� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 35, Num. 2 (2020), 137 – 184 doi:10.17398/2605-5686.35.2.137 Available online June 20, 2020 Unitary skew-dilations of Hilbert space operators V. Agniel∗ Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, France vidal.agniel@univ-lille.fr Received February 29, 2020 Presented by Mostafa Mbekhta Accepted May 12, 2020 Abstract: The aim of this paper is to study, for a given sequence (ρn)n≥1 of complex numbers, the class of Hilbert space operators possessing (ρn)-unitary dilations. This is the class of bounded linear operators T acting on a Hilbert space H, whose iterates Tn can be represented as Tn = ρnPHU n|H, n ≥ 1, for some unitary operator U acting on a larger Hilbert space, containing H as a closed subspace. Here PH is the projection from this larger space onto H. The case when all ρn’s are equal to a positive real number ρ leads to the class Cρ introduced in the 1960s by Foias and Sz.-Nagy, while the case when all ρn’s are positive real numbers has been previously considered by several authors. Some applications and examples of operators possessing (ρn)-unitary dilations, showing a behavior different from the classical case, are given in this paper. Key words: Hilbert space operators, Dilations, Compressions of linear operators, Functional calculi, Numerical radius, ρ-radii, ρ-classes, (ρn)-classes If there are too many of them, you can remove ρ-radii and (ρn)-classes. AMS Subject Class. (2010): 47A12, 47A20, 47A30, 47A60. 1. Introduction Classes Cρ have been introduced by B. Sz-Nagy and C. Foias [22] in 1966. For a complex Hilbert space H and a real number ρ > 0, a bounded linear operator T ∈L(H) is said to be in the class Cρ(H) if all powers of T can be skew-dilated to powers of a unitary operator on a Hilbert space K, containing H as a closed subspace. This means that Tn = ρPHU n|H, for all n ≥ 1, where U ∈ L(K) is a suitable unitary operator, and PH ∈ L(K) denotes the orthogonal projection onto H. Such an operator T is called a ρ-contraction, while the unitary operator U is called a ρ-dilation, or a ρ-unitary dilation, of T. ∗ http://perso.eleves.ens-rennes.fr/~VAIGN357/index.html ISSN: 0213-8743 (print), 2605-5686 (online) c©The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.35.2.137 mailto:vidal.agniel@univ-lille.fr http://perso.eleves.ens-rennes.fr/~VAIGN357/index.html https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 138 v. agniel The famous Sz.-Nagy dilation theorem (see [22]) shows that C1(H) is ex- actly the class of all Hilbert space contractions i.e., operators of norm no greater than one. It is also known (see [6]) that the class C2(H) coincides with the class of all operators T with numerical range W(T) included in the closed unit disk; equivalently, those T satisfying w(T) ≤ 1. Here the numerical range W(T) and the numerical radius w(T) of T are defined by W(T) = { 〈Tx,x〉 : ‖x‖ = 1 } ; w(T) = sup { |λ| : λ ∈ W(T) } . Let T be an operator in the class Cρ. Then (i) T is power-bounded. More precisely, we have ‖Tn‖≤ max(1,ρ), for all n ≥ 0. In particular, the spectral radius r(T) of T satisfies r(T) ≤ 1; (ii) Tk is in Cρ(H) for every k ≥ 1; (iii) For a closed subspace, F , of H which is stable by T (i.e., T(F) ⊂ F), the restriction T |F is in Cρ(F); (iv) The functional calculus map f 7→ f(T) that sends a polynomial f into f(T) can be extended in a well-defined manner to the disk algebra A(D) := C0(D) ∩ Hol(D). It is a morphism of Banach algebras, and satisfies ‖f(T)‖≤ max(1,ρ)‖f‖L∞(D); (v) T is similar to a contraction: there is an invertible operator L ∈ L(H) such that ‖LTL−1‖≤ 1. We refer the reader to [11, 12, 23, 17, 15] for proofs of these results, which mainly use several characterizations of classes Cρ(H). We record the principal ones in the following theorem. Theorem. Let T be an operator in L(H) and let ρ > 0. The following are equivalent: (i) T ∈ Cρ(H); (ii) r(T) ≤ 1 and, for all z ∈ D, we have ( 1 − 2 ρ ) I + 2 ρ Re ( (I −zT)−1 ) ≥ 0; (iii) For all z ∈ D and all h ∈ H we have ( 2 ρ −1 ) ‖zTh‖2 + ( 2− 2 ρ ) 〈zTh,h〉≤ ‖h‖2. We remark that these characterization can be expressed in terms of classes of operator-valued holomorphic functions. For instance, (ii) says that the unitary skew-dilations of hilbert space operators 139 map z 7→ (1 − 2 ρ )I + 2 ρ (I − zT)−1 is in the Caratheodory class of operator- valued holomorphic functions on D, having all real parts positive-definite operators. Item (iii) can be equivalently expressed by the membership of z 7→ zT((ρ− 1)zT −ρI)−1 to the Schur class of holomorphic maps f : D → L(H) having all norms no greater than one (i.e., ‖f(z)‖≤ 1 for every z ∈ D). J.A.R. Holbrook [11] and J.P. Williams [24] introduced the notion of ρ- radius of an operator T ∈L(H) as follows: wρ(T) := inf { u > 0 : 1 u T ∈ Cρ(H) } . This ρ-radius is a quasi-norm on the Banach space L(H), equivalent to the operator norm, whose closed unit ball is exactly Cρ(H). Recall ([13]) that a quasi-norm satisfies all properties of a norm, except that the triangular inequality holds true up to a multiplicative constant. For ρ > 2, the quasi- norm wρ satisfies ([23, 14]) wρ(T1 + T2) ≤ ρ ( wρ(T1) + wρ(T2) ) . Therefore the ρ-contractions are exactly the contractions for the ρ-radius, and many relationships between classes Cρ can be expressed more easily using the associated ρ-radii. The ρ-radius is a usual Banach-space norm for 0 < ρ ≤ 2. Some generalizations of classes Cρ have been studied, like classes CA(H) introduced by H. Langer (see [23, page 53] and its references, and [20]), or the classes C(ρn)(H) considered by several authors (see [17, 4, 18]). This latter generalization will be the main topic of study in this paper, with the novelty that we consider the general case when the ρn’s are non-zero complex scalars. This will lead to classes of operators with several new features and different behavior. 2. Hilbert space operators with (ρn)-dilations Definition and first properties. In light of the preceding discussion we introduce the following definition. Definition 2.1. (Classes C(ρn)) Let (ρn)n≥1 be a sequence of complex numbers, with ρn 6= 0 for each n. We write (ρn)n≥1 ∈ (C∗)N ∗ . Let H be a complex Hilbert space. Define now C(ρn)(H) := { T ∈L(H) : there exists a Hilbert space K and a unitary oper- ator U ∈L(K) with H ⊂ K and Tn = ρnPHUn|H, ∀n ≥ 1 } . 140 v. agniel Here PH ∈L(K) is the orthogonal projection from K onto its closed subspace H. We say in this case that T possesses (ρn)-dilations. In other words, an operator T is in the class C(ρn)(H) if and only if all its powers admit dilations of the form ρnU n for a certain unitary operator U acting on a larger Hilbert space. For the rest of this paper, we will suppose that the Hilbert space H on which T acts is fixed. If there is no ambiguity, C(ρn)(H) will be abbreviated as C(ρn). Note also that the sequence (ρn) = (ρn)n≥1 starts at n = 1: for n = 0 we have of course T 0 = IH = PHU 0|H. In the papers [17, 4, 18], the case when the ρn’s are non-negative real numbers is considered. We went for a broader choice of sequences as the main ideas do not rely heavily on the fact that ρn are in R∗+ and as this eventually allows for some interesting new phenomena for the classes C(ρn). One first difference is recorded in the following remark. Remark 2.2. The definition of C(ρn) easily gives that T ∈ C(ρn) if and only if T∗ ∈ C(ρn). Therefore, when the ρn are real scalars, the class C(ρn) is stable under the adjoint map T 7→ T∗. This is no longer true in the general case. Remark 2.3. As another basic remark, we note that if T is in C(ρn), then we have ‖Tn‖≤ |ρn|. Thus, r(T) ≤ lim infn ( |ρn| 1 n ) . This relationship implies that two different cases appear in the study of the classes C(ρn): (i) 0 < lim infn ( |ρn| 1 n ) ≤ +∞; (ii) lim infn ( |ρn| 1 n ) = 0. Although many of the proofs below work the same way in both cases, most of the results will be stated in the case (i). The study of the case (ii) is more problematic. Indeed, in case (ii), the class C(ρn) will only contain quasinilpotent operators, that is operators whose spectra reduce to {0}. We also note that when lim infn ( |ρn| 1 n ) = +∞, we trivially have r(T) < lim infn ( |ρn| 1 n ) for every operator T. We also note that the condition lim infn ( |ρn| 1 n ) = +∞ leads to small changes in the proofs below: the main difference between this condition and lim infn ( |ρn| 1 n ) < +∞ in case (i) is the fact that the quantity 1 lim infn ( |ρn| 1 n ), which exists when lim infn (|ρn|1n) ∈ ]0, +∞[, has to be replaced by 0 when lim infn ( |ρn| 1 n ) = +∞. This motivates the following convention. unitary skew-dilations of hilbert space operators 141 Convention. For the rest of this paper, we assume that 1 lim infn ( |ρn| 1 n ) = 0 whenever lim inf n ( |ρn| 1 n ) = +∞. (2.1) One of the main tools to characterize the classes C(ρn) is the following Herglotz-type theorem. Theorem 2.4. Let H a Hilbert space. Let F : D → H be an analytic function such that: (i) F(0) = I (ii) Re(F(z)) ≥ 0, ∀z ∈ D. Then, there exists a Hilbert space K containing H and U ∈L(K) an unitary operator such that F(z) = PH(I + zU)(I −zU)−1|H, ∀z ∈ D A proof of this theorem can be found in [8, pages 65 – 69]. Definition 2.5. For (ρn)n ∈ (C∗)N ∗ and for w in a complex Banach al- gebra, f(ρn) denotes the entire series given by f(ρn)(w) = ∑ n≥1 2wn ρn . For a ∈ R, we denote Re≥a the half-plane {z ∈ C, Re(z) ≥ a}, while Re>a is the half-plane {z ∈ C, Re(z) > a}. Proposition 2.6. Let (ρn)n ∈ (C∗)N ∗ and let T ∈ L(H). The following are equivalent: (i) T ∈ C(ρn); (ii) The series f(ρn)(zT) = ∞∑ n=1 2 ρn znTn is absolutely convergent in L(H) and I + Re ( f(ρn)(zT) ) ≥ 0, ∀z ∈ D. Proof. (i) ⇒ (ii) Let U be an unitary operator on a Hilbert space K, with K containing H as a closed subspace, such that Tn = ρnPHU n|H, ∀n ≥ 1. 142 v. agniel For every polynomial P(X) = a0 + · · · + anXn and every z ∈ D, we have a0I+ a1 ρ1 zT +· · ·+ an ρn (zT)n = PH ( a0I+a1zU+· · ·+an(zU)n ) |H = PH P(zU)|H. Since the series 1 + ∑ n≥1 2w n converges absolutely to f(w) = 1+w 1−w for all w ∈ D, and since U is unitary, the series I + ∑ n≥1 2(zU) n converges in norm to f(zU) = (I + zU)(I −zU)−1, ∀z ∈ D. Thus, as ∥∥∥∥Tnρn ∥∥∥∥ = ∥∥PHUn|H∥∥ ≤ ∥∥Un∥∥ ≤ 1, the series IH + ∑ n≥1 2 ρn (zT)n is absolutely convergent and converges to PH[(I + zU)(I − zU)−1]|H for all z ∈ D. As U is unitary, f(zU) is nor- mal, so the closure of its numerical range W(f(zU)) is the convex hull of its spectrum. We have σ ( f(zU) ) = f ( σ(zU) ) ⊂ f(D) ⊂ Re>0. Thus, W ( (I + zU)(I −zU)−1 ) = W ( f(zU) ) ⊂ Hull ( σ(f(zU)) ) ⊂ Re≥0. Furthermore, W(PHf(zU)|H) ⊂ W(f(zU)), so the numerical range of IH + f(ρn)(zT) is included in Re≥0. This is equivalent to Re(IH + f(ρn)(zT)) ≥ 0, so (ii) is true. (ii) ⇒ (i) We define F(z) := IH +f(ρn)(zT). Thus, F is analytic on D, F(0) = IH, and Re(F(z)) ≥ 0 for all z ∈ D. By applying Theorem 2.4, we obtain a Hilbert space K and a unitary operator U ∈L(K), such that F(z) = PH(I + zU)(I − zU)−1|H, for all z ∈ D. By developing both analytic expressions in entire series, and identifying their coefficients, we obtain 2 ρn Tn = 2PHU n|H for all n ≥ 1. Therefore T ∈ C(ρn). We will obtain most of the following results by applying Proposition 2.6. We can directly see by applying this proposition that any class C(ρn) contains 0, the null operator, so none of these classes is empty. One remark is in order. We did not consider the case where ρn = 0 for some n in Definition 2.1. Indeed, this condition does not go well with com- putations similar to the ones in the proof of Proposition 2.6. Having ρn = 0 unitary skew-dilations of hilbert space operators 143 implies Tn = 0, but it does not give any information on PHU n|H. This pre- vents us from showing that certain sums of powers of T and T∗ are positive, which is a crucial tool when dealing with operators in the class C(ρn). If we were to denote m := inf{n : ρn = 0}, then any operator T in C(ρn) would need to be nilpotent of order at most m. The following corollary treats this nilpotent case and gives a characterization that was the one we expected in the case ρm = 0. See also [5, Proposition 6.1] for another use of the positivity condition (ii) below. Corollary 2.7. Let (ρn)n ∈ (C∗)N ∗ and m ≥ 1. Let T ∈ L(H) be such that Tm = 0. Then, the following are equivalent: (i) T ∈ C(ρn); (ii) I + Re (∑m−1 n=1 z n 2 ρn Tn ) ≥ 0 for all z ∈ D. Thus, for any sequence (τn) such that ρk = τk, for all 1 ≤ k < m, we have T ∈ C(τn) if and only if T ∈ C(ρn). Proof. A direct application of Proposition 2.6 with the extra condition Tm = 0 gives the equivalence. Now we come back to Proposition 2.6. When lim infn(|ρn| 1 n ) > 0, we can see that the series ∑∞ n=1 2 ρn znTn is absolutely convergent if and only if |z|r(T) < lim infn(|ρn| 1 n ). We can thus reformulate Proposition 2.6 as follows. Theorem 2.8. Let (ρn)n∈(C∗)N ∗ with lim infn(|ρn| 1 n )>0. Let T ∈L(H). Then, the following assertions are equivalent: (i) T ∈ C(ρn); (ii) r(T) ≤ lim infn ( |ρn| 1 n ) and, for f(ρn)(zT) := ∑∞ n=1 2 ρn znTn, we have I + Re ( f(ρn)(zT) ) ≥ 0, ∀z ∈ D. Remark 2.9. Replacing the condition of absolute convergence of a series by a condition concerning the spectral radius of T is useful in several instances. We can first notice that if we take v > 0 small enough, then vT will satisfy the spectral radius condition. However, if lim infn ( |ρn|1/n ) = 0, this condition must be replaced by lim supn (‖Tn‖ |ρn| 1/n) ≤ 1, which can only be satisfied by 144 v. agniel certain quasinilpotent operators. Hence, aside from nilpotent operators and Corollary 2.7, knowing which operators can be “near” operators belonging to a class C(ρn) is a difficult problem. In this case, the map f(ρn) also has convergence radius 0, so we cannot use analytic or geometric properties related to the images of certain disks by f(ρn). Many of the following results, related to specific operators or to f(ρn) will have no meaning in this case, but others will be true under the additional condition lim sup n ( ‖Tn‖ |ρn| )1 n ≤ 1. We look now at the closure of the class C(ρn) for the operator norm. Corollary 2.10. Let (ρn)n ∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0. Then the class C(ρn) is closed for the operator norm: if (Tm)m a sequence of opera- tors converging in L(H) to T, such that Tm ∈ C(ρn), then T ∈ C(ρn). Proof. Let (Tm)m a sequence of operators converging to T such that Tm ∈ C(ρn). We have r(T) = lim m ( r(Tm) ) ≤ lim inf n ( |ρn| 1 n ) . Thus, for any z ∈ D, the series f(ρn)(zT) converges absolutely and f(ρn)(zT) = limm f(ρn)(zTm). Hence, for any h ∈ H, we have Re (〈( I + f(ρn)(zT) ) h,h 〉) = Re [ lim m 〈( I + f(ρn)(zTm) ) h,h 〉] ≥ 0. This implies that I + Re ( f(ρn)(zT) ) ≥ 0, and the proof is complete by using Theorem 2.8. Operator radii. The condition in Theorem 2.8 will be useful when studying the (ρn)-radius, which is introduced in the following definition. Definition 2.11. Let (ρn)n ∈ (C∗)N ∗ . Let T ∈ L(H). We define the (ρn)-radius of T as: w(ρn)(T) := inf { u > 0 : T u ∈ C(ρn) } ∈ [0, +∞]. unitary skew-dilations of hilbert space operators 145 The definition of the (ρn)-radius is similar to the definition of the ρ-radius that can be found in [11, 1, 3, 2]. As the classes C(ρn) and Cρ share the same type of definition, the (ρn)-radius and the ρ-radius will share the same role with some slight different variations. We will for now focus on properties of the (ρn)-radius. Lemma 2.12. Let (ρn)n ∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0. Then, the map T 7→ w(ρn)(T) takes values in [0, +∞[, is a quasi-norm, is equivalent as a quasi-norm to the operator norm ‖·‖, and its closed unit ball is the class C(ρn). We also have w(ρn)(T) ≥ ( ‖Tm‖ |ρm| ) 1 m and w(ρn)(T) ≥ r(T) lim infn ( |ρn| 1 n ). Proof. We start off by showing that the (ρn)-radius is finite while obtaining its equivalence with the operator norm ‖·‖. Let T ∈L(H). Let u > 0 be such that T u ∈ C(ρn). For any m ≥ 1, we have ‖Tm‖ um ≤ |ρm|, that is u ≥ ( ‖Tm‖ |ρm| ) 1 m . Therefore, by taking the infimum over u such that T u ∈ C(ρn), we get w(ρn)(T) ≥ ( ‖Tm‖ |ρm| ) 1 m . For m = 1 we obtain w(ρn)(T) ≥ (‖T‖ |ρ1| ) . If we also take the lim sup of the right-hand side quantity, we get w(ρn)(T) ≥ r(T) lim infn ( |ρn| 1 n ) . Now, let r < lim infn ( |ρn| 1 n ) . Therefore, the series f(|ρn|)(rz) := ∑∞ n=1 2 |ρn| rnzn is absolutely convergent for all z ∈ D, thus analytic on D. Since f(|ρn|)(0) = 0, there is a radius r0, with 1 > r0 > 0, such that |f(|ρn|)(r0w)| ≤ 1 for all |w| ≤ r. Let u > 0 be such that ‖T‖ u < r0r. Thus, we have r ( T u ) < r0r < lim inf n ( |ρn| 1 n ) , 146 v. agniel and for all z ∈ D we have∥∥∥∥f(ρn) ( z T u )∥∥∥∥≤ ∞∑ n=1 2 |ρn| |z|n ( T u )n ≤ ∞∑ n=1 2 ρn |z|n(r0r)n = ∣∣f(|ρn|)(r0|z|r)∣∣ ≤ 1. We recall that for any B ∈L(H) we have Re(B) ≥−‖Re(B)‖I = − ∥∥∥∥B + B∗2 ∥∥∥∥I ≥−‖B‖I. Thus, for any z ∈ D, f(ρn) ( zT u ) converges absolutely and we have I + Re ( f(ρn) ( z T u )) ≥ I − ∥∥∥∥f(ρn) ( z T u )∥∥∥∥I ≥ 0. This means that T u ∈ C(ρn) according to Proposition 2.6, so w(ρn)(T) ≤ u < +∞. Furthermore, since T u ∈ C(ρn) for every u such that u > ‖T‖ r0r , we get w(ρn)(T) ≤ ‖T‖ r0r . Hence, we have ‖T‖ |ρ1| ≤ w(ρn)(T) ≤ ‖T‖ r0r . With these inequalities we immediately get w(ρn)(T) = 0 ⇔ T = 0. These inequalities also imply that, for S,T ∈L(H), we have w(ρn)(S + T) ≤ ‖S + T‖ r0r ≤ ‖S‖ + ‖T‖ r0r ≤ |ρ1| r0r ( w(ρn)(S) + +w(ρn)(T) ) . In order to show that w(ρn)(·) is a quasi-norm, we still have to show that it is homogeneous, that is w(ρn)(zT) = |z|w(ρn)(T) for any z ∈ C. Let z ∈ C. The cases z = 0 and T = 0 have been treated, so we now consider z = eit|z| 6= 0 and T 6= 0. Let u ≥ w(ρn)(zT) be such that zT u ∈ C(ρn). Denote u ′ = u|z|. We can see that r ( zT u ) = r ( T u′ ) and that f(ρn) ( wzT u ) = f(ρn) ( eitw T u′ ) for any w ∈ D. Thus, the series f(ρn) ( eitw T u′ ) converges absolutely and I+Re ( f(ρn)(e itw T u′ ) ) ≥ 0, for any w ∈ D. Hence T u′ ∈ C(ρn), so u′ = u |z| ≥ w(ρn)(T). unitary skew-dilations of hilbert space operators 147 Thus, by taking the infimum for u ≥ w(ρn)(zT), we get w(ρn)(zT) ≥ |z|w(ρn)(T). Applying the same result to T ′ = zT and z′ = 1 z , we obtain w(ρn)(T) = w(ρn)(z ′T ′) ≥ |z′|w(ρn)(T ′) = 1 |z| w(ρn)(zT), which proves the desired equality. We will now prove that the closed unit ball for the (ρn)-radius is exactly C(ρn). Notice again that w(ρn)(T) = 0 reduces to T = 0. If T ∈ C(ρn), then w(ρn)(T) ≤ 1 1 = 1. Conversely, suppose that w(ρn)(T) ≤ 1 and let (um)m be a sequence, with um > 0, converging to w(ρn)(T) such that T um ∈ C(ρn). Using the fact that the class C(ρn) is closed for the operator norm, as proved in Corollary 2.10, we get T w(ρn)(T) ∈ C(ρn). Therefore, we have r(T) ≤ r ( T w(ρn)(T) ) ≤ lim inf n ( |ρn| 1 n ) and I + Re ( f(ρn)(zT) ) ≥ 0 for every z with |z| ≤ 1 w(ρn)(T) . Since 1 w(ρn)(T) ≥ 1, we can conclude that T ∈ C(ρn). The proof is now com- plete. Remark 2.13. In the case when lim infn ( |ρn| 1 n ) = 0, we have w(ρn)(T) = +∞ unless T is quasinilpotent and the sequence of ‖Tn‖ 1 n decreases to 0 fast enough. Remark 2.14. Since the (ρn)-radius is homogeneous and w(ρn)(T) ≤ 1 ⇔ T ∈ C(ρn), whenever T 6= 0, we have{ u > 0: T u ∈ C(ρn) } = [ w(ρn)(T), +∞ [ . Corollary 2.15. Let (ρn)n ∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0. Let T ∈L(H). We have 148 v. agniel (i) For any z 6= 0, 1|z|w(ρn)(T) = w(ρn)( 1 z T) = w(znρn)(T); (ii) If T ∈ C(ρn)(H), then T k ∈ C(ρkn)(H), for all k ≥ 1; (iii) w(ρkn)n(T k) ≤ w(ρn)(T) k, for all k ≥ 1; (iv) w(ρn)(T) = w(ρn)(T ∗). Proof. (i) The left-hand equality is given by the homogeneity of w(ρn)(·). For the right-hand one, we can see that( T z )n = ρnPHU n|H if and only if Tn = znρnPHUn|H. Thus T z ∈ C(ρn) if and only if T ∈ C(znρn). Lemma 2.12 implies that w(ρn) ( 1 z T ) = w(znρn)(T). (ii) By definition of the class C(ρn), if T ∈ C(ρn), then (Tk)m = ρkmPH(U k)m|H, so Tk ∈ C(ρkn)(H). (iii) The result is true when T = 0. When T 6= 0, consider T ′ = T w(ρn)(T) . By homogeneity of w(ρn)(·), we have w(ρn)(T ′) = 1, so T ′ ∈ C(ρn) according to Lemma 2.12. Thus, for any k ≥ 1, (T ′)k ∈ C(ρkn)(H). Using again the homogeneity of the (ρn)-radius, we obtain w(ρkn)n(T k) w(ρn)(T) k = w(ρkn) ( (T ′)k ) ≤ 1. This completes the proof. (iv) We use Remark 2.2 and Lemma 2.12 to obtain the equivalence w(ρn)(T) ≤ 1 ⇔ w(ρn)(T ∗) ≤ 1. Since the (ρn)-radii are homogenous, these quantities must be equal. Corollary 2.16. Let (ρn)n ∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0. Let T ∈L(H). The following assertions are true: unitary skew-dilations of hilbert space operators 149 (i) Let F be an invariant closed subspace of T. Then w(ρn)(T |F ) ≤ w(ρn)(T); (ii) For any isometry V we have w(ρn)(V TV ∗) ≤ w(ρn)(T), with equality if V is unitary; (iii) For a Hilbert space K we have w(ρn)(T ⊗ IK) = w(ρn)(T); (iv) For Tm ∈L(Hm), m ≥ 1 with supm(‖Tm‖) < +∞, we have w(ρn)(⊕m≥1Tm) = sup m≥1 ( w(ρn)(Tm) ) ; (v) If T (∞) denotes the countable orthogonal sum T ⊕ T ⊕ ··· , then w(ρn) ( T (∞) ) = w(ρn)(T). Proof. (i) We have r(T|F ) ≤ r(T). If I + Re ( f(ρn)(zT) ) is positive, then I + Re ( f(ρn)(zT |F ) ) is positive too. Thus, by using Lemma 2.12 we obtain w(ρn)(T) ≤ 1 ⇒ w(ρn)(T|F ) ≤ 1. The homogeneity of the (ρn)-radius gives the result. (ii) We have r(V TV ∗)≤r(T) and (V TV ∗)n =V TnV ∗. Thus, f(ρn)(zV TV ∗) = V f(ρn)(zT)V ∗. Hence, for any h ∈ H and any z ∈ D, we have Re (〈( I + f(ρn)(zV TV ∗) ) h,h 〉) = Re (〈( I + f(ρn)(zT) ) V ∗h,V ∗h 〉) . By applying Theorem 2.8 and Lemma 2.12, we get w(ρn)(T) ≤ 1 ⇒ w(ρn)(V TV ∗) ≤ 1. The homogeneity of the (ρn)-radii gives the desired inequality. When the isometry V is also invertible, the converse inequality is true, so both quantities are equal. (iii) Since ‖Tn‖ = ‖(T ⊗ IK)n‖, we have r(T) = r(T ⊗ IK). Let u > 0 be such that u ≥ r(T) lim infn ( |ρn| 1 n ). Thus the series f(ρn)(zT⊗IKu ) is absolutely convergent for all z ∈ D, and f(ρn) ( zT⊗IK u ) = f(ρn) ( zT u ) ⊗ IK. Since for any h1 ⊗k1,h2 ⊗k2 ∈ H ⊗K we have 〈h1 ⊗k1,h2 ⊗k2〉 = 〈h1,h2〉〈k1,k2〉, we can see that the condition〈( I + Re ( f(ρn) ( z T ⊗ IK u ))) (h⊗k),h⊗k 〉 ≥ 0, ∀h⊗k ∈ H ⊗K, 150 v. agniel is equivalent to〈( I + Re ( f(ρn) ( z T u ))) (h),h 〉 ≥ 0, ∀h ∈ H. Hence, T⊗IK u ∈ C(ρn)(H ⊗ K) is equivalent to T u ∈ C(ρn)(H), which implies that w(ρn)(T) = w(ρn)(T ⊗ IK). (iv) Since supm(‖Tm‖) < +∞, the linear map T = ⊕m≥1Tm is bounded on the Hilbert space H = ⊕m≥1Hm, and ‖T‖ = supm(‖Tm‖). Thus, r(T) = supm(r(Tm)). Let u > 0 be such that u ≥ r(T) lim infn ( |ρn| 1 n ). We have r ( Tm u ) ≤ r ( T u ) ≤ lim inf n ( |ρn| 1 n ) . Thus, the series f(ρn) ( zT u ) and f(ρn) ( zTm u ) are absolutely convergent for all z ∈ D, and f(ρn) ( z T u ) = ⊕m≥1f(ρn) ( z Tm u ) . Since for any h = (hm)m ∈ H, we have[ I + Re ( f(ρn) ( z T u ))] (h) = (( I + Re ( f(ρn) ( z Tm u ))) (hm) ) m , this implies that〈( I + Re ( f(ρn) ( z T u ))) (h),h 〉 ≥ 0, ∀h ∈ H, is equivalent to〈( I + Re ( f(ρn) ( z Tm u ))) (hm),hm 〉 ≥ 0, ∀hm ∈ Hm, ∀m ≥ 1. Hence, the assertion T u ∈ C(ρn)(H) is equivalent to Tm u ∈ C(ρn)(Hm), ∀n ≥ 1, which implies that w(ρn)(T) = supm ( w(ρn)(Tm) ) . (v) The proof is a consequence of item (iii) and [5, Remark 1.1]. The items (i) and (ii) of this corollary show that the classes C(ρn) are unitarily invariant, and stable under the restriction to an invariant closed subspace. The item (iv) is a generalization of a known property of direct unitary skew-dilations of hilbert space operators 151 sums of operators in the class C(ρ). Items (i), (ii) and (v) show that, under the condition of Corollary 2.16, the radius w(ρn) is an admissible radius in the terminology of [5, Definition 1.1]. Thus, all the results proved in [5] for admissible radii are valid for w(ρn) when lim infn ( |ρn| 1 n ) > 0. In particular, the following result is true. Corollary 2.17. Let T ∈ L(H), with ‖T‖ ≤ 1 and Tn = 0 for some n ≥ 2. Then, for each polynomial p with complex coefficients, we have w(ρn) ( p(T) ) ≤ w(ρn) ( p(S∗n) ) . Here S∗n is the nilpotent Jordan cell S∗n =   0 1 0 · · · 0 0 0 0 1 · · · 0 0 ... ... ... . . . ... ... 0 0 0 · · · 0 1 0 0 0 · · · 0 0   on the standard Euclidean space Cn. Some other consequences of the condition lim infn ( |ρn| 1 n ) > 0 are proved in the next proposition. Proposition 2.18. Let (ρn)n ∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0. The following assertions are true: (i) We have w(ρn)(I) = min ({ r ≥ lim inf n ( |ρn| 1 n )−1 : f(ρn) ( D ( 0, 1 r )) ⊂ Re≥−1 }) ; (ii) For any T ∈L(H), we have w(ρn)(T) ≥ r(T)w(ρn)(I); (iii) If T is normal, then w(ρn)(T) = ‖T‖w(ρn)(I). Proof. (i) Take u = w(ρn)(I) such that I u ∈ C(ρn). We have r( I u ) ≤ lim infn ( |ρn| 1 n ) , so 1 u is no greater than the convergence radius of f(ρn). For any z ∈ D, we have f(ρn) ( z I u ) = f(ρn)( z u )I. Thus, I + Re ( f(ρn) ( z I u )) ≥ 0 for any z ∈ D if and only if f(ρn) ( D ( 0, 1 u )) ∈ Re≥−1. (ii) Let T ∈L(H). There is nothing to prove if T = 0 or r(T) = 0. Otherwise, let u = w(ρn)(T) be such that T u ∈ C(ρn) (cf. Lemma 2.12). Since I + 152 v. agniel Re ( f(ρn) ( zT u )) ≥ 0, the spectrum of I +f(ρn) ( zT u ) lies in Re≥0. This spectrum is the set { 1 + f(ρn)(zw), w ∈ σ ( T u )} . The union of these spectra, when z describes D, is { 1 + f(ρn)(w), |w| < r(T) u } . Since r(T) u > 0, we obtain from item (i) that u r(T) ≥ w(ρn)(I). Hence w(ρn)(T) ≥ r(T)w(ρn)(I). (iii) Let T be a normal operator with T 6= 0. For u = ‖T‖.w(ρn)(I), we have r ( T u ) = ‖T‖ u = 1 w(ρn)(I) ≤ lim inf n ( |ρn| 1 n ) . Thus, we obtain that ⋃ z∈D σ ( I + f(ρn) ( z T u )) = { 1 + f(ρn)(w), |w| < 1 w(ρn)(I) } . Item (i) of this proposition tells us that this set is included in Re≥0. As T is normal, I + f(ρn) ( zT u ) is also normal, so W ( I + f(ρn) ( z T u )) ⊂ Hull ( σ ( I + f(ρn) ( z T u ))) ⊂ Re≥0, ∀z ∈ D. Hence, I + Re ( f(ρn) ( zT u )) ≥ 0, and T u ∈ C(ρn). By Lemma 2.12, we then have w(ρn)(T) ≤ u = ‖T‖w(ρn)(I). The inequality of item (ii) provides the desired equality. Remark 2.19. Since we also have w(ρn)(I) ≥ 1 lim infn ( |ρn| 1 n ), the inequality in Proposition 2.18 is better than the last one of Lemma 2.12. Thus, if there is T such that w(ρn)(T) = r(T) lim infn ( |ρn| 1 n ), the same must be true for the identity operator I. In the case when ρn = ρ, ρ > 0, this can only happen when ρ ≥ 1. unitary skew-dilations of hilbert space operators 153 3. Classes C(ρ) for ρ 6= 0 In this section, we will focus on the case where ρn = ρ, for some ρ ∈ C∗. This is an intermediate class between the classical case considered by Sz.-Nagy and Foias (classes Cτ for τ > 0) and the the general C(ρn)-classes. Thus the obtained results are already known when ρ > 0, but the generalization to the case ρ ∈ C∗ seems to be new. Nevertheless, we acknowledge the influence of [23, 1, 2, 14] for the results of this section. The results obtained here will turn out to be useful when we will look again at C(ρn)-classes in the next section. Some characterizations. Lemma 3.1. Let ρ 6= 0 and ρn = ρ, ∀n ≥ 1. Let T ∈L(H). The following are equivalent: (i) T ∈ C(ρ)(H); (ii) r(T) ≤ 1 and Re (( 1 − 2 ρ ) I + 2 ρ (I −zT)−1 ) ≥ 0, ∀z ∈ D; (iii) r(T) ≤ 1 and Re ( 2 ρ (I−zT) ) + Re ( 1− 2 ρ ) (I−zT)∗(I−zT) ≥ 0, ∀z ∈ D; (iv) Re ( 2 ρ (I −zT) ) + Re ( 1 − 2 ρ ) (I −zT)∗(I −zT) ≥ 0, ∀z ∈ D. Proof. (i) ⇔ (ii) We have lim infn ( |ρn| 1 n ) = 1. When r(T) ≤ 1, for z ∈ D, we have I + ∑ n≥1 2 ρ (zT)n = ( 1 − 2 ρ ) I + 2 ρ (I −zT)−1. Apply now Proposition 2.6. (ii) ⇔ (iii) We will use several times the known fact that for A,B ∈ L(H), with A invertible, Re(B) ≥ 0 ⇔ Re(A∗BA) ≥ 0. We obtain the equivalence (ii) ⇔ (iii) by choosing A = (I −zT), B = ( 1 − 2 ρ ) I + 2 ρ (I −zT)−1 and by rearranging the expression, using that (I −zT)∗(I −zT) is a positive self-adjoint operator and Re(A∗) = Re(A). (iii) ⇒ (iv) It is immediate. 154 v. agniel (iv) ⇒ (iii) Suppose that r(T) > 1. Thus, there exists γ ∈ C such that |γ| = r(T) > 1, and there is a sequence (hn) of vectors hn ∈ H such that ‖hn‖ = 1 and ‖(T − γI)hn‖ → 0 as n → ∞. Let 0 < � < |γ| − 1 and set gn := (T −γI)hn. Let also η = �eit, for some t that will be chosen later on. Let z := 1+η γ . Then, |z| < 1+(|γ|−1)|γ| = 1. Furthermore, we have (I −zT)hn = ( I − 1 γ T ) hn − η γ Thn + ηhn −ηhn = −zgn −ηhn. Thus, we obtain Re (〈[ 2 ρ (I −zT) + ( 1 − 2 ρ ) (I −zT)∗(I −zT) ] hn, hn 〉) ≥ 0 ⇒ Re ( 2 ρ [ −η‖hn‖2 −〈zgn,hn〉 ] + ( 1 − 2 ρ ) ‖(I −zT)hn‖2 ) ≥ 0 ⇒ Re ( 2 ρ [−η −〈zgn,hn〉] + ( 1 − 2 ρ )[ |η|2 + 2Re(〈zgn,hn〉) + |z|2‖gn‖2 ]) ≥ 0. Hence, by taking the limit as n → +∞, we obtain Re ( 2 ρ (−η) + ( 1 − 2 ρ ) |η|2 ) = Re ( −2 ρ eit ) � + Re ( 1 − 2 ρ ) �2 ≥ 0. We can then choose t ∈ R depending on arg(ρ) and sgn ( Re ( 1− 2 ρ )) to obtain either −2 |ρ| � + ∣∣∣∣Re ( 1 − 2 ρ )∣∣∣∣�2 ≥ 0 or 2|ρ|�− ∣∣∣∣Re ( 1 − 2 ρ )∣∣∣∣�2 ≤ 0. But since −2|ρ| < 0, there is some � > 0 such that −2 |ρ| + ∣∣Re(1 − 2 ρ )∣∣� is strictly negative, which is impossible. This contradiction shows that r(T) ≤ 1, which concludes the proof. Lemma 3.2. Let ρ 6= 0 and α > 0 be two scalars. Let T ∈ L(H). The following assertions are equivalent: (i) w(ρ)(T) ≤ α; (ii) r(T) ≤ α, ( (ρ − 1)zT − ραI ) is invertible and ∥∥(zT)((ρ − 1)zT − ραI )−1∥∥ ≤ 1, ∀z ∈ D; (iii) r(T) ≤ α, ( (ρ−1)T−ρwI ) is invertible and ∥∥T((ρ−1)T−ρwI)−1∥∥ ≤ 1, ∀|w| > α. unitary skew-dilations of hilbert space operators 155 Proof. (i) ⇒ (ii) When replacing T with T α , all expressions in (i) and (ii) are reduced to the case α = 1. Now, as w(ρn)(T) ≤ α = 1, we use Lemma 3.1 to have r(T) ≤ 1 and Re (( 1 − 2 ρ ) I + 2 ρ (I −zT)−1 ) ≥ 0, ∀z ∈ D. We denote Cz := ( 1 − 2 ρ ) I + 2 ρ (I − zT)−1, for z ∈ D. We recall that since Re(Cz) ≥ 0, we have (Cz + I) invertible and∥∥(Cz − I)(Cz + I)−1∥∥ ≤ 1. A computation gives Cz − I = 2 ρ zT(I −zT)−1 and Cz + I = [ 2I + ( 2 ρ − 2 ) zT ] (I −zT)−1. Thus, (Cz − I)(Cz + I)−1 = 1 ρ zT [ I + ( 1 ρ − 1 ) zT ]−1 = zT [ ρI + (1 −ρ)zT ]−1 = −zT [ −ρI + (ρ− 1)zT ]−1 . This means that all the conditions of (ii) are fulfilled. (ii) ⇒ (i) We again reduce to the case α = 1. We denote Dz = zT [ ρI − (ρ− 1)zT ]−1 , for z ∈ D. Since ‖Dz‖≤ 1, we have Dz ∈ C(1), so r(Dz) ≤ 1 and Re ( (I + wDz)(I −wDz)−1 ) ≥ 0, for all w ∈ D. We obtain: I + wDz = [ ρI + (w + 1 −ρ)zT ][ ρI − (ρ− 1)zT ]−1 and I −wDz = [ ρI + (−w + 1 −ρ)zT ][ ρI − (ρ− 1)zT ]−1 . Thus, (I + wDz)(I −wDz)−1 = [ ρI + (w + 1 −ρ)zT ][ ρI + (−w + 1 −ρ)zT ]−1 . 156 v. agniel Since r(T) ≤ 1, (I − zT) is invertible so [ρI + (−w + 1 −ρ)zT]−1 converges to 1 ρ (I −zT)−1 when w tends to 1, by continuity of the inverse map. Thus, lim w→1, w∈D (I + wDz)(I −wDz)−1 = 1 ρ ( ρI + (2 −ρ)zT ) (I −zT)−1 = Cz. Hence, Re(Cz) ≥ 0 for all z ∈ D and r(T) ≤ 1, so T ∈ C(ρ). (ii) ⇔ (iii) For z 6= 0, we take w = α z to obtain the result. The converse gives the result for all z ∈ D, z 6= 0, which extends to D by continuity. Reducing to the case ρ > 0. With this characterization of C(ρ) classes, we are now able to obtain the main relationship between (ρ)-radii and (τ)-radii, ρ ∈ C∗, τ > 0. This relationship extends the “symmetric” relationship τw(τ)(T) = (2 − τ)w(τ)(T), 0 < τ < 2, that was already known (see [2, Theorem 3]). Proposition 3.3. Let ρ 6= 0 and α > 0 be two scalars. Let T ∈ L(H). The following assertions are equivalent: (i) w(ρ)(T) ≤ α; (ii) ( (ρ − 1)zT − ραI ) is invertible and ∥∥(zT)((ρ − 1)zT − ραI)−1∥∥ ≤ 1, ∀z ∈ D; (iii) ( (ρ−1)T−ρwI ) is invertible and ∥∥T((ρ−1)T−ρwI)−1∥∥ ≤ 1, ∀|w| > α. Furthermore, we have: |ρ|w(ρ)(T) = ( 1 + |ρ− 1| ) w1+|ρ−1|(T). (3.1) Hence, the map ρ ∈ C∗ 7→ |ρ|w(ρ)(T) is constant on circles of center 1, is continuous on C∗ and can be extended continuously to 2w(2)(T) at 0. Proof. Using the results of Lemma 3.2, we can see that items (ii) and (iii) are equivalent and that item (i) implies item (ii). We only need to show that item (ii) implies r(T) ≤ α. We can reduce the proof to the case α = 1 by considering T α instead of T. We also recall that if ρ > 0, the result is valid (see [19, Thm. 1] or [7] for a proof). Let ρ 6= 0. We denote S = 1+|ρ−1||ρ| T. Suppose unitary skew-dilations of hilbert space operators 157 that [(ρ − 1)zT − ρI]−1 exists and that ‖zT[(ρ − 1)zT − ρI]−1‖ ≤ 1, for all z ∈ D. With ρ− 1 = |ρ− 1|eit, ρ = |ρ|eis and w = z.e−is+it we then have∥∥∥zT[(ρ− 1)zT −ρI]−1∥∥∥ ≤ 1 ⇔ ∥∥∥zT[|ρ− 1|eitzT −|ρ|eisI]−1∥∥∥ ≤ 1 ⇔ ∥∥∥ze−iseite−itT[|ρ− 1|ze−iseitT −|ρ|I]−1∥∥∥ ≤ 1. ⇔ ∣∣e−it∣∣∥∥∥wT[|ρ− 1|wT −|ρ|I]−1∥∥∥ ≤ 1 ⇔ ∥∥∥wT[(1 + |ρ− 1|− 1)wT −|ρ|I]−1∥∥∥ ≤ 1 ⇔ ∥∥∥∥∥w1 + |ρ− 1||ρ| T [( 1 + |ρ− 1|− 1 ) w 1 + |ρ− 1| |ρ| T − ( 1 + |ρ− 1| ) I ]−1∥∥∥∥∥ ≤ 1 ⇔ ∥∥∥wS[(1 + |ρ− 1|− 1)wS −(1 + |ρ− 1|)I]−1∥∥∥ ≤ 1. Since w describes D when z does, this is true for all w ∈ D. Therefore w(1+|ρ−1|)(S) ≤ 1 as 1 + |ρ − 1| > 0 (see the beginning of the proof and Lemma 3.2). Thus, r(S) ≤ 1, which implies r(T) ≤ |ρ| 1+|ρ−1| ≤ 1. Now that we have showed that the condition about the spectral radius of T is not necessary, we can see that the equivalences in the previous computations give w(ρ)(T) ≤ 1 ⇔ w(1+|ρ−1|) ( 1 + |ρ− 1| |ρ| T ) ≤ 1. By homogeneity of the (ρn)-radii, this is equivalent to |ρ|w(ρ)(T) = ( 1 + |ρ− 1| ) w(1+|ρ−1|)(T). The properties of the map ρ ∈ C∗ 7→ |ρ|w(ρ)(T) can now be obtained from its restriction to [1, +∞[, which is known to be continuous (see [2, Corollary 2] for example). Equation (3.1) gives a simple geometric understanding of a formula that was previously known only for real numbers ρ between 0 and 2. It also implies the following relationship between Cρ classes. Corollary 3.4. We have C(ρ) = 1 + |ρ− 1| |ρ| C(1+|ρ−1|). 158 v. agniel We conclude that complex (ρ)-radii of an operator T can be expressed in terms of the real positive ones. Corollary 3.5. Let ρ 6= 0 and let T ∈L(H). We have: (i) w(ρ)(I) = 1+|ρ−1| |ρ| , ∀ρ 6= 0; (ii) If T is normal, then w(ρ)(T) = ‖T‖ 1+|ρ−1| |ρ| ; (iii) If T 2 = 0, then w(ρn)(T) = w(ρ1)(T) = 2w(T) |ρ1| = ‖T‖ |ρ1| ; (iv) It T 2 = bI, b ∈ C, then |ρ|w(ρ)(T) = w(T)+ √ w2(T)2 + |b|(|ρ− 1|2 − 1); (v) It T 2 = aT, a ∈ C, then |ρ|wρ(T) = 2w(T) + |a||ρ− 1|. Proof. (i) It is known that w(ρ)(I) = 1 when 1 ≤ ρ. The relationship of Proposition 3.3 gives the result. (ii) When T is normal, we have w(ρn)(T) = ‖T‖w(ρn)(I). (iii) If T 2 = 0, then T ∈ C(ρn) if and only if I + Re ( 2 ρ1 zT ) ≥ 0 for all z ∈ D. By Corollary 2.7, this is equivalent to T|ρ1| ∈ C(1), to 2T |ρ1| ∈ C(2) and to T ∈ C(ρ1). Thus, Lemma 2.12 and the following facts w(2)(T) = w(T) and w(1)(T) = ‖T‖ imply that w(ρn)(T) = w(ρ1)(T) = 2w(T) |ρ1| = ‖T‖ |ρ1| . (iv), (v) We can reduce these cases to T 2 = I (respectively T 2 = T) by taking δ to be a square root of b (respectively a) and looking at T δ (respectively T δ2 ). Then, [2, Theorem 6] gives the result when ρ > 0, and we extend it to ρ ∈ C∗ by using Proposition 3.3. Computations and some applications. For the next auxiliary result we need some notation. For an operator T acting on H and for h ∈ H, define Vh := Span ( Tn(h), n ≥ 0 ) and Th := T|Vh ∈L(Vh). Lemma 3.6. Let T ∈L(H). Let ρ 6= 0. Then, with the previous notation, we have w(ρ)(T) = sup h∈H ( w(ρ)(Th) ) . unitary skew-dilations of hilbert space operators 159 If we also have P(T) = 0 for some P ∈ C[X] with deg(P) = n, then Th can be identified as some matrix S ∈ Mn(C) such that P(S) = 0, and the computation of w(ρ)(Th) can be obtained from the computation of w(ρ)(S). Proof. Let h ∈ H. We already proved in Corollary 2.16 that w(ρ)(Th) ≤ w(ρ)(T). Conversely, for 1 u = suph∈H ( w(ρ)(Th) ) , ( I − zT u ) is invertible as( I−zTh u ) is invertible for all h ∈ H and we have Re (〈( I +f(ρ) ( zT u )) g,g 〉) ≥ 0 for all g ∈ H. Thus T u ∈ C(ρ), which implies suph∈H ( w(ρ)(Th) ) ≥ w(ρ)(T) and concludes the proof. Remark 3.7. Here is an attempt to compute w(ρ)(T), ρ > 1, when T satisfies the quadratic equation T 2 + aT + bI = 0. We use some ideas from [2], which allows one to obtain an expression of w(ρ)(T) depending on w(2)(T) when a = 0 or b = 0. Up to considering reitT, we can assume that |b| = 1 and Re(āb) = 0. With α,β the roots of X2 + aX + b and η ∈ C, we want to compute w(ρ)(M), for M =  α η 0 β  . Using Lemma 3.2 (ii) and Proposition 3.3, we obtain that w(ρ)(M) is the largest (in modulus) z that is solution of 2(ρ− 1)2 + ρ2|z|2 ( |a|2 + |a2 − 4b| 2 ) + |η|2ρ2|z|2 = ∣∣∣∣(ρ− 1)2 ia|a| + a(ρ− 1)ρz + ρ2z2 ∣∣∣∣2 + 1. In the case ρ = 2, the equation simplifies into 2 + 2|z|2 ( |a|2 + |a2 − 4b| ) + 4|η|2|z|2 = ∣∣∣∣4z2 + 2az + ia|a| ∣∣∣∣2 + 1. However, unlike the case where a = 0 or b = 0 in [2], we couldn’t find a way to have an algebraic expression of w(ρ)(T) in terms of w(2)(T). Using Proposition 3.3, we can also generalize some results of [1] about characterizing unitary operators through their ρ-radii. Proposition 3.8. Let T ∈L(H) be invertible. Then (i) T is unitary if and only if σ(T) ⊂ ∂D and there exists ρ ∈ C∗ such that w(ρ)(T) ≤ w(ρ)(I). 160 v. agniel (ii) T = ‖T‖U for U unitary if and only if there exists ρ ∈ C∗ and m > 0 such that w(ρ)(T −m) w(ρ)(I) = ( w(ρ)(T) w(ρ)(I) )−m . Proof. (i) The formula of Proposition 3.3 can be rewritten as w(ρ)(S) = w(ρ)(I)w(1+|ρ−1|)(S). It allows us to obtain the same relationship between T and I for w(1+|ρ−1|), and we can then apply [1, Theorem 2.1] to get the result. (ii) The formula of Proposition 3.3 allows us to obtain the same relationship for w(1+|ρ−1|), which simplifies into: w1+|ρ−1|(T −m) = w1+|ρ−1|(T) −m. We can now apply [1, Theorem 1.1], and the proof is complete. Proposition 3.9. Let ρ 6= 0 be a complex number. Then (i) The ρ-radius w(ρ)(·) is a norm on L(H) if and only if |ρ− 1| ≤ 1; (ii) If |ρ− 1| > 1, then, for all operators T1 and T2 in L(H), we have wρ(T1 + T2) ≤ ( 1 + |ρ− 1| )( wρ(T1) + wρ(T2) ) . Proof. For two operators T1,T2, we have w(ρ)(T1 + T2) ≤ C ( w(ρ)(T1) + w(ρ)(T2) ) if and only if the same is true for w(1+|ρ−1|). It is known [23, 14] that for τ > 0, w(τ) is a norm if and only if 0 < τ ≤ 2. We conclude that w(ρ)(·) is a norm if and only if ρ lies in the closed circle of center 1 and radius 1. Moreover, when τ > 2, w(τ) is a quasi-norm with multiplicative constant (also called the modulus of concavity of the quasi-norm [13]) lower or equal to τ. We thus obtain (ii). For the next proposition we recall that for r > 0 the disc algebra over the disc D(0,r), A(D(0,r)), is the set of holomorphic functions on D(0,r) that are continuous on D(0,r). Proposition 3.10. Let ρ 6= 0 be a complex number. Let T ∈ C(ρ). Then the functional calculus map f 7→ f(T) that sends a polynomial f into f(T) unitary skew-dilations of hilbert space operators 161 can be extended continuously to the disk algebra A ( D ( 0, 1 w(ρ)(I) )) . It is a morphism of Banach algebras, and satisfies ‖f(T)‖≤ ( 1 + |ρ− 1| ) ‖f‖ L∞ ( D ( 0, 1 w(ρ)(I) )). Furthermore, for f ∈ A ( D ( 0, 1 w(ρ)(I) )) such that f(0) = 0, we have w(ρ) ( f(T) ) ≤ w(ρ)(I)‖f‖L∞ ( D ( 0, 1 w(ρ)(I) )). If f ∈ A(D) with f(0) = 0, we also have w(ρ) ( f(T) ) ≤‖f‖L∞(D). The constants in these three inequalities are optimal. Proof. We notice first that T ∈ C(ρ) is equivalent to w(ρ)(T) ≤ 1, which is equivalent to w(1+|ρ−1|)(T) ≤ |ρ| 1 + |ρ− 1| = 1 w(ρ)(I) ≤ 1. Hence, w(ρ)(I)T lies in C(1+|ρ−1|), so there exists a Hilbert space K and an unitary operator U over K such that( w(ρ)(I)T )n = ( 1 + |ρ− 1| ) PHU n|H, ∀n ≥ 1. Therefore, if we denote V := U w(ρ)(I) , for any polynomial P we get P(T) = PH [( 1 + |ρ− 1| ) P(V ) −|ρ− 1|P(0)I ] |H. Since V is a normal operator with spectral radius 1 w(ρ)(I) , we then have ‖P(T)‖≤ ∥∥(1 + |ρ− 1|)P(V ) −|ρ− 1|P(0)I∥∥ ≤ ∥∥(1 + |ρ− 1|)P −|ρ− 1|P(0)∥∥ L∞ ( D ( 0, 1 w(ρ)(I) )). As the polynomials are dense in the algebra A ( D ( 0, 1 w(ρ)(I) )) , the morphism of algebras P 7→ P(T) extends continuously on A ( D ( 0, 1 w(ρ)(I) )) . 162 v. agniel Let us estimate the norm of this map. For f in the algebra we denote g(z) := f ( z w(ρ)(I) ) . Hence, g ∈ A(D), and we have f(T) = g ( w(ρ)(I)T ) . Applying a reformulation of Theorem 2 in [15] by Ando and Okubo, we obtain ‖f(T)‖ = ∥∥g(w(ρ)(I)T)∥∥ ≤ max (1, 1 + |ρ− 1|)‖g‖L∞(D) = ( 1 + |ρ− 1| ) ‖f‖ L∞ ( D ( 0, 1 w(ρ)(I) )). We will now prove the two remaining inequalities. The fact that V is nor- mal implies that f 7→ f(V ) is well defined and bounded on A ( D ( 0, 1 w(ρ)(I) )) . Therefore f(T) = PH [( 1 + |ρ− 1| ) f(V ) −|ρ− 1|f(0)I ] |H, ∀f ∈ A ( D ( 0, 1 w(ρ)(I) )) . We now suppose that f satisfies f(0) = 0. If f ≡ 0, then f(T) = 0 and the statements are true. Otherwise, up to dividing f by its norm, we may assume that ‖f‖ L∞ ( D ( 0, 1 w(ρ)(I) )) = 1. For a fixed n ≥ 1, we get f(T)n = fn(T) = ( 1 + |ρ− 1| ) PHf n(V )|H = ( 1 + |ρ− 1| ) PHf(V ) n|H. As we have ‖f(V )‖≤ ‖f‖ L∞ ( D ( 0, 1 w(ρ)(I) )) = 1, the operator f(V ) lies in C(1) which in turns implies that f(V ) can be dilated on a larger Hilbert space as follows f(V )m = PKW m|K, ∀m ≥ 1, with W a suitable unitary operator. Combining the two dilations, we obtain f(T)n = ( 1 + |ρ− 1| ) PHW n|H, ∀n ≥ 1. Therefore f(T) lies in C(1+|ρ−1|), which is equivalent to w(1+|ρ−1|)(f(T)) ≤ 1. This inequality is in turn equivalent to w(ρ)(f(T)) ≤ w(ρ)(I), which proves the second inequality of this Proposition. Lastly, if f ∈ A(D) with f(0) = 0, we can use Schwarz’s lemma to obtain ‖f‖ L∞ ( D ( 0, 1 w(ρ)(I) )) ≤ 1 w(ρ)(I) ‖f‖L∞(D), which in turn gives w(ρ)(f(T)) ≤‖f‖L∞(D). unitary skew-dilations of hilbert space operators 163 For the optimality of these inequalities, let us take T such that T 2 = 0 and ‖T‖ = |ρ|, and f(z) = z. We then have w(ρ)(T) = ‖T‖ |ρ| = 1 = w(ρ)(I)‖f‖L∞ ( D ( 0, 1 w(ρ)(I) )) = ‖f‖L∞(D) and ‖f(T)‖ = |ρ| = ( 1 + |ρ− 1| ) ‖f‖ L∞ ( D ( 0, 1 w(ρ)(I) )). The proof is complete. When ρ does not lie in [1, +∞[, the algebra where the functional calculus is defined strictly contains the disc algebra A(D). For 0 < ρ < 1, the norm of this map is then 2 − ρ. This result differs from [15, Theorem 2] as Ando and Okubo looked in [15] at the map f 7→ f(T) on A(D) and not on a larger algebra. 4. Inequalities and parametrizations for (ρn)-radii Operator radii of products and tensor products. A useful tool, used to study the behavior of a product or sum of double-commuting opera- tors, is the following result, proved in [11, Theorem.4.2]. Proposition 4.1. Let Tn,Sn ∈ L(H), n ∈ Z, be such that for all 0 ≤ r < 1, t ∈ R, the series ∑∞ n=−∞r |n|eintTn and ∑∞ n=−∞r |n|eintSn converge absolutely and have self-adjoint non-negative sums. If, moreover, we have Tn ·Sm = Sm ·Tn, ∀m,n ∈ Z, then the series ∑∞ n=−∞r |n|eintTn ·Sn converges absolutely and has a self-adjoint non-negative sum, for all 0 ≤ r < 1, t ∈ R. Using Proposition 4.1 we can easily obtain the following auxiliary result. Lemma 4.2. Let T,S ∈L(H) be two operators that are double-commuting (i.e., TS =ST, TS∗=S∗T). Let (ρn)n, (τn)n∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0 and lim infn ( |τn| 1 n ) > 0. Then, we have w(ρnτn)(ST) ≤ w(ρn)(S)w(τn)(T). Proof. If S = 0 or T = 0, then ST = 0 and both sides of the inequality are equal to zero. If S 6= 0 and T 6= 0, then, up to dividing S and T by their 164 v. agniel respective radius, we can consider that w(ρn)(S) = w(τn)(T) = 1. Thus, we need to prove that w(ρnτn)(S.T) ≤ 1. We define Tm :=   1 ρm Tm if m ≥ 1, I if m = 0, 1 ρ|m| (T∗)|m| if m ≤−1, Sm :=   1 τm Sm if m ≥ 1, I if m = 0, 1 τ|m| (S∗)|m| if m ≤−1. The condition w(ρn)(S) = w(τn)(T) = 1, together with Lemma 2.12 and Proposition 2.6, ensure us that the conditions of Proposition 4.1 are fulfilled, since I + Re ( f(ρn)(re itS) ) = ∑ m∈Z r |m|eimtSm, for all 0 ≤ r < 1, t ∈ R. Thus, ∑ m∈Z r |m|eimtSmTm converges absolutely, is self-adjoint, and has a positive sum, for all 0 ≤ r < 1, t ∈ R. This implies that the series ∑ n≥1 2 ρnτn ( reitST )n = f(ρnτn) ( reitST ) is absolutely convergent and that I + Re ( f(ρnτn)(re itST) ) ≥ 0 for all 0 ≤ r < 1, t ∈ R. Thus ST ∈ C(ρnτn) and w(ρnτn)(ST) ≤ 1, which concludes the proof. Corollary 4.3. Let T,S ∈ L(H) and let (ρn)n, (τn)n ∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0 and lim infn ( |τn| 1 n ) > 0. (i) If T and S double-commute, then w(ρn)(ST) ≤ w(1)(S)w(ρn)(T) ≤ |τ1|w(τn)(S)w(ρn)(T). This inequality is optimal when dim(H) ≥ 4. (ii) We have w(1)(ST) ≤ w(1)(S)w(1)(T) ≤ |τ1||ρ1|w(τn)(S)w(ρn)(T). This inequality is optimal when dim(H) ≥ 2. (iii) For R ∈L(H′), we have w(ρnτn)(T ⊗R) ≤ w(ρn)(T)w(τn)(R). unitary skew-dilations of hilbert space operators 165 Proof. (i) We use Lemma 4.2 for S,T and (1)n, (ρn)n to get the left- hand side inequality. The right-hand side inequality comes from the fact w(τn)(S) ≥ ‖S‖ |τ1| (cf. Lemma 2.12). By taking S =   0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0   , T =   0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0   such that ST =   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0   , some computation show that S and T double-commute, and that ‖S‖ = ‖T‖ = ‖ST‖, S2 = T 2 = (ST)2 = 0. Corollary 3.5(iii) shows that all three quantities are equal to ‖ST‖ |ρ1| . (ii) The inequality on the right-hand side follows again from Lemma 2.12. By taking S = ( 0 1 0 0 ) , T = ( 0 0 1 0 ) such that ST = ( 1 0 0 0 ) , we have ‖S‖ = ‖T‖ = ‖ST‖ = 1, S2 = T 2 = 0, and ST is self-adjoint. Thus, w(τn)(S) = w(ρn)(T) = 1 ρ and w(1)(ST) = 1, so all quantities are equal to 1. (iii) As IH,T double-commute and IH′,R double-commute too, we can apply Lemma 4.2 to (T ⊗ IH′)(IH ⊗ R) = T ⊗ R. We then apply Corollary 2.16(iii). Although these inequalities are optimal for some operators, they tend to lose a good part of the information in the general case. For example, we have w(3)(I) = 1 ≤ w(−1)(I)w(−3)(I) = 5. Such a loss of information on the radius of the identity operator I also impacts almost every estimate of radii for other operators in L(H). Corollary 4.4. Let (ρn)n ∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0. Then, ‖T‖ |ρ1| ≤ w(ρn)(T) ≤‖T‖w(ρn)(I). Furthermore, the coefficients in this equivalence of quasi-norms are optimal. 166 v. agniel Proof. The left-hand side inequality ‖T‖ |ρ1| ≤ w(ρn)(T) has been obtained in Lemma 2.12. The equality case is obtained for T such that T 2 = 0, as seen in Corollary 3.5. The right-hand side inequality comes from Lemma 4.2, with S = I and τn = 1. It is an improvement of the one that was obtained in Lemma 2.12. The equality case is obtained for any T normal of norm 1. Operator radii as 1-parameter families. To better understand the behavior of the associated radii associated with classes of operators, it is useful to look at (ρn)-radii as 1-parameter families. This is obtained by studying the map z 7→ w(zρn). We will present results for the real parameter case (r ∈]0, +∞[) and for the complex one (z ∈ C∗). The two main ingredients we are using are the double-commuting inequal- ity of Lemma 4.2 for T,I and (ρn)n,(1)1, and the fact that f(zρn) = 1 z f(ρn). Proposition 4.5. Let T ∈ L(H) and consider (ρn)n ∈ (C∗)N ∗ with lim infn ( |ρn| 1 n ) > 0. (i) For all z 6= 0, we have: |z| 1 + |z − 1| w(zρn)(T) ≤ w(ρn)(T) ≤ w(zρn)(T) ( |z| + |z − 1| ) . (ii) The map z 7→ w(zρn)(T) is continuous on C ∗, and r 7→ w(reitρn)(T) is continuous and decreasing on ]0, +∞[, for all t ∈] −π,π]. (iii) We have 1 3 w(zρn)(T) ≤ w(|z|ρn)(T) ≤ 3w(zρn)(T), and these inequalities are optimal. Proof. (i) We use Lemma 4.2 to obtain w(zρn)(T) ≤ w(z)(I)w(ρn)(T) and w(ρn)(T) ≤ w(z−1)(I)w(zρn)(T). As w(z)(I) = 1+|z−1| |z| and w(z−1)(I) = |z| + |z − 1|, we obtain the desired inequalities. (ii) Up to changing (ρn)n by (wρn)n, the continuity must only be shown at the point w = 1, that is when z → 1. As we have w(ρn)(T) ≤ w(zρn)(T) ( |z| + |z − 1| ) ≤ w(ρn)(T) ( |z| + |z − 1| )1 + |z − 1| |z| unitary skew-dilations of hilbert space operators 167 and as (|z| + |z − 1|), 1+|z−1||z| both tend to 1 from above as z → 1, we obtain lim z→1 w(zρn)(T) = w(ρn)(T). For any t ∈ R and 0 < r < R, we have w(Reitρn)(T) ≤ w(Rr−1)(I)w(reitρn)(T) = w(reitρn)(T). Thus, r 7→ w(reitρn)(T) is decreasing on ]0, +∞[. (iii) We use the fact that w(eit)(I) = 1 + |eit − 1| has a maximum of 3 when eit = −1. The equality case for the inequality on the left-hand side is attained at T = I, ρn = 1 and z = −1, whereas the equality case for the one on the right-hand side is attained at T = I, ρn = −1, z = −1. Since r 7→ w(rρn)(T) is decreasing, the classes C(rρn) are increasing (for the usual order of inclusion of sets), for r ∈]0, +∞[. By using nilpotent operators of order 2, and item (iii) of Corollary 3.5, we can also immediately show that these inclusions are always strict. For the following propositions, we recall that 1 lim infn ( |ρn| 1 n ) = 0 if lim infn ( |ρn| 1 n ) = +∞. Proposition 4.6. Let (ρn)n ∈ (C∗)N ∗ and T ∈ L(H) be such that lim infn ( |ρn| 1 n ) > r(T) ≥ 0. Then, there is r > 0 such that for all z with |z| = r, r(T) lim infn ( |ρn| 1 n ) ≤ w(zρn)(T) ≤ 1. Proof. Let s > 1 be such that r(sT) < lim infn ( |ρn| 1 n ) . As lim sup n→∞ ( 2sn‖Tn‖ |ρn| )1 n = r(sT) lim infn ( |ρn| 1 n ) < 1, there is B > 0 such that 2sn‖Tn‖ |ρn| ≤ B. Thus, for all w ∈ D, we have ∥∥f(zρn)(wT)∥∥ ≤ ∑ n≥1 2‖Tn‖ |z||ρn| ≤ ∑ n≥1 B |z|sn = 1 |z| sB 1 −s < +∞. By taking |z| large enough, we have ‖f(zρn)(wT)‖ < 1, which implies that I + Re ( f(zρn)(wT) ) ≥ 0, ∀w ∈ D. 168 v. agniel Thus w(zρn)(T) ≤ 1. The left-hand side inequality comes from items (i) and (ii) of Proposition 2.18: we have w(zρn)(T) ≥ r(T)w(zρn)(I) and w(zρn)(I) ≥ 1 lim infn ( |zρn| 1 n ). Proposition 4.7. Let T ∈ L(H) and let (ρn)n ∈ (C∗)N ∗ be such that lim infn ( |ρn| 1 n ) > 0. Then lim |z|→+∞ ( w(zρn)(T) ) = r(T) lim infn ( |ρn| 1 n ). Proof. According to Proposition 4.5 and Proposition 2.18, the map r 7→ w(reitρn)(T) is decreasing on ]0, +∞[ and w(ρn)(T) ≥ r(T)w(ρn)(I) ≥ r(T) lim infn ( |ρn| 1 n ). We will show that w(zρn)(T) is as close to this lower bound as we want when z is large enough. Let � > 0. If r(T) = 0, then r ( 1 � T ) = 0, so Proposition 4.6 implies the existence of r > 0 such that w(zρn) ( T � ) ≤ 1 for all z with |z| = r. Thus, w(zρn)(T) ≤ �. If r(T) 6= 0, for 0 < R < lim infn ( |ρn| 1 n ) we have r ( RT (1 + �)r(T) ) ≤ lim inf n ( |ρn| 1 n ) . Thus, by Proposition 4.6, there exists r > 0 such that w(zρn) ( RT (1+�)r(T) ) ≤ 1 for all z with |z| = r. Hence, r(T) lim infn ( |ρn| 1 n ) ≤ w(zρn)(T) ≤ (1 + �)r(T)R . We then obtain the result by taking R = lim infn ( |ρn| 1 n ) (1−�) if lim infn ( |ρn| 1 n ) is finite, or R = 1 � if lim infn ( |ρn| 1 n ) = +∞. Proposition 4.8. Let T ∈ L(H). Let (ρn)n ∈ (C∗)N ∗ be such that lim infn ( |ρn| 1 n ) > 0. We have: unitary skew-dilations of hilbert space operators 169 (i) z 7→ w(zρn)(T) is uniformly continuous on C\D(0,�), for all � > 0. This maps tends to +∞ as |z|→ 0, and to r(T) lim infn ( |ρn| 1 n ) as |z|→ +∞; (ii) For any t ∈ R, the map r 7→ w(reitρn)(T) is log-convex on ]0, +∞[. Proof. (i) On the closed set C\D(0,�), the function z 7→ w(zρn)(T) is continuous, decreasing on every half-line of the form eit[�, +∞[, and con- verges to r(T) lim infn ( |ρn| 1 n ) as |z|→ +∞. Thus, a standard argument (considering two cases, � ≤ |z| ≤ R and |z| ≥ R) shows that this map is uniformly continuous. One can also use the double-commuting inequality of Lemma 4.2 for T and IH, as well as the uniform continuity of the map z 7→ w(z)(I) on C \ D(0,η), in order to prove the uniform continuity of z 7→ w(zρn)(T). The limit as |z| → +∞ has been obtained in Proposition 4.7, while the limit as |z| → 0 comes from the fact that w(zρn)(T) ≥ ‖T‖ |z||ρ1| , as remarked in Lemma 2.12. (ii) Let t ∈ R. Denote G′(z) := −e−itf(ρn)(zT). For any α > 0, we have w(reitρn)(T) ≤ α if and only if f(eitρn) ( zT α ) is analytic on D and I + Re ( 1 r f(eitρn) ( zT α )) ≥ 0, for all z ∈ D. By taking w = z α , this is equivalent to G′(w) being analytic on D ( 0, 1 α ) and Re(G′(w)) ≤ rI, for all w ∈ D ( 0, 1 α ) . The result is then obtained by mimicking the proof of [2, Theorem 1] by Ando and Nishio and replacing G with G′. Even though the expression of f(zρn) is more complex than f(z)(w) = 2 z w 1−w , the main regularity properties remain valid due to its analyticity. Proposition 4.9. Let (ρn)n ∈ (C∗)N ∗ be such that lim infn ( |ρn| 1 n ) > 0. If one of the following assertions is true (i) lim infn ( |ρn| 1 n ) < 1; (ii) |ρn| < 1 for some n ≥ 1; (iii) w(ρn)(I) > 1; (iv) ρn = M + xn, (xn)n ∈ `2(C), 170 v. agniel then all operators in C(ρn)(H) are similar to contractions. If, on the contrary, we have: (i’) w(ρn)(I) < 1, then C(ρn)(H) contains operators that are not similar to contractions. Both statements remain true if the conditions are only fulfilled for the subsequence (ρkn)n, for some fixed k ≥ 1. Proof. (i), (ii), (iii) We can see that (i) ⇒ (ii) ⇒ (iii). If (iii) is true, then for T ∈ C(ρn), we have r(T) ≤ w(ρn)(T) w(ρn)(I) < 1, so T is similar to a contraction. (iv) It has been shown in [17, Chapter 2] (see also [4, Corollary 5.2.1]) that when ρn = M + xn, (xn)n ∈ `2(C), all operators in C(ρn) are similar to contractions. (i’) On the contrary, when w(ρn)(I) < 1, 1 w(ρn)(I) I ∈ C(ρn) and this operator is not similar to a contraction. The last assertion of the theorem follows from two facts. The first one is that T ∈ C(ρn) implies T k ∈ C(ρkn). The second one is that T k is sim- ilar to a contraction if and only if T is similar to a contraction: see [10, Problem 6 (ii)] for a proof when k = 2 that extends to any k by taking ((f,g)) := ∑k−1 j=1〈A jf,Ajg〉. Proposition 4.10. Let (ρn)n ∈ (C∗)N ∗ be such that lim infn ( |ρn| 1 n ) > 0. (i) If lim infn ( |ρn| 1 n ) = +∞, then ⋃ r>0 C(rρn)(H) = L(H). (ii) If lim infn ( |ρn| 1 n ) < +∞, then we have{ T : r(T) < lim inf n ( |ρn| 1 n )} ⊂ ⋃ r>0 C(rρn)(H) ⊂ { T : r(T) ≤ lim inf n ( |ρn| 1 n )} . (iii) Moreover, we have{ T : r(T) < lim inf n ( |ρn| 1 n )} = ⋃ r>0 C(rρn)(H) unitary skew-dilations of hilbert space operators 171 if and only if w(rρn) ( lim inf n ( |ρn| 1 n ) I ) > 1, ∀r > 0. Proof. (i) By using Proposition 4.6, for any T there exists r > 0 such that w(rρn)(T) ≤ 1. (ii) We use again Proposition 4.6 in order to obtain the left-hand side inclu- sion. The other inclusion follows from Proposition 2.6. (iii) Suppose that there is a number r > 0 and an operator T with r(T) = lim infn ( |ρn| 1 n ) such that w(rρn)(T) ≤ 1. Then 1 ≥ w(rρn)(T) ≥ r(T)w(rρn)(I) ≥ r(T) lim infn ( |ρn| 1 n ) = 1. Thus all inequalities are equalities, and w(rρn) ( lim infn(|ρn| 1 n ).I ) = 1. Hence, the union of all C(rρn) strictly contains { T : r(T) < lim infn ( |ρn| 1 n )} if and only if it contains lim infn ( |ρn| 1 n ) I. The proof is complete. Remark 4.11. Replacing (ρn)n by (e itρn)n leaves unchanged the quantity lim infn ( |ρn| 1 n ) . However, the union of all classes C(rρn) can become a different set. With ρn = ρ, we have lim infn ( |ρn| 1 n ) = 1 and w(ρ)(I) = 1 if and only if ρ ∈ [1, +∞[. Thus,⋃ r>0 C(reit)n(H) = { T : r(T) < 1 } if t 6= 0 [2π]. This is not an equality if t = 0 (look at the identity operator I). However, the set ⋃ r>0 C(r)(H) does not contain all operators with spectral radius one. In- deed, it has been proven in [17, Chapter 2] (see also [4, Corollary 5.2.1]) that all operators contained in this union are all similar to contractions. Furthermore, all operators similar to a contraction are not in this union. For a counterex- ample, any non-orthogonal projection T (that is T 2 = T and ‖T‖ > 1) is not in this union since Corollary 4.3,(v), says that w(ρ)(T) = ‖T‖+|ρ−1| |ρ| > 1. - For a sequence (ρn)n that satisfies α = lim infn ( |ρn| 1 n ) ∈]0, +∞[, we can go back to the case lim infn ( |ρn| 1 n ) = 1 by considering the sequence ( ρn αn ) n . As this normalization is equivalent to a dilation by a factor 1 α on the class 172 v. agniel C(ρn), we can then try to see if in this case the class C(ρn) is always included in the set of operators that are similar to contractions. This question is motivated by Properties 4.10 and 4.9. The answer is true when w(ρn)(I) > 1, but Corollary 4.18 will give a negative answer in many remaining cases, even if we consider the inclusion in the set of power-bounded operators. At this point we would like to mention that, for every k ≥ 2, there is ([9]) a Hilbert space operator T /∈∪ρ>0Cρ but with Tk belonging to C(τ) for every τ ≥ 1. Related results are given in the next proposition. Proposition 4.12. Let (ρn)n ∈ (C∗)N ∗ be such that lim infn ( |ρn| 1 n ) > 0. Let H a Hilbert space of dimension at least 2. (i) For T ∈ L(H) with T 2 = 0 and ‖T‖ > |ρ1|, Tk is in the class C(ρn) for every k ≥ 2, but T is not. (ii) If lim infn ( |ρn| 1 n ) > 1, then Tk ∈ ⋃ r>0 C(rρn) for some k ≥ 2 implies that T ∈ ⋃ r>0 C(rρn). (iii) If lim infn ( |ρn| 1 n ) < 1, then there exists T ∈L(H) such that Tk lies in⋃ r>0 C(rρn) for every k ≥ 2 whereas T does not. (iv) If lim infn ( |ρn| 1 n ) = 1 and I /∈ ⋃ r>0 C(rρn), then T k ∈ ⋃ r>0 C(rρn) for some k ≥ 2 implies that T ∈ ⋃ r>0 C(rρn). Proof. (i) As we have ‖T‖ > |ρ1|, T cannot lie in C(ρn), whereas T k = 0 does. (ii) Let T be such that Tk ∈ ⋃ r>0 C(rρn) for some k ≥ 2. Then, r(T k) ≤ lim infn ( |ρn| 1 n ) . Hence, r(T) ≤ lim inf n ( |ρn| 1 n )1 k < lim inf n ( |ρn| 1 n ) , that is T ∈ ⋃ r>0 C(rρn) according to Proposition 4.10, (i) and (ii). (iii) Take r > 0 such that lim inf n ( |ρn| 1 n ) < r < lim inf n ( |ρn| 1 n )1 2 , and denote T = rI. Thus, using item (ii) of Proposition 4.10, we can see that since for every k ≥ 2 we have r(Tk) ≤ r(T 2) < lim inf n ( |ρn| 1 n ) < r(T), unitary skew-dilations of hilbert space operators 173 T doesn’t lie in ⋃ r>0 C(rρn) whereas T k does. (iv) If Tk ∈ ⋃ r>0 C(rρn), then r(T k) < 1 according to item (iii) of Proposition 4.10. This implies that r(T) < 1, which implies in turn that T ∈ ⋃ r>0 C(rρn). Remark 4.13. As the classes C(rρn) are increasing for the inclusion of sets, the assertion T ∈ ⋃ r>0 C(rρn) is equivalent to the existence of R > 0 such that T ∈ C(rρn) for every r ≥ R. When lim infn ( |ρn| 1 n ) = 1 and I ∈ ⋃ r>0 C(rρn), which is the case when ρn = ρ > 0, we do not know if the result of Găvruţa [9] stays true, as the type of operators he used in his proof is not suited in this setting: since there are sequences (ρn) such that ⋃ r>0 C(rρn) contains all power-bounded operators (see Corollary 4.18), taking a T such that Tk = I will not work. Example 4.14. For ρn = 2(n!), we have I + f(ρn)(zT) = exp(zT), and a quick computation gives w(2(n!))(I) = 2 π < 1 (see item (iv) of Example 4.20 for another proof). Therefore π 2 I ∈ C(2(n!))(H) and this class contains an operator not similar to a contraction. We can also try to obtain some relationships between the (γnρn)-radii of an operator, for sequences (γn)n ∈ ∂DN ∗ , in order to see for which sequences (γn)n the maximal or minimal radii are attained. The following lemma answers the question for the maximal radii when T = I. Lemma 4.15. Let (ρn)n ∈ (C∗)N ∗ be such that α = lim infn ( |ρn| 1 n ) > 0. If limx→α− f(|ρn|)(x) > 1, then f(|ρn|)(x) = 1 has a unique solution, r1, on ]0,α[. Otherwise, denote r1 = α. We then have: (i) w(−|ρn|)(I) = 1 r1 ; (ii) w(−|ρn|)(I) ≥ w(γnρn)(I) ≥ 1 α , for any (γn)n ∈ ∂DN ∗ ; (iii) The condition w(rγnρn)(I) = 1 α , ∀r ≥ 1, ∀γn ∈ ∂D is equivalent to lim x→α− f(|ρn|)(x) ≤ 1 and to w(−|ρn|)(I) = 1 α . 174 v. agniel Proof. (i), (ii) The right-hand side inequality of (ii) is the last inequality of Lemma 2.12. For any z ∈ D(0,α) and γn ∈ ∂D, we have ∣∣f(γnρn)(z)∣∣ ≤ ∑ n≥1 2|z|n |ρn| = f(|ρn|)(|z|). Also, the map x 7→ f(|ρn|)(x) is strictly increasing on ]0,α[, as f(|ρn|) is non- constant with positive Taylor coefficients, so if limx→α− f(|ρn|)(x) > 1 the real number r1 is indeed unique. Let u > 0 be such that u ≥ 1r1 ≥ 1 α . Then 1 u ≤ r1 and lim x→1 u − f|ρn|(x) ≤ 1. Since we have f(γnρn) ( D ( 0, 1 u )) ⊂ D ( 0, lim x→1 u − f(|ρn|)(x) ) ⊂ D, Proposition 2.18 implies that w(γnρn)(I) ≤ 1 r1 . When γn = − ρn|ρn|, we have f(γnρn)(x) = f(−|ρn|)(x) = −f(|ρn|)(x). Thus, the negative number lim x→1 u − ( − f(|ρn|)(x) ) lies in the adherence of f(−|ρn|) ( D ( 0, 1 u )) , and the smallest u ≥ 1 α such that f(−|ρn|) ( D ( 0, 1 u )) ⊂ Re≥−1 is 1 r1 . Hence, w(−|ρn|)(I) = 1 r1 ≥ w(γnρn)(I). (iii) By (ii) and using that r 7→ w(rγnρn)(I) is decreasing, we have w(rγnρn)(I) = 1 α , ∀r ≥ 1, ∀(γn)n ∈ ∂DN ∗ if and only if w(−|ρn|)(I) = 1 α . This equation is equivalent to r1 = α, that is limx→α− ( f(|ρn|)(x) ) ≤ 1. unitary skew-dilations of hilbert space operators 175 We do not know if the (|ρn|)-radius of I is always the minimal one. The idea of the proof of Lemma 4.15 can be transported to any operator T if we add a summability condition to the sequence (ρn)n. Proposition 4.16. Let a = (an)n ∈ (C∗)N ∗ be such that ∑ n≥1 1 |an| ≤ 1. Let T ∈L(H) and define ρn := { 2an‖Tn‖ if Tn 6= 0, 1 otherwise, (i) If r(T) > 0 or if T is nilpotent, then T ∈ C(ρn). (ii) If r(T) > 0 and lim infn ( |an| 1 n ) = 1, then w(znρn)(T) = 1, for all zn such that |zn| ≥ 1 and limn ( |zn| 1 n ) = 1. Proof. (i) Suppose first that r(T) > 0. Since ∑ n 1 |an| < +∞, we have lim infn(|an| 1 n ) ≥ 1, thus lim infn ( |ρn| 1 n ) ≥ r(T) > 0. We also have: ∥∥f(ρn)(zT)∥∥ ≤ ∑ n≥1 2|z|n‖Tn‖ 2|an|‖Tn‖ ≤ ∑ n≥1 1 |an| ≤ 1. Thus, I + Re ( f(ρn)(zT) ) ≥ ( 1 − ∥∥f(ρn)(zT)∥∥)I ≥ 0 for all z ∈ D, so T ∈ C(ρn). If T is nilpotent then f(ρn)(zT) becomes a finite sum and the same computation gives the result, as lim infn ( |ρn| 1 n ) > 0. (ii) When r(T) > 0 and lim infn ( |an| 1 n ) = 1, we have r(T) = lim infn ( |ρn| 1 n ) , so 1 ≥ w(ρn)(T) ≥ r(T) lim infn ( |ρn| 1 n ) = 1. Thus w(ρn)(T) = 1. If we multiply each an by a complex number zn with |zn| ≥ 1 and limn ( |zn| 1 n ) = 1, the sum ∑ n≥1 1 |znan| decreases, while lim infn ( |znan| 1 n ) = 1. Thus, we can apply the previous result to (znρn)n and obtain w(znρn)(T) = 1. Remark 4.17. For any T with r(T) > 0, if we take a sequence (ρn)n as in item (ii) of the previous Proposition, then the result says that z 7→ w(zρn)(T) is constant and equal to 1 on C\D. - The choice of (ρn)n only depends on ‖Tn‖. For example, with any T normal 176 v. agniel with ‖T‖ = 1, by taking an = π 2 6 n2, we have w(2anzn)(T) = 1 for any sequence (zn)n such that 1 ≤ |zn| and sup |zn| < +∞. - If T is quasinilpotent but not nilpotent, we have lim infn ( |ρn| 1 n ) = 0. How- ever, the statement of item (i) holds true for such a T, with a very similar proof. Using the ideas in the proof of Proposition 4.16, we can show that some sets⋃ r>0 C(rρn) largely differ from ⋃ ρ>0 C(ρ) or {T : r(T) < 1} even if lim infn ( |ρn| 1 n ) = 1. Corollary 4.18. Let (ρn)n be such that lim infn ( |ρn| 1 n ) = 1. The fol- lowing assertions are true: (i) If ( 1 ρn ) ∈ `1, then ⋃ r>0 C(rρn) contains all power-bounded operators. (ii) If f(ρn) ∈ H ∞(D) and f ′ (ρn) ∈ H∞(D), then ⋃ r>0 C(rρn) contains an operator that is not power-bounded. (iii) If nk+1+� = O(|ρn|) for k ∈ N∗ and some � > 0, then ⋃ r>0 C(rρn) contains all operators T such that ‖Tn‖ = O(nk). Proof. (i) Let T be a power-bounded operator with ‖Tn‖≤ C. Let r > 0 and z ∈ D. We have ∥∥f(rρn)(zT)∥∥ ≤ ∑ n≥1 2 r|ρn| |z|n‖Tn‖≤ 2C r ∑ n≥1 1 |ρn| < +∞. Hence, for r large enough, we have ∥∥f(rρn)(zT)∥∥ ≤ 1 for every z ∈ D. This implies that I + Re ( f(rρn)(zT) ) ≥ 0, ∀z ∈ D. This in turn implies that T ∈ C(rρn) since we also know that r(T) ≤ 1 = lim infn ( |rρn| 1 n ) . (ii) We first note that both the entire series f( ρn n )(z) = ∑ n≥1 2n ρn zn and f(ρn) have radii of convergence 1, so their sum is analytic on D. We also have f( ρn n )(z) = zf ′ (ρn) (z). Let N be a nilpotent operator of order 2 and set T = I +N. Since Tn = I +nN, we have ‖Tn‖' n‖N‖ so T is not power-bounded. We will show that T belongs to a class C(rρn) for large enough r > 0. Let unitary skew-dilations of hilbert space operators 177 r > 0 and z ∈ D. We have: ∥∥f(rρn)(zT)∥∥ = ∥∥∥∥∥∑ n≥1 2 rρn zn(I + nN) ∥∥∥∥∥ = ∥∥∥∥1rf(ρn)(z)I + 1rzf ′(ρn)(z)N ∥∥∥∥ ≤ 1 r (∥∥f(ρn)∥∥H∞ + ∥∥f ′(ρn)∥∥H∞‖N‖) < +∞. Hence, for r large enough, we have ∥∥f(rρn)(zT)∥∥ ≤ 1 for every z ∈ D, which implies that I + Re ( f(rρn)(zT) ) ≥ 0, ∀z ∈ D. This in turn implies that T ∈ C(rρn) since we also know that r(T) = 1 = lim infn ( |rρn| 1 n ) . (iii) Let T be such that ‖Tn‖ = O(nk) and let z ∈ D. We have ‖T n‖ |ρn| = O ( 1 n1+� ) , so this sequence is in `1. If T is nilpotent, then T is power-bounded and we can apply (i) to get a positive r > 0 such that T ∈ C(rρn). Otherwise, we can consider the complex numbers an := ρn ‖Tn‖ ∥∥∥∥ ( ‖Tn‖ |ρn| ) n ∥∥∥∥ `1 . We have ∑ n≥1 1 |an| = ∥∥∥∥ ( ‖Tn‖ |ρn| ) n ∥∥∥∥−1 `1 ∑ n≥1 ‖Tn‖ |ρn| = 1. Thus, for τn := 2an‖Tn‖, we can use Proposition 4.16 to obtain T ∈ C(τn). Since τn = 2ρn ∥∥∥(‖Tn‖|ρn| )n ∥∥∥ `1 , we have τn = rρn for some r > 0, which concludes the proof. The condition f ′ (ρn) ∈ H∞(D) implies that the sequence ( n ρn ) n is bounded, but it does not imply the condition ( 1 ρn ) ∈ `1 from (i). Thus, for a sequence (ρn) satisfying the conditions of item (ii), the set ⋃ r>0 C(rρn) may not contain every power-bounded operator. Some examples. We conclude this paper by providing a computation of w(zρn)(I) in two examples, where sequences (ρn)n were chosen to match some common analytic maps. The difficulty lies in the computation of the boundary of f(zρn) ( D ( 1, 1 u )) , as some specific points on the boundary do not always have an explicit expression. 178 v. agniel Example 4.19. Let R > 0 and −π < t ≤ π. We have: (i) I + f(Reitn)(zI) = I − 2 Reit log(1 −zI); (ii) w(Reitn)(I) = 1 if t = 0 and R ≥ 2 log(2); (iii) w(Reitn)(I) = 1 exp( R 2 )−1 > 1 if t = 0 and 0 < R < 2 log(2); (iv) w(Reitn)(I) = 1 1−exp(−R 2 ) > 1 if t = π; (v) w(Reitn)(I) = 1 if t = ± π 2 and R ≥ π; (vi) w(Reitn)(I) = 1 sin( R 2 ) if t = ±π 2 and 0 < R < π; (vii) w(Reitn)(I) = 1 if 0 < |t| < π 2 and R ≥ 2 cos(t) log(2 cos(t)) + 2 sin(t)t; (viii) If we have 0 < |t| < π 2 and 0 < R < 2 cos(t) log(2 cos(t)) + 2 sin(t)t, then w(Reitn)(I) = inf { u > 1 : 1 − 2 R gt(u) ≥ 0 } > 1 with gt(u) := cos(t) log (√ u2 − sin(t)2 + cos(t) u ) + arcsin ( sin(t) u ) sin(t). The same holds if π 2 < |t| < π. Proof. Let R > 0, t∈ ]−π,π]. As n ∈ R, we have w(Re−itn)(I) = w(Reitn)(I), so we can restrict the study to t ∈ [0,π]. A direct computation gives: f(Reitn)(zT) = − 2 Reit log(1 −zT). As lim infn ( |n| 1 n ) = 1, we have w(Reitn)(I) ≥ 1. Thus, we consider those u > 1 such that I + Re ( f(Reitn) ( zI u )) is positive for every z ∈ D. It is equivalent to look at the positivity of 1 + Re ( f(Reitn) (z u )) = 1 − 2 R Re ( e−it log ( 1 − z u )) . We start off by studying the boundary of log ( D ( 1, 1 u )) . By analyticity, we have ∂ log ( D ( 1, 1 u )) ⊂ log ( ∂D ( 1, 1 u )) . As log ( eisR ∩ D ( 1, 1 u )) is a horizontal interval that is non-empty if and only if |s| ≤ arcsin ( 1 u ) , the previous sets unitary skew-dilations of hilbert space operators 179 are equal and log ( D ( 1, 1 u )) is convex. Thus, the set log ( ∂D ( 1, 1 u )) can be parameterized by two arcs depending on the imaginary part of its elements: s 7→ log ( cos(s) ± 1 u √ 1 − sin(s)2u2 ) + is, s ∈ [ − arcsin (1 u ) ; arcsin (1 u )] . We want to compute the minimum of 1 − 2 R Re ( e−it log ( 1 − e is u )) in order to find for which u > 1 this minimum is non-negative. For the cases t = 0, t = π, and t = π 2 , computing this minimum amounts to finding the extrema of the real or imaginary part of the elements in log ( ∂D ( 1, 1 u )) . As these extrema are log ( 1 ± 1 u ) for the real part and ±arcsin ( 1 u ) for the imaginary part, an easy computation gives all the u > 1 such that inf w∈R ( 1 − 2 R Re ( e−it log ( 1 − eiw u ))) ≥ 0 in all three cases, which proves the items (ii), (iii), (iv), (v), (vi). For 0 < t < π 2 , computing this minimum leads to searching the lower bound of f1(s) := cos(π − t) log ( cos(s) − 1 u √ 1 − sin(s)2u2 ) −s sin(π − t). For π 2 < t < π, computing this minimum leads to searching the lower bound of f2(s) := cos(π − t) log ( cos(s) + 1 u √ 1 − sin(s)2u2 ) −s sin(π − t). The derivatives of these maps are: f ′1(s) = sin(s)u cos(π − t)√ 1 − sin(s)2u2 −sin(π−t), f ′2(s) = − sin(s)u cos(π − t)√ 1 − sin(s)2u2 −sin(π−t). Both of them only have one zero, at s = −arcsin (sin(t) u ) . And in both cases the searched minimum for 1 − 2 R Re ( e−it log ( 1 − e is u )) is: 1− 2 R [ cos(t) log (√ u2 − sin(t)2 + cos(t) u ) +arcsin (sin(t) u ) sin(t) ] = 1− 2 R gt(u). If 0 < t < π 2 , this minimum decreases towards 1 − 2 R gt(1) := 1 − 2 R [ cos(t) log ( 2 cos(t) ) + t sin(t) ] 180 v. agniel when u → 1+. So I u ∈ C(Reitn) for every u > 1 if and only if 1 − 2 R gt(1) ≥ 0, that is R ≥ 2gt(1). This proves item (vii) and half of item (viii). If π 2 < t < π, this minimum decreases towards −∞ as u → 1+, so the smallest u for which this minimum is non-negative verifies u > 1 and w(Reitn)(I) = u. This gives the other half of item (viii) and concludes the proof. Example 4.20. Let R > 0 and −π < t ≤ π. We have: (i) I + f(Reitn!)(zI) = I + 2 R.eit ( exp(zI) − I ) ; (ii) w(Reitn!)(I) = 1 log( R 2 +1) if t = π; (iii) w(Reitn!)(I) = 1 log( 2 2−R ) if t = 0 and 0 < R ≤ 2 − 2 e ; (iv) w(Reitn!)(I) = 1 π 2 −t if 0 ≤ |t| < π 2 and R = 2 cos(t); (v) w(Reitn!)(I) ≤ 1 log( R 2 −cos(t)) for R > 2 + 2 cos(t); (vi) w(Reitn!)(I) ≥ 1√ π2+log ( R 2 cos(t) −1 )2 if 0 ≤ |t| < π2 and R > 4 cos(t); (vii) In general, we have w(Reit.n!)n(I) = inf { u > 0 : ∀θ ∈ [−π,π] with θ + sin(θ) u = t + kπ, k ∈ Z, we have (−1)ke cos(θ) u cos(θ) ≥ cos(t) − R 2 } . For R ≥ 2eπ/2 − 2, we can restrict the infimum after u in ]0, 2 π ] and to the smallest θ ∈ ]−π 2 , 0] such that θ + sin(θ) u = t + kπ. Proof. Let R > 0, t ∈ [−π,π] and u > 0. As n ∈ R, we have w(Re−itn!)(I) = w(Reitn!)(I), so we restrict the study to t ∈ [0,π]. A computation gives I + f(Reitn!)(zI) = I + 2 R.eit ( exp(zI) − I ) . We will first use Lemma 4.15 to compute w(−Rn!)(I) and rule out the case t = π. As f(Rn!)(x) = 2 R ( exp(x) − 1 ) , we get f(Rn!)(x) = 1 ⇔ x = log ( R 2 + 1 ) . unitary skew-dilations of hilbert space operators 181 Hence, w(−Rn!)(I) = 1 log( R 2 +1) and item (ii) is proved. As lim infn ( |n!| 1 n ) = +∞, we have I u ∈ C(Reitn!) if and only if u ≥ w(Reitn!)(I), if and only if I + Re ( f(Reitn!) ( z I u )) for every z ∈ D. Thus, we need to study the positivity of 1 + Re ( f(Reitn!) (z u )) = 1 + 2 R Re ( e−it ( exp (z u ) − 1 )) , for every z ∈ D and for any u > 0. By analyticity, we only need to make the computations for z ∈ ∂D. We have 1 + 2 R Re ( exp (z u − it ) −e−it ) ≥ 0 ⇔ exp ( Re (z u )) cos ( Im(z) u − t ) ≥− R 2 + cos(t). Denote, for s ∈ [−π,π], gu(s) := e cos(s) u cos ( t− sin(s) u ) . Thus, I u ∈ C(Reitn!) is equivalent to min s∈[−π,π] (gu(s)) ≥− R 2 + cos(t). Therefore, this inequality will be verified if and only if u ≥ w(Reitn!)(I). Also, since min s ( gu(s) ) = min |w|= 1 u ( Re ( exp(w − it) )) = min |w|< 1 u ( Re ( exp(w − it) )) , we can see that mins ( gu(s) ) is the minimum of a harmonic non-constant map over the disc D ( 0, 1 u ) . The maximum principle implies that the map u 7→ mins ( gu(s) ) is strictly increasing. Hence, w(Reitn!)(I) is the only number u > 0 such that mins∈[−π,π] ( gu(s) ) = −R 2 + cos(t). Let us focus now on the minimum of gu. The derivative of gu is g′u(s) = 1 u e cos(s) u sin ( t− sin(s) u −s ) . 182 v. agniel Hence, the minimum of gu will be reached for a s0 such that hu(s0) := t − s0 − sin(s0) u = kπ, for some k ∈ Z. For such a s0, we will also have gu(s0) = (−1)ke cos(s0) u cos(s0). If u ≥ 1, the map hu is strictly decreasing, with range [t−π,t + π]. Hence, there will only be 2 (resp. 3) values of s such that hu(s) = kπ if t ∈]0,π[ (resp. t = 0). If t = 0 and u ≥ 1, these values of s will be −π, 0, π, and the minimum of gu will be gu(π) = exp (−1 u ) . Thus, if t = 0 and w(Rn!)(I) ≥ 1, we will have exp ( −1 w(Rn!)(I) ) = − R 2 + 1, which is equivalent to 0 < R ≤ 2 − 2 e . Thus w(Rn!)(I) = 1 log( 2 2−R ) , proving item (iii). When t ∈ ]0,π[ and u ≥ 1, we have however no explicit formula for the two values of s mentioned above. For t ∈ [0, π 2 [ and R = 2 cos(t), we will have mins ( gw (Reitn!) (I)(s) ) = 0. As e cos(s) u cos(s) = 0 if and only if s = ±π 2 , this minimum will be attained at π 2 or −π 2 , and w(Reitn!)(I) will be the largest u > 0 such that gu( π 2 ) = 0 or gu(−π2 ) = 0. The latter condition is equivalent to 1 u ± t = π 2 + kπ, that is 1 u = π 2 ± t + kπ. Since we have 0 ≤ t < π 2 , the integer k needs to be positive. By looking at the smallest possible value for 1 u we get w(Reitn!)(I) = 1 π 2 −t, proving item (iv). In general, we can see that mins ( gu(s) ) ≥ −e 1 u . When R > 2 + 2 cos(t), the inequality −e 1 u ≥ cos(t)−R 2 is equivalent to u ≥ 1 log( R 2 −cos(t)) , which proves item (v). If uπ < 1, we have uπ = sin(α) for some α > 0, and gu(α) = −cos(t)e √ 1−u2π2 u . When R > 4 cos(t), the inequality gu(α) ≤ cos(t) − R2 is equivalent to u ≤ 1√ π2 + log ( R 2 cos(t) − 1 )2 . Item (vi) is now proved. unitary skew-dilations of hilbert space operators 183 Taking R ≥ 2eπ/2 − 2 = R0, we get −R2 + 1 ≤ 2 −e π/2 < −1 and w(Reitn!)(I) ≤ w(−Rn!)(I) ≤ w(−R0n!)(I) = 2 π , for every 0 ≤ t ≤ π, according to Lemma 4.15. We can then see that for u = w(Reitn!)(I) and for a number s0 such that gu(s0) = mins ( gu(s) ) and hu(s0) = kπ, the relationship (−1)ke cos(s0) u cos(s0) = gu(s0) = − R 2 + cos(t) < −1 implies that cos(s0) > 0 and that k is odd. In this case, |s0| will be the smallest real s in [0, π 2 [ such that hu(s) or hu(−s) is equal to kπ with k odd. As we also have hu( −π 2 ) = t + π 2 + 1 u ≥ π ≥ t = hu(0) ≥ 0, we can see that s0 lies in ]−π 2 , 0]. This gives all the announced results. 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