Operator inequalities, functional models and ergodicity

arXiv:1908.05032v3 [math.FA] 2 Jul 2020 OPERATOR INEQUALITIES, FUNCTIONAL MODELS AND ERGODICITY LUCIANO ABADIAS, GLENIER BELLO, AND DMITRY YAKUBOVICH Abstract. We discuss when an operator, subject to a rather general inequality in hereditary form, admits a unitarily equivalent functional model of Agler type in the reproducing kernel Hilbert space associated to the inequality. To the contrary to the previous work, the kernel need not be of Nevanlinna-Pick type. We derive some consequences concerning the ergodic behavior of the operator. 1. Introduction 1.1. Motivation. Let α(t) be a function representable by the power series P∞ n=0 αntnin D:= {|t|<1}, where the coefficients αnare real numbers, and let T∈L(H) be a bounded linear operator on a Hilbert1space H. Put (1.1) α(T∗, T) := ∞ X n=0 αnT∗nTn, where the series is assumed to converge in the strong operator topology SOT in L(H). When αis a polynomial, the series above is just a finite sum, and there is no convergence problem. In particular, when α(t) = 1 −t, the right hand side of (1.1) is I−T∗T, so T∈L(H) is a contraction if and only if (1 −t)(T∗, T)≥0. In the 1960’s Sz.-Nagy and Foias developed a beautiful spectral theory of contractions (see [62]) based on the construction of their functional model. In his landmark paper [5], Agler showed that if Thas spectrum σ(T) contained in the unit disc Dand α(T∗, T)≥0, then it is natural to model Tby parts of B⊗IE, where Bis a suitable weighted backward shift and IEis the identity operator on some auxiliary Hilbert space E. (By a part of an operator we mean its restriction to an invariant subspace.) More generally, when σ(T)⊂D, it has been found in various particular cases that instead of B⊗IE one should consider operators of the form (B⊗IE)⊕S, where Sis an isometry or a unitary operator. This representation is called a coanalytic model. As Agler proved in [6], it holds, in particular, for m-hypercontractions, i.e., operators T∈L(H) such that (1 −t)j(T∗, T)≥0 for j= 1,...,m. Agler’s theorem was generalized in [46] by M¨uller and Vasilescu to tuples Date: July 6, 2020. Key words and phrases. dilation; functional model; operator inequality; ergodic properties. 1All Hilbert spaces will be assumed to be separable 1 2L. Abadias, G. Bello, and D. Yakubovich of operators. The first results on Agler model techniques are exposed in the book [7] by Agler and McCarthy. In [48], Olofsson obtained operator formulas for wandering subspaces, relevant in the models of m-hypercontractions. His results were generalized by Eschmeier in [31] to tuples of commuting operators, and by Ball and Bolotnikov in [10] to what they call β-hypercontractions. M¨uller studied the case where α=pis a polynomial in [45]. He considers the class C(p) of operators T∈L(H) such that p(T∗, T)≥0. He proves that any contraction T∈ C(p) has a coanalytic model whenever p(1) = 0,1/p(t) is analytic in D, and 1/p( ¯wz) is a reproducing kernel. This last condition is equivalent to the fact that all Taylor coefficients of 1/p(t) at the origin are positive. M¨uller also considers some operator inequalities for Twith infinitely many terms, with the same property of positivity. This permits him to show that any operator T is unitarily equivalent to a part of a backward weighted shift with the same spectral radius (see [45, Corollary 2.3]). In [50], Olofsson deals with the case where αis not a polynomial. His assumptions are that αis analytic on D, does not vanish on D, and 1/α has positive Taylor coefficients at the origin. Under this setting, he studies contractions Ton Hsuch that α(rT∗, rT )≥0 for every r∈[0,1). With more assumptions, he obtains the coanalytic model for this class of operators. In [11], the last two authors considered functions αin the Wiener algebra AWof analytic functions in the unit disc with summable sequence of Taylor coefficients, subject to certain conditions. It was assumed that the series PαnT∗nTnconverges in norm. The operators studied there turn out to be similar to contractions (see [11, Theorem I]). This will no longer be true in the setting of the present paper (see Example 7.3). In [11], an explicit model in the spirit of Sz.-Nagy and Foias model was constructed for the class of operators considered there. The roles of the defect operator and the defect space were played by (1.2) D:= (α(T∗, T))1/2,D:= DH, where the non-negative square root is taken. 1.2. Our setting. Here the operator Dand the space D, defined by (1.2) whenever α(T∗, T)≥ 0, will also play an important role. Recall that now we consider the convergence of (1.1) in SOT. As it will be seen from Example 7.3, this is the appropriate convergence in this context. Our assumptions are the following. Hypotheses 1.1. Suppose αis a function in AWwhich does not vanish on D. We put k(t) = 1/α(t) = ∞ X n=0 kntnt∈D, with α0=k0= 1, and assume that kn>0 for every n≥1. Operator Inequalities, Functional Models and Ergodicity 3 Under Hypotheses 1.1, we denote by Hkthe weighted Hilbert space of power series f(t) = P∞ n=0 fntnwith finite norm kfkHk:= ∞ X n=0 |fn|2kn1/2 . Let Bkbe the backward shift on Hk, defined by (1.3) Bkf(t) = f(t)−f(0) t. Definition 1.2. Fix a function αsatisfying Hypotheses 1.1, and let Tbe an operator in L(H). We say that Tis α-modelable if Tis unitarily equivalent to a part of an operator of the form (Bk⊗IE)⊕S, where Sis an isometry. We remark that Bk⊗IEacts on the Hilbert space Hk⊗ E, which can be identified with the weighted Hilbert space of E-valued power series f(t) = P∞ n=0 fntnwith norm given by kfkHk⊗E =∞ X n=0 kfnk2 Ekn1/2 . It acts according to the same formula (1.3). It is natural to pose the following question. Question 1.3. Given a function αsatisfying Hypotheses 1.1, give a good sufficient condition for an operator T∈L(H) to be α-modelable. One of the strongest results in this direction is contained in the recent papers by Bickel, Hartz and McCarthy [17] and by Clouˆatre and Hartz [21]. It is stated for spherically symmetric tuples of operators. For the case of a single operator, their result can be formulated as follows. Theorem 1.4 ([21, Theorem 1.3]).Let αbe a function with α0= 1 and αn≤0for all n≥1. Suppose that k= 1/α has radius of convergence 1,kn>0for every n≥0and (1.4) lim n→∞ kn kn+1 = 1. Then Bkis bounded, and a Hilbert space operator Tis α-modelable if and only if α(T∗, T )≥0. It is easy to see that the hypotheses of Theorem 1.4 imply Hypotheses 1.1. This theorem concerns the Nevanlinna-Pick case, that is, when α0= 1 and αn≤0 for n≥1. Alternatively, we say that kis a Nevanlinna-Pick kernel. In the recent work [22], Clouˆatre, Hartz and Schillo establish a Beurling–Lax–Halmos theorem for reproducing kernel Hilbert spaces in the Nevanlinna-Pick context. We refer the reader to [26, 49, 56, 57] for more results in the Nevanlinna-Pick case. In the recent preprint [32], Eschmeier and Toth extend previous results by Eschmeier [31] to all complete Nevanlinna-Pick kernels, in the context of operator tuples. 1.3. Main results. The following result gives a new answer to Question 1.3. 4L. Abadias, G. Bello, and D. Yakubovich Theorem 1.5. Assume Hypotheses 1.1. If k∈AW, and its Taylor coefficients {kn}satisfy k1/n n→1,sup kn/kn+1 <∞and (1.5) lim m→∞ sup n≥2mX m≤j≤n/2 kjkn−j kn = 0, then Bkis bounded, and the operator T∈L(H)is a part of Bk⊗IE(for some Hilbert space E) if and only if both P|αn|T∗nTnand PknT∗nTnconverge in SOT and α(T∗, T)≥0. Moreover, in this case one can take E=D. As it will be seen later, the SOT-convergence of P|αn|T∗nTnimplies the the SOTconvergence of PαnT∗nTn. Notice that in Theorem 1.5, the isometric part Sis unnecessary (see Theorem 1.12 (ii) below for more information). This theorem shows that Tis α-modelable in many cases when kis not a Nevanlinna-Pick kernel, and so Theorem 1.4 does not apply. Not much about these kernels has been known previously. Given an integer N≥2, there are examples of functions ksatisfying the hypotheses of Theorem 1.5 with whatever prescribed signs of the coefficients α2,...,αN(see Example 5.1). Note that α1=−k1is always negative. Remark 1.6. Suppose that k∈AWand the sequence kn kn+1 1 + 1 n+ 1a is increasing for some a > 1. Then (1.5) holds. This is close to [60, Proposition 34]. Indeed, put k∗ j:= (j+ 1)−a, and define ρj:= kj/k∗ j. Then our condition reduces to the condition ρn+2/ρn+1 ≥ρn+1/ρn, for all n, which implies that ρjρn−j/ρn≤C, for 0 ≤j≤n. Since kjkn−j kn =ρjρn−j ρn k∗ jk∗ n−j k∗ n and {k∗ n}satisfies (1.5), it follows that {kn}also satisfies (1.5). Hence, for sufficiently regular sequences {kn}, the condition (1.5) is rather close to the condition Pkn<∞. It can be added that, in fact, in Theorem 1.5 {kn}need not be regular; moreover, the quotients kn/kn+1 need not converge (see Remark 5.2). The techniques employed in the proof are different from [21]. We use, basically, a combination of M¨uller’s arguments in [45] and Banach algebras techniques. The above theorems open the question of describing invariant subspaces of Bk⊗IEand of constructing a functional model of operators under the study, which certainly would be interesting. We do not address this question in this paper. Given an operator C:H→ E, where Eis an auxiliary Hilbert space, we define (1.6) VCx(z) = C(IH−zT)−1x, x ∈H, z ∈D. Operator Inequalities, Functional Models and Ergodicity 5 The next result shows that whenever Tis α-modelable, the operator VD:H→ Hk⊗Dis a contraction, and we can give an explicit model for T(that is, give explicitly E,Sand the transform which sends the initial space into the model space). First we need to state one more technical hypothesis, whose meaning will be clear later. Hypotheses 1.7. Let αbe a function satisfying Hypothesis 1.1. Put (1.7) β(t) = X n≥0 βntn,where βn=|αn|, and γ(t) = β(t)k(t). We assume that kn/kn+1 ≤C′and γn≤C′′knfor all n≥0. The condition kn/kn+1 ≤C′is equivalent to boundedness of Bk. If it holds, then the second condition is satisfied whenever there is some Nsuch that either αn≥0 for n≥N, or αn≤0 for n≥N. Theorem 1.8 (Explicit model).Assume Hypotheses 1.1 and 1.7. Let Tbe α-modelable. Then α(T∗, T)≥0,VDis a contraction, and hence we can define W= (IH−V∗ DVD)1/2,W=WH. Moreover, S:W → W, given by SW x := WTx, is an isometry and the operator (VD, W) : H→(Hk⊗D)⊕W,(VD, W )h= (VDh, Wh) provides a model of T, in the sense that (VD, W)is isometric and ((Bk⊗ID)⊕S)·(VD, W) = (VD, W)·T. Remark 1.9. Suppose αsatisfies the above two hypotheses, and suppose that Tis an αmodelable operator, which is given already by its model without the isometric part. That is, there is an invariant subspace Lof an operator Bk⊗IE, acting on Hk⊗ E, such that Tis the restriction of this operator to L. Then D=E(identified with the constant functions in Hk⊗E), and VDis the identity operator on L. This follows from Corollary 2.13 below. Similarly, in the general case, if Tis a part of an operator (Bk⊗IE)⊕S, where Sis an isometry, there is a unitary operator usuch that the transform (VD, uW ) is just the identity. If it is known that Tis α-modelable, one can ask about the uniqueness of the model. For answering this question, we need the following definitions. Definition 1.10. Let Lbe an invariant subspace of (Bk⊗IE)⊕S, where S:W → W is an isometry. We will say that the corresponding model operator (Bk⊗IE)⊕S|L is minimal if the following two conditions hold. (i) Lis not contained in (Hk⊗E′)⊕W for any E′$E. 6L. Abadias, G. Bello, and D. Yakubovich (ii) Lis not contained in (Hk⊗E)⊕W′for any W′$Winvariant by S. In Remark 3.5 we show that the explicit model obtained in Theorem 1.8 is indeed minimal. Note that under Hypotheses 1.1, αis defined on the closed unit disc Dand does not vanish on the interval [0,1). Since α(0) = α0= 1, we obtain that α(1) ≥0. We distinguish the following two cases. This distinction appears already in [21, Subsection 2.3] for the Nevanlinna-Pick case. Definition 1.11. Suppose that αmeets Hypotheses 1.1. We will say that αis of critical type (or, alternatively, that we have the critical case) if α(1) = 0. If α(1) >0, we will say that α is of subcritical type (or, alternatively, that we have the subcritical case). Theorem 1.12 (Uniqueness of the minimal model).Suppose that αmeets Hypotheses 1.1 and 1.7. Let Tbe an α-modelable operator. (i) In the critical case, the minimal model of Tis unique. More precisely, the pair of transforms (VD, W0), where W0= (I−V∗ DVD) : H→ W0and W0:= Ran(I−V∗ DVD), gives rise to a minimal model, and any minimal model is provided by (VC, W), where C=vD,W=wW0:H→ W and v, w are unitary isomorphisms. (ii) In the subcritical case, the minimal model of Tis not unique, in general. However, there always exists a minimal model given by V=VD, in the sense that VD:H→ Hk⊗IDis an isometry such that (Bk⊗ID)VD=VDT. Note that in this case the isometry Sis absent. We remark that there are other works that give answers to the above Question 1.3. In particular, Pott [52] gave a model for operators satisfying two inequalities (1 −p)(T∗, T)≥0 and (1 −p)m(T∗, T)≥0, where pis a polynomial with nonnegative coefficients, m≥1 and p(0) = 0 (this class is a generalization of m-hypercontractions). In fact, she treats tuples of operators. In [10], Ball and Bolotnikov consider a function α(t) in the Wiener algebra such that k= 1/α has positive coefficients satisfying 0 < ε ≤kn/kn+1 ≤1 for all n(so that Bk is a contraction). They show that an operator Tis α-modelable, with absent isometric part, if and only if if α(T∗, T)≥0 as well as infinitely many additional inequalities hold (Tis βhypercontractive, where βn= 1/kn), and Tis what they call β-stable. See [10], Theorem 4.3. In [10], Theorem 7.2, Ball and Bolotnikov give a model of Tin terms of their generalization of the characteristic function, which is an infinite family of operator-valued functions. Whereas these authors treat both subcritical and critical cases, Theorem 1.5 only concerns the subcritical case (because of the condition k∈AW). 1.4. Consequences of the model. If an operator Tis α-modelable, it is natural to study what consequences can be derived from the model. Here we obtain two types of consequences: (1) when the defect operator Dhas finite rank (that is, dim D<∞), and (2) ergodic consequences when α(t) = (1 −t)awith 0 < a < 1. Operator Inequalities, Functional Models and Ergodicity 7 We will use the space Rk=H˜ k, where ˜ kn= 1/kn. It is easy to see that it is the reproducing kernel Hilbert space, corresponding to the positive definite kernel k(z, w) := k( ¯wz). The pairing hf, gi=Pfn¯gn(f∈ Hk, g ∈ Rk) makes Rknaturally dual to Hκ. In this interpretation, the adjoint operator to Bkis the operator g(z)7→ zg(z), acting on Rk. If αis of subcritical type, we have the following result related to the Carleson condition. Theorem 1.13. Let Tbe an operator similar to a part of Bk⊗ID, acting on the space Hk⊗D, where Rkis a Banach algebra and Dis finite dimensional. Suppose that lim n→∞inf j≥0 kj kn+j1/n = lim n→∞k1/n n= 1, and also that (1.8) ∞ X n=N kn≤CN−ε∀N≥0, for some positive constants Cand εwhich do not depend on N. Suppose that the spectrum σ(T)does not cover D. Put E:= (σ(T)∩D)∩T and let {lν}denote the lengths of the finite complementary intervals of E(in T). Then the Lebesgue measure of Eis 0, and the Carleson condition holds: X ν lνlog 2π lν <∞. Some of the arguments employed in the proof of this theorem are related with the so-called index of an invariant subspace of Rk⊗E; see Section 6 for more details. In the critical case, an important family of functions αare those of the form α(t) := (1−t)a, for a > 0. Note that they satisfy Hypotheses 1.1. When a=mis a positive integer, it is said that T∈L(H) is an m-contraction if (1 −t)m(T∗, T)≥0, and that Tis an m-isometry if (1 −t)m(T∗, T) = 0. The papers [13, 15, 16, 35, 55] (among others) study m-isometries. The paper [43] is dedicated to a profound study of 2-isometries. In [36], Gu treats a more general class of (m, p)-isometries on Banach spaces, and in [37], he discusses m-isometric tuples of operators on a Hilbert space. In [20], Chavan and Sholapurkar study another interesting class of operators: Tis a joint complete hyperexpansion of order mif (1 −t)n(T∗, T)≤0 for every integer n≥m. That work, in fact, is devoted to tuples of commuting operators. Here we introduce the case when the exponent ais not an integer. The definitions of a-contraction and a-isometries are the natural ones: we say that Tis an a-contraction if (1 −t)a(T∗, T)≥0, and Tis an a-isometry if (1 −t)a(T∗, T) = 0. Note that α(t) := (1 −t)ais of Nevanlinna-Pick type when 0 < a < 1. In this case, with the help of the model given by Theorem 1.4, we will get the following two ergodic results. 8L. Abadias, G. Bello, and D. Yakubovich Theorem 1.14. If Tis an a-contraction, with 0< a < 1, then Tis quadratically (C, b)- bounded for any b > 1−a. That Tis quadratically (C, b)-bounded (where the letter C stands for Ces`aro) means that there exists a constant c > 0 such that sup n≥0 1 kb+1(n) n X j=0 kb(n−j)kTjxk2≤ckxk2(∀x∈H), where the numbers k−s(n), called Ces`aro numbers, are defined by (1 −t)s=: ∞ X n=0 k−s(n)tn. As we will show (see Example 7.3), for any a∈(0,1), the class of a-contractions on His strictly larger than the class of contractions. It is obvious that any contraction is quadratically (C, b)-bounded (due to the equality Pn j=0 kb(n−j) = kb+1(n) for any b > 0). The meaning of the above fact is that some ergodic properties of contractions still hold true for a-contractions. Theorem 1.15. Let Tbe an a-contraction with 0< a < 1and let b > 1−a. Then the following statements are equivalent. (i) The isometry Sdoes not appear in the (1 −t)a-model of T. (ii) For every x∈H, (1.9) ∃lim n→∞ 1 kb+1(n) n X j=0 kb(n−j)kTjxk2= 0. (iii) For every x∈H, lim inf n→∞ kTnxk= 0. Remark 1.16. For any a∈(0,1), there are a-contractions which are not contractions. This follows from Theorem 7.2 below. The same holds for a > 1. Indeed, if m < a ≤m+ 1, where mis an integer, then it is easy to get (see our forthcoming paper [1]) that any (m+1)-isometry Tis also an a-isometry, which means that (1 −t)a(T∗, T) = 0. There are (m+ 1)-isometries that are not contractions, and each of them is an example of this type. 1.5. Contents. The paper is organized as follows. In Section 2 we introduce two families of operators in L(H) depending on a fixed function α(t) = Pn≥0αntn: Admw αand Cw α. Essentially, Admw αis the family of operators Tfor which we can define α(T∗, T), and its subfamily Cw αconsists of those Tfor which α(T∗, T )≥0. We use the superscript notation “w” in Admw αand Cw αto make it easier to compare the results from [11] and from the present paper. Notice that in [11], only the convergence of We obtain some interesting properties of these families and characterize the membership of backward and forward weighted shifts to them. In Section 3, we prove Theorems 1.8 and 1.12. Operator Inequalities, Functional Models and Ergodicity 9 The proof of Theorem 1.5 is given in Section 4. In Section 5 we study the scope of Theorem 1.5. There we present examples satisfying the hypothesis of Theorem 1.5, where Theorem 1.4 does not apply. In Section 6 we prove Theorem 1.13. The proofs of Theorems 1.14 and 1.15 are given in Section 7. In our forthcoming paper [1], we will study models up to similarity (instead of unitary equivalence). There we will consider functions αthat may have zeroes in D. We will prove that under certain hypotheses, any operator in Cw αis similar to an a-contraction if α(t) “behaves like” (1 −t)ain a neighborhood of 1. We will also study a-contractions in more detail. 2. Preliminaries on classes defined by operator inequalities In this section we introduce the operator classes Admw αand Cw αassociated to a function α(t) = Pn≥0αntn, with αn∈R. After studying them, we analyze why Hypotheses 1.1 are natural. Finally, at the end of the section we discuss the membership of weighted shifts in the classes Admw αand Cw α. 2.1. The classes Admw αand Cw α.Before entering into the definitions and basic properties of these classes, let us mention the following well known result that will be used repeatedly. Lemma 2.1 (see [38, Problem 120]).If an increasing sequence {An}of selfadjoint Hilbert space operators satisfies An≤CI for all n, where Cis a constant, then {An}converges in the strong operator topology. Definition 2.2. Given a function α(t) = Pn≥0αntnwith αn∈R, we put (2.1) Admw α:= T∈L(H) : ∞ X n=0 |αn|kTnxk2<∞for every x∈H. Note that this class of operators is not affected if we change the signs of some coefficients αn’s. If Xand Yare two quantities (typically non-negative), then X.Y(or Y&X) will mean that X≤CY for some absolute constant C > 0. If the constant Cdepends on some parameter p, then we write X.pY. We write X≍Ywhen both X.Yand Y.X. Proposition 2.3. The following statements are equivalent. (i) T∈Admw α. (ii) P∞ n=0 |αn|kTnxk2.kxk2for every x∈H. (iii) The series P∞ n=0 |αn|T∗nTnconverges in the strong operator topology in L(H). 16 L. Abadias, G. Bello, and D. Yakubovich for every x∈Hand every non-negative integer n. Fix a positive integer 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 get τ0= 1 and τ1=···=τN= 0. Moreover, τN+i=k0αN+i+···+kNαi<0 for every i≥1, because all the αj’s above are negative or zero and the kj’s are positive. Therefore N X n=0 knkDT nxk2≤ kxk2 for every Nand hence the series PknkDTnxk2converges for every x∈H. This gives kVDxk2=∞ X n=0 knkDT nxk2≤ kxk2, as we wanted to prove.  The following fact is simple and well-known . Proposition 3.2. Let T∈L(H)with σ(T)⊂D, and let Ebe a Hilbert space. A bounded transform V:H→ Hκ⊗E satisfies (3.1) V T = (Bκ⊗IE)V if and only if there is a bounded linear operator C:H→ E such that V=VC(see (1.6)). Proof. It is well-known (and straightforward) that any bounded transform VCsatisfies (3.1). Conversely, suppose that V T = (Bκ⊗IE)V. Define 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. Therefore an+1(x) = an(Tx). The statement follows, putting C:= a0, which has to be a bounded linear operator.  Proposition 3.3. Let C:H→ E be a bounded operator and let T∈ Cw α. Then there exists a bounded operator W:H→ W such that the operator (VC, W )is isometric and transforms Operator Inequalities, Functional Models and Ergodicity 17 Tinto a part of the operator (Bk⊗IE)⊕S, where S∈L(W)is an isometry, if and only if the following conditions hold. (i) VC:H→ Hk⊗E is a contraction. (ii) For every x∈H, kxk2−kVCxk2=kTxk2−kVCTxk2. Proof. Let us suppose first the existence of such operator W. Since (VC, W ) is an isometry, (i) holds. Notice that (ii) is equivalent to proving that kWxk2=kWTxk2for every x∈H. But this is also immediate since SWx =W T x and Sis an isometry. Conversely, suppose now that (i) and (ii) are true. By (i), we can put W:= (I−V∗ CVC)1/2 and W:= Ran W. Using (ii) we have (3.2) kW xk2=kxk2−kVCxk2=kTxk2−kVCTxk2=kWT xk2. We define S(Wx) := W T x, for every x∈H. Note that Sis well defined, since kSW xk=kWxkby (3.2). Since WH is dense in W,Scan be extended to an isometry on W. By the definition of W, we know that (VC, W) is an isometry and it is immediate that (Bk⊗ID)VC=VCTand SW =WT. This completes the converse implication.  Proposition 3.4. Let T∈ Cw α. Assume that C:H→ E and W:H→ W are any bounded operators such that (VC, W )is isometric on (Hk⊗E)⊕W and transforms Tinto a part of (Bk⊗IE)⊕S, where S∈B(W)is an isometry. Then Cand Dare related by (3.3) kDxk2=kCxk2+α(1)kWxk2,∀x∈H. Proof. Since (VC, W) is isometric, we have (3.4) kxk2=kVCxk2+kW xk2=∞ X n=0 knkCTnxk2+kWxk2, for every x∈H. Substituting xby Tjxabove and multiplying by αj, we obtain that 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. Yakubovich where we have used that kWxk2=kWTxk2. Therefore 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, the only non-vanishing summand in the last series above is for m= 0 and we obtain (3.3). Finally, note that the rearrangement in (⋆) is correct as ∞ X j=0 ∞ X n=0 |αj|knCTn+jx2≤∞ X j=0 |αj|Tjx2<∞, where we have used (3.4) and that T∈ Cw α. Recall the definition of the minimal model (Definition 1.10). Remark 3.5. Suppose that Tis α-modelable. Then Tis unitarily equivalent to (Bk⊗IE)⊕ S|L, where L=Ran (VC, W). This model is minimal if and only if (a) Ran C=E; and (b) Ran W=W. Indeed, in this case, it is easy to see that (a) is equivalent to (i), and (b) is equivalent to (ii) in Definition 1.10. Proof of Theorem 1.12. Suppose that the hypotheses are satisfied. First we notice that α(T∗, T)≥0, as it follows from Corollary 2.13 and Proposition 2.6. Therefore Dis welldefined. (i) In the critical case (i.e., α(1) = 0), (3.3) gives kDxk=kCxk ∀x∈H, so there exists a unitary operator vsuch that C=vD. This implies the statement. (ii) Suppose we are in the subcritical case (i.e., α(1) >0). First, we remark that the model is not unique in general. For instance, take T=Uany unitary operator. Using Proposition 2.3 and that α∈AW, we obtain that T∈Admw α. Since ∞ X n=0 αnkTnxk2= ∞ X n=0 αn!kxk2∀x∈H, we get that α(T∗, T) = α(1)I≥0. Obviously, T=Uis a minimal model for T(where E= 0, and W=H). Moreover, if k= 1/α fits (1.5), then Theorem 1.5 (which is proved in the next section, but its proof is completely independent) gives another model for T. (See Example 5.1 and Remark 5.2.) Operator Inequalities, Functional Models and Ergodicity 19 Now suppose that Tis any α-modelable operator and (VC, W ) provides its model. Let us see that there exists a minimal model of Twith V=VDand Wabsent. Changing xby Tnx in (3.3) we obtain kDT nxk2=kCT nxk2+α(1) kWxk2, where we have used that kWT xk=kW xk. Therefore 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 isometry and therefore provides a model of T. The space Lis just Ran VDin Hk⊗D(which is closed). This model is minimal, because Ran Dis dense in D. (See Remark 3.5.) This gives all statements of (ii).  Proof of Theorem 1.8. It is an immediate consequence of Theorem 1.12 and Proposition 3.3 (i) that VDis a contraction. Finally, for proving that (VD, W) gives a model, we just need to use the same argument employed in the reciprocal implication of Proposition 3.3.  Results close to Theorems 1.8 and 1.12 appear in Schillo’s PhD thesis [58]. He deals with the generality of tuples of commuting operators, but for the case of one operator, the hypotheses needed there are more restrictive than ours. For example, in [58, Theorem 5.16], the uniqueness of the coextension is proved when T is what he calls a strong k-contraction. For one single operator Tand using our notations, these are operators such that α(T∗, T)≥0, the limit Σ(T) := IH−lim N→∞ N X n=0 knT∗nα(T∗, T)Tn exists (in SOT), Σ(T)≥0, and Σ(T) = T∗Σ(T)T. In [58, Corollary 5.17], he gives an explicit model involving the defect space DT. His assumptions are somewhat technical (see [58, Assumption 5.8]). He also assumes the existence of α(B∗ k, Bk), for which [58, Proposition 2.10] says that a sufficient condition is that the coefficients {αn}of the function αhave eventually the same sign. Recall that our Theorem 1.12 (ii) says that for the subcritical case the model is not unique in general. Therefore, since Schillo obtains uniqueness of the coextension, it seems that his assumptions exclude the subcritical case. Schillo’s thesis also contains a result on the description of invariant subspaces of a backward shift, analogous to Bk⊗IE, in his setting of operator tuples. Notice that in Theorems 1.8 and 1.12 we are only assuming that Tis α-modelable. In particular, we do not impose any restriction about the signs of the Taylor coefficients of the function α. 20 L. Abadias, G. Bello, and D. Yakubovich 4. Proof of Theorem 1.5 In this section we prove Theorem 1.5. For that, we need to cite some results concerning Banach algebras. For any sequence ω={ωn}∞ n=0 of positive weights, define the weighted space ℓ∞(ω) := (f(t) = ∞ X n=0 fntn: sup n≥0|fn|ωn<∞). In general, its elements are formal power series. We will also use the separable version of this space: ℓ∞ 0(ω) := (f(t) = ∞ X n=0 fntn: lim n→∞|fn|ωn= 0). Proposition 4.1 (see [47]).ℓ∞(ω)is a Banach algebra (with respect to the formal multiplication of power series) if and only if (4.1) sup n≥0 n X j=0 ωn ωjωn−j <∞. Theorem 4.2. Let ωn>0and ω1/n n→1. If supnωn+1/ωn<∞and (4.2) lim m→∞ sup n≥2mX m≤j≤n/2 ωn ωjωn−j = 0, then the following is true. (i) ℓ∞(ω)is a Banach algebra. (ii) If f∈ℓ∞(ω)does not vanish on D, then 1/f ∈ℓ∞(ω). Proof. The hypotheses imply (4.1), so that (i) follows from Proposition 4.1. To get (ii), we apply the results of the paper [29] by El-Fallah, Nikolski and Zarrabi. We use the notation of this paper. Put ω′(n) = ω(n)/(n+ 1), A=ℓ∞(ω) and A0=ℓ∞ 0(ω). The hypotheses imply that A (and hence A0) is compactly embedded into the multiplier convolution algebra mult(ℓ∞(ω′)), see [29, Lemma 3.6.3]. Hence, by [29, Theorem 3.4.1], for any f∈A0,δ1(A0,M(A0)) = 0, see [29, Subsection 0.2.3] for the definition of this quantity. This means that for any δ > 0 there is a constant c1(δ)<∞such that the conditions f∈A0,kfkA= 1 and |f|> δ on D imply that 1/f ∈A0and k1/fkA≤c1(δ). In particular, (ii) holds for fin A0. To get (ii) in the general case, suppose that f∈Aand |f|> δ > 0 on D. Since f(rt)∈A0for all r < 1, we get that the norms of the functions 1/f(rt) in Aare uniformly bounded by c1(δ) for all r < 1. When r→1−, each Taylor coefficient of 1/f(rt) tends to the corresponding Taylor coefficient of 1/f(t). It follows that 1/f is in A(and k1/fkA≤c1(δ)).  Proof of Theorem 1.5. Put ωn:= 1/kn. Operator Inequalities, Functional Models and Ergodicity 21 The first part of Theorem 1.5 (that Bkis bounded) is straightforward. Also, by Theorem 4.2 (i), ℓ∞(ω) is an algebra. First suppose that Tis a part of Bk⊗IE, and let us prove that Bk∈ Cw α∩Admw k. By Theorem 2.12 (i), we know that Bk∈Admw kif and only if m X j=0 kjkm−j.km, which follows from Theorem 4.2 (i) and Proposition 4.1. Now let us see that Bk∈ Cw α. By Theorem 4.2 (ii), α= 1/k belongs to ℓ∞(ω), and therefore |αn|.kn. Then, since Bk∈Admw k, we obtain that Bk∈Admw α. Finally, Theorem 2.12 (ii) gives that Bk∈ Cw α(because αk = 1 has non-negative Taylor coefficients). Hence Talso is in Cw α∩Admw k. Conversely, let us assume now that T∈ Cw α∩Admw k. We want to prove that Tis a part of Bk⊗IE. We adapt the argument of [45, Theorem 2.2] (where the convergence of the series of operators is in the uniform operator topology). By Proposition 3.2, (4.3) (Bk⊗ID)VD=VDT. Moreover, 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, where we have used that Pn+m=jknαmis equal to 1 if j= 0 and is equal to 0 if j≥1. The re-arrangement of the series is correct since, using that T∈Admw α∩Admw k, we have ∞ X n=0 kn ∞ X m=0 |αm|Tn+mx2.∞ X n=0 knkTnxk2.kxk2 and the series converges absolutely. Hence VDis an isometry. Joined to (4.3), this proves that Tis unitarily equivalent to a part of Bk⊗ID. Notice that in particular, we showed that the hypotheses of Theorem 1.5 imply Hypotheses 1.7. 5. Discussion of Theorem 1.5 In this section we discuss the scope of Theorem 1.5 and give a series of examples where it applies, whereas Theorem 1.4 does not. We also will give a direct proof of a particular case of Theorem 4.2, which does not use the results of [29]. 22 L. Abadias, G. Bello, and D. Yakubovich Given an analytic function f(t) = Pfntn, we denote by [f]Nits truncated polynomial of degree N, that is, [f]N:= f0+f1t+...+fNtN. Example 5.1. Let σ2,...,σNbe an arbitrary sequence of signs (that is, a sequence of numbers ±1). We assert that there are functions α, k meeting all the hypotheses of Theorem 1.5 such that sign(αn) = σn, for n= 2,...,N. This is in contrast with Theorem 1.4, where the Nevanlinna-Pick condition was assumed: αn≤0 for n≥2. To prove the existence of αand kas above, take a polynomial eαof degree Nsuch that eα0= 1,eα1<0. For n= 2,...,N, we set eαn<0 if σn=−1 and eαn= 0 if σn= 1. Put ek:= [1/eα]N. The formula (5.1) ekn=X s≥1 n1+···+ns=n (−1)seαn1···eαns shows that all the coefficients of ekare positive. We also require that neither eαnor the polynomial ekvanish on D. It is so if, for instance, |αn|are sufficiently small for n= 2,...,N. Now perturb the coefficients eαjthat are equal to zero, obtaining a new polynomial bαsuch that bαj:= (εif σj= 1 eαjotherwise (2 ≤j≤N). By continuity, if ε > 0 is small enough, we can guarantee that the polynomial bk= [1/bα]N also has positive Taylor coefficients, and we can also guarantee that bk(which is a slight perturbation of ek) does not vanish on D. Finally, take as kany function in AWwith real Taylor coefficients such that the first ones are k0=bk0= 1, k1=bk1, . . . , kN=bkN, and kn−j kn≤C0(∀n≥2j), for some constant C0. For instance, one can put kn=An−bfor n > N, with A > 0 (small enough) and b > 1. Then k∈AWdoes not vanish on D. Then obviously ksatisfies (1.5) and hence all the hypotheses of Theorem 1.5. The function α:= 1/k in AWhas the desired pattern of signs. Finally, it is important to note that α1=−k1is always negative. Remark 5.2. It is also easy to see that whenever {kn}satisfies (1.5), any other sequence {˜ kn}with k0= 1 and c < ˜ kn/kn< C for n > 1, where c, C are positive constants, also satisfies this condition. In particular, if {kn}satisfies (1.5) and {˜ kn}is as above, where C is sufficiently small, then k(t) is invertible in AW, so that all hypotheses of Theorem 1.5 are Operator Inequalities, Functional Models and Ergodicity 23 fulfilled. So there are many examples of functions k(t) meeting these hypotheses, such that the quotients kn/kn+1 do not converge. Let us mention now some remarks on Theorem 4.2. Remark 5.3. It is immediate that the condition (5.2) ωn ωjωn−j≤τj(∀n≥2j),where ∞ X j=0 τj<∞, implies (4.2) and (4.1) (in particular, supnωn+1/ωn<∞). Let us give a direct proof of Theorem 4.2 for this particular case. Statement (i) follows using Proposition 4.1. (ii) Put g:= 1/f. Suppose that g6∈ ℓ∞(ω). This means that sup n≥0|gn|ωn=∞. Hence, it is clear that there exists a sequence {ρ0 n}in [0,1] such that ρ0 n→0 (slowly) and (5.3) sup n≥0|gn|ωnρ0 n=∞. Claim. There exists a sequence {ρn}with (5.4) ρ0 n≤ρn≤1 and ρn→0 such that eωn:= ρnωndefines a Banach algebra ℓ∞(eω). Indeed, since Pτj<∞, there exists a sequence of positive numbers {cj}such that cjր ∞ and still Pcjτj<∞. Take any sequence {ρn}that decreases, tends to zero, and satisfies ρn≥max(ρ0 n,1/cn). Then, for eωn:= ρnωnwe have eωn eωjeωn−j =ωn ωjωn−j ρn ρjρn−j≤ωn ωjωn−j 1 ρj≤τjcj(∀n≥2j). Since Pτjcj<∞, Proposition 4.1 implies that ℓ∞(eω) is a Banach algebra, and the proof of the claim is completed. Now fix {eωn}as in the claim. We may assume that (ρ0 n)1/n →1 and therefore (ρn)1/n →1. Since the polynomials are dense in the Banach algebra ℓ∞ 0(eω), any complex homomorphism χon ℓ∞ 0(eω) is determined by its value on the power series t. So the map χ7→ χ(t) is injective and continuous from the spectrum (the maximal ideal space) of ℓ∞ 0(eω) to C. Since eω1/n n→1, its image contains Dand is contained in D. Hence the spectrum of ℓ∞ 0(eω) is exactly the set {χλ:λ∈D}, where χλ(f) = f(λ). (We borrow this argument from [29].) As fneωn= (fnωn)ρn→0, we have f∈ℓ∞ 0(eω). Then, using the Gelfand theory (see, for instance, [54, Chapter 10]), we get that g= 1/f ∈ℓ∞ 0(eω), which contradicts (5.3). Therefore, the assumption g /∈ℓ∞(ω) is false, as we wanted to prove. 24 L. Abadias, G. Bello, and D. Yakubovich Remark 5.4. Notice that the above characterization of the spectrum of the algebra ℓ∞ 0(eω) (see the above Remark 5.3) implies the following fact: the conditions (4.1) and ω1/n n→1 imply that Pn1/ωn<∞. This can be proved in an elementary way, without recurring to the Gelfand theory. Indeed, by (4.1), there exists a constant C > 0 such that n X j=1 ωn ωjωn−j≤C for every n≥1. Fix a positive integer L. Then obviously, for every n≥L, (5.5) L X j=1 ωn ωjωn−j≤C. Let us see that (5.6) lim sup n→∞ min 1≤j≤L ωn ωn−j≥1. Indeed, if (5.6) were false, then there would exist some r < 1 and a positive integer Nsuch that min 1≤j≤L ωn ωn−j≤rfor n≥N. From this, it is easy to see that ωn≤rsnmax 0≤k≤Nωk, sn:= n−N L+ 1, where [a] denotes the integer part of a. Since snbehaves asymptotically as n/L, it follows that lim supn→∞ ω1/n n≤r1/L <1, which contradicts the hypothesis that ω1/n n→1. Therefore, (5.6) is true. Now, using (5.5), it follows that 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 get that P1/ωjconverges. The following statement shows that in the subcritical case, the hypotheses of Theorem 1.5 imply that the radius of convergence of the series for αis equal to one. Proposition 5.5. If lim k1/n n= 1 and αis of subcritical type, then αdoes not continue analytically to any disc RD, where R > 1. Proof. Since k(t) has nonnegative Taylor coefficients, we have |k(t)| ≤ k(1) for all t∈D. Using that k= 1/α, it follows that in the subcritical case, |α(t)| ≥ α(1) >0 for any t∈D. So, αcannot continue analytically to any disc RD, where R > 1, because in this case, the radius of convergence of the Taylor series for kwould be greater than 1.  Operator Inequalities, Functional Models and Ergodicity 25 6. Finite Defect It is well-known that in the classical Sz.-Nagy-Foias model, the case of a finite rank (or Hilbert-Schmidt) defect operator is an important one, where much more tools and results are available. In this section, we derive some consequences of our model theorems for the case when an operator T∈ Cw αis α-modelable and the defect operator D= (α(T∗, T))1/2is of finite rank. We will assume that the reproducing kernel Hilbert space Rkis a Banach algebra with respect to the multiplication of power series. By [60, Proposition 32], it suffices to assume that sup n n X j=0 k2 jk2 n−j k2 n <∞; compare with the condition (4.1). Put mn= inf j kj kn+j , r1= lim n→∞m1/n n. This limit exists, see [60, Proposition 12]. We will assume that (6.1) r1= lim n→∞k1/n n= 1. Both equalities hold, in particular, if lim kn+1/kn= 1. The same is true if, for instance, the last limit does not exist, but 0 < σ < kn< C < ∞for all nand there is some m≥2 such that limnkn+m/kn= 1. We also are assuming here that the isometric part Sis not present in the model of T. Hence, Tis unitarily equivalent to the restriction of the backward shift Bk⊗IDon the space Hk⊗Dto an invariant subspace L. More generally, this applies to similarity instead of the unitary equivalence (we bear in mind models of linear operators up to similarity, which are established in [1]). Here we prove the following result. Theorem 6.1. Suppose that Tis similar to a part of Bk⊗ID, acting on the space Hk⊗D, where Rkis a Banach algebra and Dis finite dimensional. If the spectrum σ(T)does not cover the open disc D, then σ(T)∩Dis contained in the zero set of a non-zero function in Rk. Let us start with some preliminary remarks. Suppose that Tis as in the above Theorem 6.1. That is, Tis similar to (Bk⊗ID)|L, where L ⊂ Hk⊗Dis an invariant subspace of Bk⊗ID. By fixing a basis in D, we may assume that D=Cd, where d= dim D. We will identify the space Hk⊗Dwith Hd k=⊕d 1Hk, whose elements are columns with entries in Hk. The adjoint of Bkon the space Hd kis the multiplication operator Mzon the space Rd k; this later space can be seen as a Banach module over the Banach algebra Rk. Put J=L⊥⊂ Rk⊗D. 32 L. Abadias, G. Bello, and D. Yakubovich the Ces`aro means of order aof T. When this family of operators is uniformly bounded, that is, sup n≥0kMa T(n)k<∞, we say that Tis (C, a)-bounded. Remarks 7.5. (i) Note that Pn j=0 ka(j) = ka+1(n) for any a≥0. Also, if a≥0, then ka(j)≥0 for every j≥0. (ii) If a= 0, then M0 T(n) = Tn. Hence (C, 0)-boundedness is just power boundedness. (iii) If a= 1, then M1 T(n) = (n+ 1)−1Pn j=0 Tj. Hence (C, 1)-boundedness is just Ces`aro boundedness. (iv) It is well-known that if 0 ≤a < b, then (C, a)-boundedness implies (C, b)-boundedness. The converse is not true in general. For example, the Assani matrix T= −1 2 0−1! is (C, 1)-bounded, but since Tn= (−1)n(−1)n+12n 0 (−1)n! it is not power bounded (see [30, Section 4.7]). Definition 7.6. If the sequence of operators {Ma T(n)}n≥0given in Definition 7.4 converges in the strong operator topology, we say that Tis (C, a)-mean ergodic. If Tis (C, 1)-mean ergodic, it is conventional just to say that Tis mean ergodic. There is a well established literature on (C, a)-bounded operators, which explores quite a number of properties and their interplays. Properties, characterization through functional calculus and ergodic results for (C, a)-bounded operators can be found in [4, 9, 25, 27, 28, 30, 41] and references therein. The connection of these operators and ergodicity dates back to the fourties of last century, see [24] and [40]. In the latter paper, E. Hille studies (C, a)-mean ergodicity in terms of Abel convergence (that is, via the resolvent operator). As application, the well known mean ergodic von Neumann’s theorem for unitary groups on Hilbert spaces is extended to (C, a)-mean ergodicity for every a > 0 [40, p. 255]. Also, the (C, a)-ergodicity on L1(0,1) of fractional (Riemann-Liouville) integrals is elucidated in [40, Theorem 11]. In particular, if Vis the Volterra operator then TV:= I−V, as operator on L1(0,1), is not power-bounded, and it is (C, a)-mean ergodic if and only if a > 1/2 [40, Theorem 11]. This result can be extended to TVacting on Lp(0,1), 1 < p < ∞, using estimates given in [44], see [3, Section 10]. Operator Inequalities, Functional Models and Ergodicity 33 In [42], Luo and Hou introduced a new notion of boundedness: a bounded linear operator Ton a Banach space Xis said to be absolutely Ces`aro bounded if sup n≥0 1 n+ 1 n X j=0 Tjx.kxk for every x∈X. In [14], the authors study the ergodic behaviour for this class of operators. The above definition has been extended recently by Abadias and Bonilla in [2]: Tis said to be absolutely (C, a)-Ces`aro bounded for some a > 0 if sup n≥0 1 ka+1(n) n X j=0 ka(n−j)Tjx.kxk for every x∈X. Note that for a= 1 the definition of Luo and Hou is recovered. Remark 7.7. It is well-known that the following implications hold: Power bounded ⇒Absolutely (C, a)-bounded ⇒(C, a)-bounded ⇒ kTnk=O(na). The first two implications are straightforward. For the sake of completeness, we give a proof of the last one. Suppose Tis (C, a)-bounded for some a≥0. We denote by [a] the integer part of a. Then, for n > [a], we have 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 following extension of the above definitions will be important for us. Definition 7.8. Let a > 0 and p≥1. We say that a bounded linear operator Ton a Banach space Xis (C, a, p)-bounded if sup n≥0 1 ka+1(n) n X j=0 ka(n−j)kTjxkp.kxkp, 34 L. Abadias, G. Bello, and D. Yakubovich for all x∈X. Note that for p= 1 this definition is just the absolute (C, a)-boundedness. The case a= 1 has been recently considered in [23]. We will use the term quadratically (C, a)-bounded instead of (C, a, 2)-bounded. Using the asymptotics ka(n)≍(n+1)a−1given in (7.3), it is easy to see that Tis (C, a, p)- bounded if and only if (7.7) sup n≥0 1 (n+ 1)a n X j=0 (n+ 1 −j)a−1Tjxp.kxkp(∀x∈X). The following observation will be essential for the proof of Theorem 1.14. Lemma 7.9. The following holds. (i) If Tis (C, a, p)-bounded, then any part of Tis also (C, a, p)-bounded. (ii) If T1and T2are (C, a, p)-bounded, then any direct sum T1∔T2is also (C, a, p)-bounded. (iii) Let Tbe a bounded linear operator on a Hilbert space. If Tis quadratically (C, a)- bounded, then T⊗IEis also quadratically (C, a)-bounded, where IEis the identity operator on some Hilbert space E. Proof. (i) and (ii) are immediate. For (iii) note that if d= dim E ≤ ∞, then the orthogonal sum of dcopies of Tis clearly quadratically (C, a)-bounded (by the Pythagoras Theorem).  The following result is very useful. Its proof is simple, and we omit it. Lemma 7.10. Let 0≤a < b. Then (C, a, p)-boundedness implies (C, b, p)-boundedness. This lemma shows an inclusion of classes of operators. By [2, Corollaries 2.2 and 2.3], if Tis (C, a, 1)-bounded then kTnk=o(na) for 0 < a ≤1 and kTnk=O(n) for a > 1. The following result explains why the case a= 1 is special. Theorem 7.11. If a > 1and p≥1, then (C, a, p)-boundedness is equivalent to (C, 1, p)- boundedness. Proof. Fix a > 1 and p≥1. By the above Lemma, we only need to prove that any (C, a, p)- bounded operator Tis (C, 1, p)-bounded. Let Tis (C, a, p)-bounded. Then (7.8) 1 ka+1(2n) 2n X j=0 ka(2n−j)Tjxp.kxkp, for every n≥0, and every x∈X. Since a > 1, ka(m) is an increasing function of m. In particular, ka(n)≤ka(2n−j) for j= 0,...,n. Hence (7.9) ka(n) n X j=0 Tjxp≤ 2n X j=0 ka(2n−j)Tjxp, Operator Inequalities, Functional Models and Ergodicity 35 By (7.9) and (7.8), n X j=0 Tjxp.ka+1(2n) ka(n)kxkp.(n+ 1) kxkp, which means that Tis (C, 1, p)-bounded.  Theorem 7.12. Let a > 0and 1≤q < p. If Tis (C, a, p)-bounded, then it is also (C, b, q)- bounded for each b > qa/p. In particular, (C, a, p)-boundedness implies (C, a, q)-boundedness. Proof. Let us first recall that if r > −1, then (7.10) m X j=1 jr.mr+1 (∀m≥1). Let Tbe (C, a, p)-bounded and let b > qa/p. Suppose first that b6= 1, and put s:= p p−q, s′:= p q, γ := q(a−1) p(b−1) . Note that sand s′are positive and satisfy 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¨older’s inequality and (7.10) it follows that 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 for every x∈Xand every non-negative integer n. Hence the statement follows using (7.7). Now suppose that b= 1. Take any b′∈(qa/p, 1). We have already proved that T is (C, b′, p)-bounded. Then, by Lemma 7.10, it follows that Tis (C, 1, p)-bounded. This completes the proof.  Lemma 7.13. Let a > 0and p≥1. Then every isometry Sis (C, a, p)-bounded. 36 L. Abadias, G. Bello, and D. Yakubovich Proof. This is immediate, 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 for every x∈X. Lemma 7.14. Let 0< s < 1and let a > 0. Then Bsis quadratically (C, a)-bounded if and only if 1−s < a. Moreover, for 1−s < a we have (7.12) lim n→∞ 1 ka+1(n) n X j=0 ka(n−j)kBj sxk2= 0 (∀x∈ Hs). Proof. Recall the notation en=tn∈ Hk=Hs, where k(t) = (1−t)−s. Suppose that 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) for every n. Therefore Bsis not quadratically (C, 1−s)-bounded, and by Lemma 7.10 we obtain that Bsis not quadratically (C, a)-bounded for a < 1−s. Let us assume now that 1 −s < a ≤1 and fix x∈ Hs. Write xin the form x=Pxmem, where xm∈C. Then Bj sx2=∞ X m=j ks(m−j)|xm|2.∞ X m=j (m+ 1 −j)s−1|xm|2, for every 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). Operator Inequalities, Functional Models and Ergodicity 37 In (I), note that since 1 −s < a ≤1, and m≤n, we have (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, where in the last estimate we used (7.10). Therefore (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 that m > n and s−1<0, we have 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. Therefore (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 have that 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. Therefore (III).∞ X m=2n+1 |xm|2(m+ 1)s−1−→ 0 (as n→ ∞). Hence (7.12) follows when 1 −s < a ≤1. Finally, suppose that 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 this is the case of a= 1 in (7.12) (already proved). Note that (7.12) implies quadratical (C, a)-boundedness, so the proof is complete.  This lemma allows us to prove the following more general result. 38 L. Abadias, G. Bello, and D. Yakubovich Theorem 7.15. Let 0< s < 1and 1≤q≤2. Then Bsis (C, b, q)-bounded if and only if b > q(1 −s)/2. Moreover, for b > q(1 −s)/2we have (7.15) lim n→∞ 1 kb+1(n) n X j=0 kb(n−j)Bj sxq= 0 (∀x∈H). Proof. Note that q= 2 is precisely Lemma 7.14. So we assume that 1 ≤q < 2. If b= q(1 −s)/2, taking x=en, we get, as in (7.13), that 1 kb+1(n) n X j=0 kb(n−j)Bj senq&log(n+ 2) kenk2 for every n. Therefore Bsis not (C, q(1 −s)/2, q)-bounded, and by Lemma 7.10 we get that Bsis not (C, b, q)-bounded for b < q(1 −s)/2. Now suppose that b > q(1 −s)/2. Then b=qa/2 for some a > 1−s. Using H¨older’s inequality as in the proof of Theorem 7.12, we obtain 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) follows.  Proof of Theorem 1.14. Let T∈ Cw awith 0 < a < 1 and let b > 1−a. By Theorem 1.4 and Theorem 1.12 (i), Tis unitarily equivalent to a part of (Ba⊗ID)⊕S. Hence, by Lemma 7.9 (i), it is enough to prove that (Ba⊗ID)⊕Sis quadratically (C, b)-bounded. But this is immediate using Lemma 7.9 (ii) and (iii), and Lemmas 7.13 and 7.14.  For the proof of Theorem 1.15 we need the following lemma, which is in the spirit of Lemma 7.9. Lemma 7.16. The following holds. (i) If Tsatisfies (1.9), then any part of Talso satisfies (1.9). (ii) If T1and T2satisfy (1.9), then any direct sum T1∔T2also satisfies (1.9). (iii) Let Tbe a bounded linear operator on a Hilbert space. If Tsatisfies (1.9), then the operator T⊗IEalso satisfies (1.9), where IEis the identity operator on some Hilbert space E. Proof. (i) and (ii) are immediate. For (iii) we use the same argument as in Lemma 7.9 (iii) and a simple application of Lebesgue’s Dominated Convergence Theorem.  Proof of Theorem 1.15. As in the proof of Theorem 1.14, we have that Tis unitarily equivalent to (Ba⊗ID)⊕S|L, Operator Inequalities, Functional Models and Ergodicity 39 where Lis a subspace of (Ha⊗D)⊕W invariant by (Ba⊗ID)⊕S. Let us prove the circle of implications (i) ⇒(ii) ⇒(iii) ⇒(i). Suppose that (i) is true. That is, Tis unitarily equivalent to (Ba⊗ID)|L, where Lis a subspace of Ha⊗Dinvariant by Ba⊗ID. Then (ii) follows using Lemmas 7.14 and 7.16. Suppose now that lim inf n→∞ kTnxk>0 for some x∈H. Then, obviously, kTnxk> ε > 0 for every n≥0. Hence for this vector x (1.9) does not hold. Therefore we have proved that (ii) ⇒(iii). Finally, suppose that the isometry Sappears in the minimal model. Then for some vector ℓ= (ℓ1, ℓ2)∈ L, its second component ℓ2∈ W is not 0. Therefore 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 limit is kℓ2k26= 0 because of (7.11). Hence we obtain that (iii) ⇒(i).  Remark 7.17. In the same way, we get that if Tis an a-contraction and 0 < a ≤1, then lim inf n→∞ kTnxk ≤ kxk. In particular, this lower limit is finite for any x. Since Cw 1is just the set of all contractions on H,T∈ Cw 1iff T∗∈ Cw 1. However, this is no longer true for a∈(0,1). Proposition 7.18. If a∈(0,1), then there is an operator T∈ Cw asuch that T∗/∈ Cw a. Proof. Note that B∗ ais a forward weighted shift such that kB∗n af0k → ∞ as ngoes to ∞. So Ba∈ Cw a, whereas its adjoint cannot belong to Cw a, because B∗ ais not quadratically (C, b)- bounded for any b(see Lemma 7.14).  It is natural to pose the following question. Question 7.19. For which functions α, satisfying Hypotheses 1.1, is it true that T∈ Cw α implies T∗∈ Cw α? 40 L. Abadias, G. Bello, and D. Yakubovich It is so for α(t) = 1 −tand, more generally, for α(t) = 1 −tn,n≥1. The authors do not know other examples. Remarks 7.20. (i) If Tis an operator in Cw awith 0 < a < 1, and 0 < q < 2, then by Theorem 1.14 and Theorem 7.12, it follows that Tis (C, b, q)-bounded for all b > q(1−a) 2. (ii) An m-isometry T, which is not an isometry, cannot be (C, a, p)-bounded, because there are vectors xsuch that the norms kTnxkgo to infinity. The possibility for these operators to have weaker ergodic properties, such as the Ces`aro boundedness and weak ergodicity, have been studied in [13]. (iii) Let Tbe an operator in Cw awith 0 < a < 1. Using Theorem 1.14 (i) and Theorem 7.12 (with p= 2 and q= 1) we obtain that Tis (C, b, 1)-bounded for every b > (1 −a)/2. By [2, Corollary 3.1], we get that Tis (C, b)-mean ergodic, that is, there exists Pbx:= lim n→∞Mb T(n)x, x ∈H. Therefore, by [3, Theorem 3.3], we have H= Ker(I−T)⊕Ran(I−T). In fact, Ker(I−T) = RanPband Ran(I−T) = KerPb. Also note that Mb T(n)x=xfor x∈Ker(I−T),and lim n→∞Mb T(n)x= 0 for x∈Ran(I−T). Let now 0 < γ < 1,by [3, Proposition 4.8 and Remark 4.9], one can define a bounded operator (I−T)γby means of a certain functional calculus, and Ker(I−T) = Ker(I−T)γ,Ran(I−T) = Ran(I−T)γ, with Ran(I−T)⊆Ran(I−T)γ. Furthermore if γ < 1−b, for x∈Ran(I−T), x∈Ran(I−T)γ⇐⇒ ∞ X n=1 1 n1−γTnxconverges, see [3, Theorem 9.2]. (iv) By [2, Theorem 3.1], if Tis an operator in Cw awith 0 < a < 1 and b > (1 −a)/2, then lim n→∞kMb T(n+ 1) −Mb T(n)k= 0. Acknowledgments The authors thank T. Bhattacharyya and N. Nikolski and D. Schillo for their useful remarks, and A. Bonilla for his advice concerning Theorem 7.11. The first author has been Operator Inequalities, Functional Models and Ergodicity 41 partly supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS, Project E2617R, D.G. Arag´on, and Project for Young Researchers, Fundaci´on Ibercaja and Universidad de Zaragoza, Spain. The second author has been partially supported by La Caixa-Severo Ochoa grant (ICMAT Severo Ochoa project SEV-2011-0087, MINECO). Both second and third authors acknowledge partial support by Spanish Ministry of Science, Innovation and Universities (grant no. PGC2018-099124-B-I00) and the ICMAT Severo Ochoa project SEV-2015-0554 of the Spanish Ministry of Economy and Competitiveness of Spain and the European Regional Development Fund, through the “Severo Ochoa Programme for Centres of Excellence in R&D”. Both second and third authors also acknowledge financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554) and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). References [1] L. Abadias, G. Bello, and D. 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