no unique in gene al. The e o e, since Schillo ob ains uniqueness o he coex ension, i seems ha his assump ions exclude he subc i ical case. Schillo’s hesis also con ains a esul on he desc ip ion o in a ian subspaces o a backwa d shi , analogous o Bk⊗IE, in his se ing o ope a o uples. No ice ha in Theo ems 1.8 and 1.12 we a e only assuming ha Tis α-modelable. In pa icula , we do no impose any es ic ion abou he signs o he Taylo coe icien s o he unc ion α. 20 L. Abadias, G. Bello, and D. Yakubo ich 4. P oo o Theo em 1.5 In his sec ion we p o e Theo em 1.5. Fo ha , we need o ci e some esul s conce ning Banach algeb as. Fo any sequence ω={ωn}∞ n=0 o posi i e weigh s, de ine he weigh ed space ℓ∞(ω) := ( ( ) = ∞ X n=0 n n: sup n≥0| n|ωn<∞). In gene al, i s elemen s a e o mal powe se ies. We will also use he sepa able e sion o his space: ℓ∞ 0(ω) := ( ( ) = ∞ X n=0 n n: lim n→∞| n|ωn= 0). P oposi ion 4.1 (see [47]).ℓ∞(ω)is a Banach algeb a (wi h espec o he o mal mul ipli- ca ion o powe se ies) i and only i (4.1) sup n≥0 n X j=0 ωn ωjωn−j <∞. Theo em 4.2. Le ωn>0and ω1/n n→1. I supnωn+1/ωn<∞and (4.2) lim m→∞ sup n≥2mX m≤j≤n/2 ωn ωjωn−j = 0, hen he ollowing is ue. (i) ℓ∞(ω)is a Banach algeb a. (ii) I ∈ℓ∞(ω)does no anish on D, hen 1/ ∈ℓ∞(ω). P oo . The hypo heses imply (4.1), so ha (i) ollows om P oposi ion 4.1. To ge (ii), we ap- ply he esul s o he pape [29] by El-Fallah, Nikolski and Za abi. We use he no a ion o his pape . Pu ω′(n) = ω(n)/(n+ 1), A=ℓ∞(ω) and A0=ℓ∞ 0(ω). The hypo heses imply ha A (and hence A0) is compac ly embedded in o he mul iplie con olu ion algeb a mul (ℓ∞(ω′)), see [29, Lemma 3.6.3]. Hence, by [29, Theo em 3.4.1], o any ∈A0,δ1(A0,M(A0)) = 0, see [29, Subsec ion 0.2.3] o he de ini ion o his quan i y. This means ha o any δ > 0 he e is a cons an c1(δ)<∞such ha he condi ions ∈A0,k kA= 1 and | |> δ on D imply ha 1/ ∈A0and k1/ kA≤c1(δ). In pa icula , (ii) holds o in A0. To ge (ii) in he gene al case, suppose ha ∈Aand | |> δ > 0 on D. Since ( )∈A0 o all < 1, we ge ha he no ms o he unc ions 1/ ( ) in Aa e uni o mly bounded by c1(δ) o all < 1. When →1−, each Taylo coe icien o 1/ ( ) ends o he co esponding Taylo coe icien o 1/ ( ). I ollows ha 1/ is in A(and k1/ kA≤c1(δ)).  P oo o Theo em 1.5. Pu ωn:= 1/kn. Ope a o Inequali ies, Func ional Models and E godici y 21 The i s pa o Theo em 1.5 ( ha Bkis bounded) is s aigh o wa d. Also, by Theo- em 4.2 (i), ℓ∞(ω) is an algeb a. Fi s suppose ha Tis a pa o Bk⊗IE, and le us p o e ha Bk∈ Cw α∩Admw k. By Theo em 2.12 (i), we know ha Bk∈Admw ki and only i m X j=0 kjkm−j.km, which ollows om Theo em 4.2 (i) and P oposi ion 4.1. Now le us see ha Bk∈ Cw α. By Theo em 4.2 (ii), α= 1/k belongs o ℓ∞(ω), and he e o e |αn|.kn. Then, since Bk∈Admw k, we ob ain ha Bk∈Admw α. Finally, Theo em 2.12 (ii) gi es ha Bk∈ Cw α(because αk = 1 has non-nega i e Taylo coe icien s). Hence Talso is in Cw α∩Admw k. Con e sely, le us assume now ha T∈ Cw α∩Admw k. We wan o p o e ha Tis a pa o Bk⊗IE. We adap he a gumen o [45, Theo em 2.2] (whe e he con e gence o he se ies o ope a o s is in he uni o m ope a o opology). By P oposi ion 3.2, (4.3) (Bk⊗ID)VD=VDT. Mo eo e , kVDxk2=∞ X n=0 knkDTnxk2=∞ X n=0 kn ∞ X m=0 αmTn+mx2 =∞ X j=0 X n+m=j knαmTjx2=kxk2, whe e we ha e used ha Pn+m=jknαmis equal o 1 i j= 0 and is equal o 0 i j≥1. The e-a angemen o he se ies is co ec since, using ha T∈Admw α∩Admw k, we ha e ∞ X n=0 kn ∞ X m=0 |αm|Tn+mx2.∞ X n=0 knkTnxk2.kxk2 and he se ies con e ges absolu ely. Hence VDis an isome y. Joined o (4.3), his p o es ha Tis uni a ily equi alen o a pa o Bk⊗ID. No ice ha in pa icula , we showed ha he hypo heses o Theo em 1.5 imply Hypo he- ses 1.7. 5. Discussion o Theo em 1.5 In his sec ion we discuss he scope o Theo em 1.5 and gi e a se ies o examples whe e i applies, whe eas Theo em 1.4 does no . We also will gi e a di ec p oo o a pa icula case o Theo em 4.2, which does no use he esul s o [29]. 22 L. Abadias, G. Bello, and D. Yakubo ich Gi en an analy ic unc ion ( ) = P n n, we deno e by [ ]Ni s unca ed polynomial o deg ee N, ha is, [ ]N:= 0+ 1 +...+ N N. Example 5.1. Le σ2,...,σNbe an a bi a y sequence o signs ( ha is, a sequence o numbe s ±1). We asse ha he e a e unc ions α, k mee ing all he hypo heses o Theo em 1.5 such ha sign(αn) = σn, o n= 2,...,N. This is in con as wi h Theo em 1.4, whe e he Ne anlinna-Pick condi ion was assumed: αn≤0 o n≥2. To p o e he exis ence o αand kas abo e, ake a polynomial eαo deg ee Nsuch ha eα0= 1,eα1<0. Fo n= 2,...,N, we se eαn<0 i σn=−1 and eαn= 0 i σn= 1. Pu ek:= [1/eα]N. The o mula (5.1) ekn=X s≥1 n1+···+ns=n (−1)seαn1···eαns shows ha all he coe icien s o eka e posi i e. We also equi e ha nei he eαno he polynomial ek anish on D. I is so i , o ins ance, |αn|a e su icien ly small o n= 2,...,N. Now pe u b he coe icien s eαj ha a e equal o ze o, ob aining a new polynomial bαsuch ha bαj:= (εi σj= 1 eαjo he wise (2 ≤j≤N). By con inui y, i ε > 0 is small enough, we can gua an ee ha he polynomial bk= [1/bα]N also has posi i e Taylo coe icien s, and we can also gua an ee ha bk(which is a sligh pe u ba ion o ek) does no anish on D. Finally, ake as kany unc ion in AWwi h eal Taylo coe icien s such ha he i s ones a e k0=bk0= 1, k1=bk1, . . . , kN=bkN, and kn−j kn≤C0(∀n≥2j), o some cons an C0. Fo ins ance, one can pu kn=An−b o n > N, wi h A > 0 (small enough) and b > 1. Then k∈AWdoes no anish on D. Then ob iously ksa is ies (1.5) and hence all he hypo heses o Theo em 1.5. The unc ion α:= 1/k in AWhas he desi ed pa e n o signs. Finally, i is impo an o no e ha α1=−k1is always nega i e. Rema k 5.2. I is also easy o see ha whene e {kn}sa is ies (1.5), any o he sequence {˜ kn}wi h k0= 1 and c < ˜ kn/kn< C o n > 1, whe e c, C a e posi i e cons an s, also sa is ies his condi ion. In pa icula , i {kn}sa is ies (1.5) and {˜ kn}is as abo e, whe e C is su icien ly small, hen k( ) is in e ible in AW, so ha all hypo heses o Theo em 1.5 a e Ope a o Inequali ies, Func ional Models and E godici y 23 ul illed. So he e a e many examples o unc ions k( ) mee ing hese hypo heses, such ha he quo ien s kn/kn+1 do no con e ge. Le us men ion now some ema ks on Theo em 4.2. Rema k 5.3. I is immedia e ha he condi ion (5.2) ωn ωjωn−j≤τj(∀n≥2j),whe e ∞ X j=0 τj<∞, implies (4.2) and (4.1) (in pa icula , supnωn+1/ωn<∞). Le us gi e a di ec p oo o Theo em 4.2 o his pa icula case. S a emen (i) ollows using P oposi ion 4.1. (ii) Pu g:= 1/ . Suppose ha g6∈ ℓ∞(ω). This means ha sup n≥0|gn|ωn=∞. Hence, i is clea ha he e exis s a sequence {ρ0 n}in [0,1] such ha ρ0 n→0 (slowly) and (5.3) sup n≥0|gn|ωnρ0 n=∞. Claim. The e exis s a sequence {ρn}wi h (5.4) ρ0 n≤ρn≤1 and ρn→0 such ha eωn:= ρnωnde ines a Banach algeb a ℓ∞(eω). Indeed, since Pτj<∞, he e exis s a sequence o posi i e numbe s {cj}such ha cjր ∞ and s ill Pcjτj<∞. Take any sequence {ρn} ha dec eases, ends o ze o, and sa is ies ρn≥max(ρ0 n,1/cn). Then, o eωn:= ρnωnwe ha e eωn eωjeωn−j =ωn ωjωn−j ρn ρjρn−j≤ωn ωjωn−j 1 ρj≤τjcj(∀n≥2j). Since Pτjcj<∞, P oposi ion 4.1 implies ha ℓ∞(eω) is a Banach algeb a, and he p oo o he claim is comple ed. Now ix {eωn}as in he claim. We may assume ha (ρ0 n)1/n →1 and he e o e (ρn)1/n →1. Since he polynomials a e dense in he Banach algeb a ℓ∞ 0(eω), any complex homomo phism χon ℓ∞ 0(eω) is de e mined by i s alue on he powe se ies . So he map χ7→ χ( ) is injec i e and con inuous om he spec um ( he maximal ideal space) o ℓ∞ 0(eω) o C. Since eω1/n n→1, i s image con ains Dand is con ained in D. Hence he spec um o ℓ∞ 0(eω) is exac ly he se {χλ:λ∈D}, whe e χλ( ) = (λ). (We bo ow his a gumen om [29].) As neωn= ( nωn)ρn→0, we ha e ∈ℓ∞ 0(eω). Then, using he Gel and heo y (see, o ins ance, [54, Chap e 10]), we ge ha g= 1/ ∈ℓ∞ 0(eω), which con adic s (5.3). The e o e, he assump ion g /∈ℓ∞(ω) is alse, as we wan ed o p o e. 24 L. Abadias, G. Bello, and D. Yakubo ich Rema k 5.4. No ice ha he abo e cha ac e iza ion o he spec um o he algeb a ℓ∞ 0(eω) (see he abo e Rema k 5.3) implies he ollowing ac : he condi ions (4.1) and ω1/n n→1 imply ha Pn1/ωn<∞. This can be p o ed in an elemen a y way, wi hou ecu ing o he Gel and heo y. Indeed, by (4.1), he e exis s a cons an C > 0 such ha n X j=1 ωn ωjωn−j≤C o e e y n≥1. Fix a posi i e in ege L. Then ob iously, o e e y n≥L, (5.5) L X j=1 ωn ωjωn−j≤C. Le us see ha (5.6) lim sup n→∞ min 1≤j≤L ωn ωn−j≥1. Indeed, i (5.6) we e alse, hen he e would exis some < 1 and a posi i e in ege Nsuch ha min 1≤j≤L ωn ωn−j≤ o n≥N. F om his, i is easy o see ha ωn≤ snmax 0≤k≤Nωk, sn:= n−N L+ 1, whe e [a] deno es he in ege pa o a. Since snbeha es asymp o ically as n/L, i ollows ha lim supn→∞ ω1/n n≤ 1/L <1, which con adic s he hypo hesis ha ω1/n n→1. The e o e, (5.6) is ue. Now, using (5.5), i ollows ha C≥ L X j=1 ωn ωjωn−j≥min 1≤j≤L ωn ωn−jL X j=1 1 ωj . Taking lim sup when n→ ∞, and using (5.6), we ge ha P1/ωjcon e ges. The ollowing s a emen shows ha in he subc i ical case, he hypo heses o Theo em 1.5 imply ha he adius o con e gence o he se ies o αis equal o one. P oposi ion 5.5. I lim k1/n n= 1 and αis o subc i ical ype, hen αdoes no con inue analy ically o any disc RD, whe e R > 1. P oo . Since k( ) has nonnega i e Taylo coe icien s, we ha e |k( )| ≤ k(1) o all ∈D. Using ha k= 1/α, i ollows ha in he subc i ical case, |α( )| ≥ α(1) >0 o any ∈D. So, αcanno con inue analy ically o any disc RD, whe e R > 1, because in his case, he adius o con e gence o he Taylo se ies o kwould be g ea e han 1.  Ope a o Inequali ies, Func ional Models and E godici y 25 6. Fini e De ec I is well-known ha in he classical Sz.-Nagy-Foias model, he case o a ini e ank (o Hilbe -Schmid ) de ec ope a o is an impo an one, whe e much mo e ools and esul s a e a ailable. In his sec ion, we de i e some consequences o ou model heo ems o he case when an ope a o T∈ Cw αis α-modelable and he de ec ope a o D= (α(T∗, T))1/2is o ini e ank. We will assume ha he ep oducing ke nel Hilbe space Rkis a Banach algeb a wi h espec o he mul iplica ion o powe se ies. By [60, P oposi ion 32], i su ices o assume ha sup n n X j=0 k2 jk2 n−j k2 n <∞; compa e wi h he condi ion (4.1). Pu mn= in j kj kn+j , 1= lim n→∞m1/n n. This limi exis s, see [60, P oposi ion 12]. We will assume ha (6.1) 1= lim n→∞k1/n n= 1. Bo h equali ies hold, in pa icula , i lim kn+1/kn= 1. The same is ue i , o ins ance, he las limi does no exis , bu 0 < σ < kn< C < ∞ o all nand he e is some m≥2 such ha limnkn+m/kn= 1. We also a e assuming he e ha he isome ic pa Sis no p esen in he model o T. Hence, Tis uni a ily equi alen o he es ic ion o he backwa d shi Bk⊗IDon he space Hk⊗D o an in a ian subspace L. Mo e gene ally, his applies o simila i y ins ead o he uni a y equi alence (we bea in mind models o linea ope a o s up o simila i y, which a e es ablished in [1]). He e we p o e he ollowing esul . Theo em 6.1. Suppose ha Tis simila o a pa o Bk⊗ID, ac ing on he space Hk⊗D, whe e Rkis a Banach algeb a and Dis ini e dimensional. I he spec um σ(T)does no co e he open disc D, hen σ(T)∩Dis con ained in he ze o se o a non-ze o unc ion in Rk. Le us s a wi h some p elimina y ema ks. Suppose ha Tis as in he abo e Theo em 6.1. Tha is, Tis simila o (Bk⊗ID)|L, whe e L ⊂ Hk⊗Dis an in a ian subspace o Bk⊗ID. By ixing a basis in D, we may assume ha D=Cd, whe e d= dim D. We will iden i y he space Hk⊗Dwi h Hd k=⊕d 1Hk, whose elemen s a e columns wi h en ies in Hk. The adjoin o Bkon he space Hd kis he mul iplica ion ope a o Mzon he space Rd k; his la e space can be seen as a Banach module o e he Banach algeb a Rk. Pu J=L⊥⊂ Rk⊗D. 32 L. Abadias, G. Bello, and D. Yakubo ich he Ces`a o means o o de ao T. When his amily o ope a o s is uni o mly bounded, ha is, sup n≥0kMa T(n)k<∞, we say ha Tis (C, a)-bounded. Rema ks 7.5. (i) No e ha Pn j=0 ka(j) = ka+1(n) o any a≥0. Also, i a≥0, hen ka(j)≥0 o e e y j≥0. (ii) I a= 0, hen M0 T(n) = Tn. Hence (C, 0)-boundedness is jus powe boundedness. (iii) I a= 1, hen M1 T(n) = (n+ 1)−1Pn j=0 Tj. Hence (C, 1)-boundedness is jus Ces`a o boundedness. (i ) I is well-known ha i 0 ≤a < b, hen (C, a)-boundedness implies (C, b)-boundedness. The con e se is no ue in gene al. Fo example, he Assani ma ix T= −1 2 0−1! is (C, 1)-bounded, bu since Tn= (−1)n(−1)n+12n 0 (−1)n! i is no powe bounded (see [30, Sec ion 4.7]). De ini ion 7.6. I he sequence o ope a o s {Ma T(n)}n≥0gi en in De ini ion 7.4 con e ges in he s ong ope a o opology, we say ha Tis (C, a)-mean e godic. I Tis (C, 1)-mean e godic, i is con en ional jus o say ha Tis mean e godic. The e is a well es ablished li e a u e on (C, a)-bounded ope a o s, which explo es qui e a numbe o p ope ies and hei in e plays. P ope ies, cha ac e iza ion h ough unc ional calculus and e godic esul s o (C, a)-bounded ope a o s can be ound in [4, 9, 25, 27, 28, 30, 41] and e e ences he ein. The connec ion o hese ope a o s and e godici y da es back o he ou ies o las cen u y, see [24] and [40]. In he la e pape , E. Hille s udies (C, a)-mean e godici y in e ms o Abel con e gence ( ha is, ia he esol en ope a o ). As applica ion, he well known mean e godic on Neumann’s heo em o uni a y g oups on Hilbe spaces is ex ended o (C, a)-mean e godici y o e e y a > 0 [40, p. 255]. Also, he (C, a)-e godici y on L1(0,1) o ac ional (Riemann-Liou ille) in eg als is elucida ed in [40, Theo em 11]. In pa icula , i Vis he Vol e a ope a o hen TV:= I−V, as ope a o on L1(0,1), is no powe -bounded, and i is (C, a)-mean e godic i and only i a > 1/2 [40, Theo em 11]. This esul can be ex ended o TVac ing on Lp(0,1), 1 < p < ∞, using es ima es gi en in [44], see [3, Sec ion 10]. Ope a o Inequali ies, Func ional Models and E godici y 33 In [42], Luo and Hou in oduced a new no ion o boundedness: a bounded linea ope a o Ton a Banach space Xis said o be absolu ely Ces`a o bounded i sup n≥0 1 n+ 1 n X j=0 Tjx.kxk o e e y x∈X. In [14], he au ho s s udy he e godic beha iou o his class o ope a o s. The abo e de ini ion has been ex ended ecen ly by Abadias and Bonilla in [2]: Tis said o be absolu ely (C, a)-Ces`a o bounded o some a > 0 i sup n≥0 1 ka+1(n) n X j=0 ka(n−j)Tjx.kxk o e e y x∈X. No e ha o a= 1 he de ini ion o Luo and Hou is eco e ed. Rema k 7.7. I is well-known ha he ollowing implica ions hold: Powe bounded ⇒Absolu ely (C, a)-bounded ⇒(C, a)-bounded ⇒ kTnk=O(na). The i s wo implica ions a e s aigh o wa d. Fo he sake o comple eness, we gi e a p oo o he las one. Suppose Tis (C, a)-bounded o some a≥0. We deno e by [a] he in ege pa o a. Then, o n > [a], we ha e kTnk= n X j=0 k−a(j) n−j X m=0 ka(n−j−m)Tm . n X j=0 |k−a(j)|ka+1(n−j) = [a] X j=0 (−1)jk−a(j)ka+1(n−j) + n X j=[a]+1 (−1)[a]+1k−a(j)ka+1(n−j) = [a] X j=0 (−1)j+ (−1)[a]k−a(j)ka+1(n−j) + (−1)[a]+1 n X j=0 k−a(j)ka+1(n−j) . [a] X j=0 |k−a(j)|ka+1(n−j) + k1(n).ka+1(n)≍(n+ 1)a. The ollowing ex ension o he abo e de ini ions will be impo an o us. De ini ion 7.8. Le a > 0 and p≥1. We say ha a bounded linea ope a o Ton a Banach space Xis (C, a, p)-bounded i sup n≥0 1 ka+1(n) n X j=0 ka(n−j)kTjxkp.kxkp, 34 L. Abadias, G. Bello, and D. Yakubo ich o all x∈X. No e ha o p= 1 his de ini ion is jus he absolu e (C, a)-boundedness. The case a= 1 has been ecen ly conside ed in [23]. We will use he e m quad a ically (C, a)-bounded ins ead o (C, a, 2)-bounded. Using he asymp o ics ka(n)≍(n+1)a−1gi en in (7.3), i is easy o see ha Tis (C, a, p)- bounded i and only i (7.7) sup n≥0 1 (n+ 1)a n X j=0 (n+ 1 −j)a−1Tjxp.kxkp(∀x∈X). The ollowing obse a ion will be essen ial o he p oo o Theo em 1.14. Lemma 7.9. The ollowing holds. (i) I Tis (C, a, p)-bounded, hen any pa o Tis also (C, a, p)-bounded. (ii) I T1and T2a e (C, a, p)-bounded, hen any di ec sum T1∔T2is also (C, a, p)-bounded. (iii) Le Tbe a bounded linea ope a o on a Hilbe space. I Tis quad a ically (C, a)- bounded, hen T⊗IEis also quad a ically (C, a)-bounded, whe e IEis he iden i y ope a o on some Hilbe space E. P oo . (i) and (ii) a e immedia e. Fo (iii) no e ha i d= dim E ≤ ∞, hen he o hogonal sum o dcopies o Tis clea ly quad a ically (C, a)-bounded (by he Py hago as Theo em).  The ollowing esul is e y use ul. I s p oo is simple, and we omi i . Lemma 7.10. Le 0≤a < b. Then (C, a, p)-boundedness implies (C, b, p)-boundedness. This lemma shows an inclusion o classes o ope a o s. By [2, Co olla ies 2.2 and 2.3], i Tis (C, a, 1)-bounded hen kTnk=o(na) o 0 < a ≤1 and kTnk=O(n) o a > 1. The ollowing esul explains why he case a= 1 is special. Theo em 7.11. I a > 1and p≥1, hen (C, a, p)-boundedness is equi alen o (C, 1, p)- boundedness. P oo . Fix a > 1 and p≥1. By he abo e Lemma, we only need o p o e ha any (C, a, p)- bounded ope a o Tis (C, 1, p)-bounded. Le Tis (C, a, p)-bounded. Then (7.8) 1 ka+1(2n) 2n X j=0 ka(2n−j)Tjxp.kxkp, o e e y n≥0, and e e y x∈X. Since a > 1, ka(m) is an inc easing unc ion o m. In pa icula , ka(n)≤ka(2n−j) o j= 0,...,n. Hence (7.9) ka(n) n X j=0 Tjxp≤ 2n X j=0 ka(2n−j)Tjxp, Ope a o Inequali ies, Func ional Models and E godici y 35 By (7.9) and (7.8), n X j=0 Tjxp.ka+1(2n) ka(n)kxkp.(n+ 1) kxkp, which means ha Tis (C, 1, p)-bounded.  Theo em 7.12. Le a > 0and 1≤q < p. I Tis (C, a, p)-bounded, hen i is also (C, b, q)- bounded o each b > qa/p. In pa icula , (C, a, p)-boundedness implies (C, a, q)-boundedness. P oo . Le us i s ecall ha i > −1, hen (7.10) m X j=1 j .m +1 (∀m≥1). Le Tbe (C, a, p)-bounded and le b > qa/p. Suppose i s ha b6= 1, and pu s:= p p−q, s′:= p q, γ := q(a−1) p(b−1) . No e ha sand s′a e posi i e and sa is y 1/s + 1/s′= 1. Since (b−1)(1 −γ)s=pb −qa p−q−1>−1 and (b−1)γs′=a−1, using H¨olde ’s inequali y and (7.10) i ollows ha 1 (n+ 1)b n X j=0 (n+ 1 −j)b−1Tjxq ≤1 (n+ 1)b  n X j=0 (n+ 1 −j)(b−1)(1−γ)s  1/s   n X j=0 (n+ 1 −j)(b−1)γs′Tjxqs′  1/s′ .(n+ 1)−qa/p   n X j=0 (n+ 1 −j)a−1Tjxp  q/p = 1 (n+ 1)a n X j=0 (n+ 1 −j)a−1Tjxp  q/p o e e y x∈Xand e e y non-nega i e in ege n. Hence he s a emen ollows using (7.7). Now suppose ha b= 1. Take any b′∈(qa/p, 1). We ha e al eady p o ed ha T is (C, b′, p)-bounded. Then, by Lemma 7.10, i ollows ha Tis (C, 1, p)-bounded. This comple es he p oo .  Lemma 7.13. Le a > 0and p≥1. Then e e y isome y Sis (C, a, p)-bounded. 36 L. Abadias, G. Bello, and D. Yakubo ich P oo . This is immedia e, since indeed (7.11) 1 ka+1(n) n X j=0 ka(n−j)kSjxkp=1 ka+1(n)  n X j=0 ka(n−j) kxkp=kxkp o e e y x∈X. Lemma 7.14. Le 0< s < 1and le a > 0. Then Bsis quad a ically (C, a)-bounded i and only i 1−s < a. Mo eo e , o 1−s < a we ha e (7.12) lim n→∞ 1 ka+1(n) n X j=0 ka(n−j)kBj sxk2= 0 (∀x∈ Hs). P oo . Recall he no a ion en= n∈ Hk=Hs, whe e k( ) = (1− )−s. Suppose ha a= 1−s. Then 1 (n+ 1)a n X j=0 (n+ 1 −j)a−1Bj sen2&1 (n+ 1)1−s n X j=0 (n+ 1 −j)−s(n+ 1 −j)s−1 =1 (n+ 1)1−s n+1 X j=1 j−1&log(n+ 2) kenk2 (7.13) o e e y n. The e o e Bsis no quad a ically (C, 1−s)-bounded, and by Lemma 7.10 we ob ain ha Bsis no quad a ically (C, a)-bounded o a < 1−s. Le us assume now ha 1 −s < a ≤1 and ix x∈ Hs. W i e xin he o m x=Pxmem, whe e xm∈C. Then Bj sx2=∞ X m=j ks(m−j)|xm|2.∞ X m=j (m+ 1 −j)s−1|xm|2, o e e y j≥0. Hence 1 (n+ 1)a n X j=0 (n+ 1 −j)a−1Bj sx2 .1 (n+ 1)a n X j=0 (n+ 1 −j)a−1∞ X m=j (m+ 1 −j)s−1|xm|2 =1 (n+ 1)a n X m=0 |xm|2 m X j=0 (n+ 1 −j)a−1(m+ 1 −j)s−1 +1 (n+ 1)a 2n X m=n+1 |xm|2 n X j=0 (n+ 1 −j)a−1(m+ 1 −j)s−1 +1 (n+ 1)a ∞ X m=2n+1 |xm|2 n X j=0 (n+ 1 −j)a−1(m+ 1 −j)s−1 =: (I) + (II) + (III). Ope a o Inequali ies, Func ional Models and E godici y 37 In (I), no e ha since 1 −s < a ≤1, and m≤n, we ha e (7.14) m X j=0 (n+ 1 −j)a−1(m+ 1 −j)s−1≤ m+1 X j=0 (m+ 1 −j)a+s−2.(m+ 1)a+s−1, whe e in he las es ima e we used (7.10). The e o e (I).1 (n+ 1)a n X m=0 |xm|2(m+ 1)a+s−1=   [√n] X m=0 + n X m=[√n]+1  |xm|2(m+ 1)a+s−1 (n+ 1)a .kxk2 √na+ n X m=[√n]+1 |xm|2(m+ 1)s−1−→ 0 (as n→ ∞). In (II), using ha m > n and s−1<0, we ha e n X j=0 (n+ 1 −j)a−1(m+ 1 −j)s−1≤ n X j=0 (n+ 1 −j)a+s−2.(n+ 1)a+s−1. The e o e (II).1 (n+ 1)a 2n X m=n+1 |xm|2(n+ 1)a+s−1= (n+ 1)s−1 2n X m=n+1 |xm|2 . 2n X m=n+1 |xm|2(m+ 1)s−1−→ 0 (as n→ ∞). Finally, in (III), since m > 2nwe ha e ha n X j=0 (n+ 1 −j)a−1(m+ 1 −j)s−1.(m+ 1)s−1 n X j=0 (n+ 1 −j)a−1.(m+ 1)s−1(n+ 1)a. The e o e (III).∞ X m=2n+1 |xm|2(m+ 1)s−1−→ 0 (as n→ ∞). Hence (7.12) ollows when 1 −s < a ≤1. Finally, suppose ha 1 < a. Then 1 (n+ 1)a n X j=0 (n+ 1 −j)a−1Bj sx2≤1 n+ 1 n X j=0 Bj sx2−→ 0 (as n→ ∞), since his is he case o a= 1 in (7.12) (al eady p o ed). No e ha (7.12) implies quad a ical (C, a)-boundedness, so he p oo is comple e.  This lemma allows us o p o e he ollowing mo e gene al esul . 38 L. Abadias, G. Bello, and D. Yakubo ich Theo em 7.15. Le 0< s < 1and 1≤q≤2. Then Bsis (C, b, q)-bounded i and only i b > q(1 −s)/2. Mo eo e , o b > q(1 −s)/2we ha e (7.15) lim n→∞ 1 kb+1(n) n X j=0 kb(n−j)Bj sxq= 0 (∀x∈H). P oo . No e ha q= 2 is p ecisely Lemma 7.14. So we assume ha 1 ≤q < 2. I b= q(1 −s)/2, aking x=en, we ge , as in (7.13), ha 1 kb+1(n) n X j=0 kb(n−j)Bj senq&log(n+ 2) kenk2 o e e y n. The e o e Bsis no (C, q(1 −s)/2, q)-bounded, and by Lemma 7.10 we ge ha Bsis no (C, b, q)-bounded o b < q(1 −s)/2. Now suppose ha b > q(1 −s)/2. Then b=qa/2 o some a > 1−s. Using H¨olde ’s inequali y as in he p oo o Theo em 7.12, we ob ain 1 (n+ 1)b n X j=0 (n+ 1 −j)b−1Bj sxq. 1 (n+ 1)a n X j=0 (n+ 1 −j)a−1Bj sx2  q/2 −−−→ n→∞ 0, by Lemma 7.14. Hence (7.15) ollows.  P oo o Theo em 1.14. Le T∈ Cw awi h 0 < a < 1 and le b > 1−a. By Theo em 1.4 and Theo em 1.12 (i), Tis uni a ily equi alen o a pa o (Ba⊗ID)⊕S. Hence, by Lemma 7.9 (i), i is enough o p o e ha (Ba⊗ID)⊕Sis quad a ically (C, b)-bounded. Bu his is immedia e using Lemma 7.9 (ii) and (iii), and Lemmas 7.13 and 7.14.  Fo he p oo o Theo em 1.15 we need he ollowing lemma, which is in he spi i o Lemma 7.9. Lemma 7.16. The ollowing holds. (i) I Tsa is ies (1.9), hen any pa o Talso sa is ies (1.9). (ii) I T1and T2sa is y (1.9), hen any di ec sum T1∔T2also sa is ies (1.9). (iii) Le Tbe a bounded linea ope a o on a Hilbe space. I Tsa is ies (1.9), hen he ope a o T⊗IEalso sa is ies (1.9), whe e IEis he iden i y ope a o on some Hilbe space E. P oo . (i) and (ii) a e immedia e. Fo (iii) we use he same a gumen as in Lemma 7.9 (iii) and a simple applica ion o Lebesgue’s Domina ed Con e gence Theo em.  P oo o Theo em 1.15. As in he p oo o Theo em 1.14, we ha e ha Tis uni a ily equi a- len o (Ba⊗ID)⊕S|L, Ope a o Inequali ies, Func ional Models and E godici y 39 whe e Lis a subspace o (Ha⊗D)⊕W in a ian by (Ba⊗ID)⊕S. Le us p o e he ci cle o implica ions (i) ⇒(ii) ⇒(iii) ⇒(i). Suppose ha (i) is ue. Tha is, Tis uni a ily equi alen o (Ba⊗ID)|L, whe e Lis a subspace o Ha⊗Din a ian by Ba⊗ID. Then (ii) ollows using Lemmas 7.14 and 7.16. Suppose now ha lim in n→∞ kTnxk>0 o some x∈H. Then, ob iously, kTnxk> ε > 0 o e e y n≥0. Hence o his ec o x (1.9) does no hold. The e o e we ha e p o ed ha (ii) ⇒(iii). Finally, suppose ha he isome y Sappea s in he minimal model. Then o some ec o ℓ= (ℓ1, ℓ2)∈ L, i s second componen ℓ2∈ W is no 0. The e o e lim n→∞ 1 kb+1(n) n X j=0 kb(n−j)((Ba⊗ID)⊕S)jℓ2 = lim n→∞ 1 kb+1(n) n X j=0 kb(n−j)(Ba⊗ID)jℓ1⊕Sjℓ22 = lim n→∞ 1 kb+1(n) n X j=0 kb(n−j)(Ba⊗ID)jℓ12+ lim n→∞ 1 kb+1(n) n X j=0 kb(n−j)Sjℓ22. The second limi is kℓ2k26= 0 because o (7.11). Hence we ob ain ha (iii) ⇒(i).  Rema k 7.17. In he same way, we ge ha i Tis an a-con ac ion and 0 < a ≤1, hen lim in n→∞ kTnxk ≤ kxk. In pa icula , his lowe limi is ini e o any x. Since Cw 1is jus he se o all con ac ions on H,T∈ Cw 1i T∗∈ Cw 1. Howe e , his is no longe ue o a∈(0,1). P oposi ion 7.18. I a∈(0,1), hen he e is an ope a o T∈ Cw asuch ha T∗/∈ Cw a. P oo . No e ha B∗ ais a o wa d weigh ed shi such ha kB∗n a 0k → ∞ as ngoes o ∞. So Ba∈ Cw a, whe eas i s adjoin canno belong o Cw a, because B∗ ais no quad a ically (C, b)- bounded o any b(see Lemma 7.14).  I is na u al o pose he ollowing ques ion. Ques ion 7.19. Fo which unc ions α, sa is ying Hypo heses 1.1, is i ue ha T∈ Cw α implies T∗∈ Cw α? 40 L. Abadias, G. Bello, and D. Yakubo ich I is so o α( ) = 1 − and, mo e gene ally, o α( ) = 1 − n,n≥1. The au ho s do no know o he examples. Rema ks 7.20. (i) I Tis an ope a o in Cw awi h 0 < a < 1, and 0 < q < 2, hen by Theo em 1.14 and Theo em 7.12, i ollows ha Tis (C, b, q)-bounded o all b > q(1−a) 2. (ii) An m-isome y T, which is no an isome y, canno be (C, a, p)-bounded, because he e a e ec o s xsuch ha he no ms kTnxkgo o in ini y. The possibili y o hese ope a o s o ha e weake e godic p ope ies, such as he Ces`a o boundedness and weak e godici y, ha e been s udied in [13]. (iii) Le Tbe an ope a o in Cw awi h 0 < a < 1. Using Theo em 1.14 (i) and Theo em 7.12 (wi h p= 2 and q= 1) we ob ain ha Tis (C, b, 1)-bounded o e e y b > (1 −a)/2. By [2, Co olla y 3.1], we ge ha Tis (C, b)-mean e godic, ha is, he e exis s Pbx:= lim n→∞Mb T(n)x, x ∈H. The e o e, by [3, Theo em 3.3], we ha e H= Ke (I−T)⊕Ran(I−T). In ac , Ke (I−T) = RanPband Ran(I−T) = Ke Pb. Also no e ha Mb T(n)x=x o x∈Ke (I−T),and lim n→∞Mb T(n)x= 0 o x∈Ran(I−T). Le now 0 < γ < 1,by [3, P oposi ion 4.8 and Rema k 4.9], one can de ine a bounded ope a o (I−T)γby means o a ce ain unc ional calculus, and Ke (I−T) = Ke (I−T)γ,Ran(I−T) = Ran(I−T)γ, wi h Ran(I−T)⊆Ran(I−T)γ. Fu he mo e i γ < 1−b, o x∈Ran(I−T), x∈Ran(I−T)γ⇐⇒ ∞ X n=1 1 n1−γTnxcon e ges, see [3, Theo em 9.2]. (i ) By [2, Theo em 3.1], i Tis an ope a o in Cw awi h 0 < a < 1 and b > (1 −a)/2, hen lim n→∞kMb T(n+ 1) −Mb T(n)k= 0. Acknowledgmen s The au ho s hank T. Bha acha yya and N. Nikolski and D. Schillo o hei use ul e- ma ks, and A. Bonilla o his ad ice conce ning Theo em 7.11. The i s au ho has been Ope a o Inequali ies, Func ional Models and E godici y 41 pa ly suppo ed by P ojec MTM2016-77710-P, DGI-FEDER, o he MCYTS, P ojec E26- 17R, D.G. A ag´on, and P ojec o Young Resea che s, Fundaci´on Ibe caja and Uni e sidad de Za agoza, Spain. The second au ho has been pa ially suppo ed by La Caixa-Se e o Ochoa g an (ICMAT Se e o Ochoa p ojec SEV-2011-0087, MINECO). Bo h second and hi d au- ho s acknowledge pa ial suppo by Spanish Minis y o Science, Inno a ion and Uni e si ies (g an no. PGC2018-099124-B-I00) and he ICMAT Se e o Ochoa p ojec SEV-2015-0554 o he Spanish Minis y o Economy and Compe i i eness o Spain and he Eu opean Regional De elopmen Fund, h ough he “Se e o Ochoa P og amme o Cen es o Excellence in R&D”. Bo h second and hi d au ho s also acknowledge inancial suppo om he Span- ish Minis y o Science and Inno a ion, h ough he “Se e o Ochoa P og amme o Cen es o Excellence in R&D” (SEV-2015-0554) and om he Spanish Na ional Resea ch Council, h ough he “Ayuda ex ao dina ia a Cen os de Excelencia Se e o Ochoa” (20205CEX001). Re e ences [1] L. Abadias, G. Bello, and D. 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