Approximation numbers of weighted composition operators

arXiv:1612.01177v2 [math.FA] 24 Dec 2017 Approximation numbers of weighted composition operators G. Lechner∗ , D. Li† , H. Queffélec‡ , L. Rodríguez-Piazza§ December 24, 2017 Abstract We study the approximation numbers of weighted composition operators f7→ w·(f◦ϕ)on the Hardy space H2on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight wcan improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples). 1 Introduction In the study of composition operators Cϕ:f7→ f◦ϕacting on a Hilbert space Hof analytic functions (on the unit disk D), one is typically interested in understanding how function-theoretic properties of ϕare related to operator-theoretic properties of Cϕ. Basic properties such as boundedness or compactness of Cϕare by now well characterized in terms of ϕin many cases [35, 16]. More recently, also the membership of Cϕin various smaller ideals Iof bounded operators on H(such as the p-Schatten class), ∗Cardiff University, School of Mathematics, Lec[email protected] †Université d’Artois, Laboratoire de Mathématiques de Lens (LML), [email protected] ‡Université Lille Nord de France, Herve.Qu[email protected]r §Universidad de Sevilla, [email protected] 1 and more precisely the behavior of the approximation numbers an(Cϕ)of Cϕ, was studied in depth in several papers (see e.g. [27, 25, 26, 28]). If M(H)denotes the space of multipliers of H(those w∈Hsuch that wf ∈Hfor each f∈H), we can twist a composition operator Cϕ, assumed to map Hto itself, by composing it on the left with the operator Mwof multiplication by w∈ M(H). We then get a so-called weighted composition operator T=MwCϕ(see e.g. [24] or [12, 21]). A careful distinction must be made between the multipliers of H, denoted M(H), and those of Cϕ(H), denoted M(H, ϕ), namely those functions w∈ Hsuch that wf ∈Hfor each fbelonging to the range Cϕ(H), not necessarily to the whole of H. For example, if H=H2is the Hardy space, then M(H) = H∞consists of all bounded analytic functions on D. It can be proved that (see [2, 14] and [19], respectively): M(H2, ϕ) = H∞⇔ϕis a finite Blaschke product, M(H2, ϕ) = H2⇔ kϕk∞<1. In this paper, we study approximation numbers of weighted composition operators in the case of the Hardy space H=H2of the disc, and a weight w∈ M(H)⊂ M(H, ϕ). Then, since we are dealing with ideals, twisting with the bounded operator Mwcan but reinforce the membership in I, and improve the rate of decay of approximation numbers. There are (at least) three motivations for considering weighted composition operators: First, they form a natural and non-trivial generalization of composition operators on H2(D). In this context, it is natural to ask how much faster the approximation numbers an(MwCϕ)can decay in comparison to the an(Cϕ). For example, can MwCϕbe compact when Cϕis non-compact, or can the an(MwCϕ)decay quite fast when the an(Cϕ)decay rather slowly? We will address these questions in the body of the text1. As a second motivation, suppose CG ϕis a composition operator on a Hardy space H2(G)over a simply connected region properly contained in C. Then a choice of Riemann map τ:D→Ginduces a unitary between H2(D)and H2(G)[17], and we can equivalently formulate CG ϕas an operator on H2(D). This operator on H2(D), however, turns out to be a weighted composition operator MwτCD ϕτin general (see [36] and Section 3). Thus composition operators on domains other than Dautomatically produce weighted composition operators on H2(D). 1Another application of weighted composition operators to the study of composition operators on spaces of several complex variables can be found in [29]. 2 A third motivation for studying weighted composition operators comes from applications in a completely different field, namely inclusions of von Neumann algebras, used in mathematical physics to model quantum field theories [20]. For Na von Neumann algebra with a cyclic and separating vector Ωon a Hilbert space H, we will consider the Hilbert space Dobtained by closing the domain of the modular operator of (N,Ω) [38] in its graph norm. If Ncarries additional structure (a Borchers triple), this setting is related to complex analysis because an irreducible component of Dcan be naturally identified with a Hardy space H2(S)on a strip region S⊂C, bounded by two lines parallel to R(see Section 3). In applications in mathematical physics, one is interested in specific inclusions ˜ N ⊂ N and their relative commutants, the size of which can be controlled if a map built from the modular operator has sufficiently quickly decaying approximation numbers [8]. On the level of the irreducible component giving rise to the Hardy space H2(S), this condition translates to a weighted restriction operator Rw:H2(S)→L2(R),f7→ (w·f)|R, where the real line Rlies in the interior of the strip, and w∈H∞(S)is an inner function on Sobtained from the inclusion ˜ N ⊂ N. For the application in physics, sharp upper bounds on the approximation numbers of Rware desirable [1]. Mapping the strip Sto the disc, Rwcan be formulated as a Carleson embedding operator (the definition of which we recall in Section 2.1). These operators are often used in estimating approximation numbers of composition operators [25, 26]. In turn, we find from the strip picture that the embedding operator can be estimated from above by special weighted composition operators on the disc, namely those whose symbol is a lens map ϕ=ϕλ,0< λ < 1(see Section 4 for the definition). This closes the connection to composition operators on H2(D), where the Cϕλare among the best studied examples [25]. Given these motivations, this article is organized as follows. In Section 2, we introduce our notation and setup, and study weighted composition operators with general symbols ϕand weights won the disc. After deriving a simple upper bound, we give an example how a weight can turn a non-compact composition operator into a compact one. Regarding lower bounds, we show that the worst possible behavior of the an(Cϕ)(exponential if kϕk∞<1, subexponential if kϕk∞= 1) is the same for the weighted operators MwCϕ. In Section 3, we explain the links between modular theory of von Neumann algebras, Hardy spaces on strips, and weighted restriction operators. In that section we also show how weighted lens map composition operators 3 appear. Section 3 can be read independently of the other parts of the article. Finally, we consider in Section 4 the specific case of weighted lens map composition operators MwCϕλas our primary example. In the case without weight, the approximation numbers of Cϕλare known to decay like e−c√n[26]. A natural question in this context is how close the decay rate of the an(MwCϕλ)can come to exponential decay e−cn (the optimal one in the context of composition operators). For the weights motivated by the considerations in Section 3, we show that an(MwCϕλ)decays like e−cn log n. 2 Weighted composition operators on D 2.1 Preliminaries We begin by recalling a few operator-theoretic and function-theoretic facts. The approximation numbers an(T) = anof an operator T:H→H (with Ha Hilbert space) are defined by an= inf rankR<n kT−Rk, and Tis compact if and only if limn→∞ an(T) = 0. According to a result of Allahverdiev [11, p. 155], an=sn, the n-th singular number of T. We have the following alternative definition (a variant of Kolmogorov numbers) of an(T)[28], in which BHdenotes the closed unit ball of Hand d(g, A)the distance of gto A⊂H: (2.1) an(T) = inf dim E<n hsup f∈BH d(Tf, TE)i. The definition of an(T)also makes sense for T:X→Yan operator between Banach spaces (see Theorem 2.8 to come). Coming back to the hilbertian setting, two other useful alternative definitions (respectively in terms of Bernstein and Gelfand numbers) are, denoting by SEthe unit sphere of a subspace Eof H(see [11, Chapter 2], or [27]): an(T) = sup dim E=nhinf f∈SEkTfki,(2.2) an(T) = inf codimE<n kT|Ek= inf codimE<n hsup f∈SEkTfki.(2.3) The following parameters 0≤β−(T)≤β+(T)≤1were used in [28]: β+(T) = lim sup n→∞ an(T)1/n, β−(T) = lim inf n→∞ an(T)1/n. 4 When the limit exists, we denote it by β(T). It is proved in [28] that this is the case for Ta composition operator on the Hardy, Bergman, or Dirichlet space. Observe that β−(T) = 1 signifies a subexponential decay for an(T), namely an(T)≥e−nεnwhere εn>0and εn→0. Let now Dbe the open unit disk of the complex plane and H2the usual Hardy space of D. Recall [35, p. 12] that the norm of f(z) = P∞ n=0 fnzn∈H2 is defined by kfk2 2=P∞ n=0 |fn|2, or alternatively by (2.4) kfk2 2=ZT|f∗(u)|2dm(u) where mdenotes the Haar measure of the unit circle Tand f∗(u)is the (malmost everywhere existing by Fatou’s theorem) radial limit limr→1−f(ru), often again denoted f(u). The space of multipliers of H2is isometrically isomorphic to the space H∞ of functions analytic and bounded on D[18]. This means that any function w∈H∞defines a bounded multiplication operator Mw:H2→H2by the formula Mw(f) = wf and that kMwk:= sup kfk2≤1kwfk2=kwk∞:= sup z∈D|w(z)|. If ϕis a non-constant and analytic self-map of D(often called a symbol), the associated composition operator Cϕ:H2→H2is defined by Cϕ(f) = f◦ϕ. The fact that Cϕboundedly maps H2to itself for any symbol ϕis the wellknown subordination principle of Littlewood ([16, p. 29], [35, p. 16]). Next, a positive and bounded measure µon Dis called a Carleson measure (for H2) if the identity map Rµ, Rµ(f) = fmaps H=H2to L2(µ), that is if there exists a constant Csuch that: ZD|f(z)|2dµ(z)≤Ckfk2 2∀f∈H2. The best constant Cis called the Carleson-norm of µand is denoted kµkC. That is kµkC=kRµk2. Let us set ρµ(h) := sup ξ∈T µ[S(ξ, h)] where, for ξ∈T=∂D,S(ξ, h)is the Carleson box S(ξ, h) = {z∈D:|z−ξ| ≤ h}. 5 With those notations, the Carleson embedding theorem [16, p. 37] gives a geometric characterization of Carleson measures: Theorem 2.1. Let µbe a positive and bounded measure on D. Then, µis a Carleson measure if and only if, for some constant K, ρµ(h)≤Kh ∀h∈]0,1]. In this case, kµkC≤aK where a > 0is an absolute constant. Let again ϕ∗(u) = limr→1−ϕ(ru). Littlewood’s subordination principle implies that mϕ=ϕ∗(m), the image under ϕ∗of this Haar measure, is a Carleson measure for H2, and we will write ρϕinstead of ρmϕ. We will also write k.kinstead of k.k2when there is no ambiguity. The main subject of this article are weighted composition operators on H2, defined in terms of a weight w(typically w∈H∞) and a symbol ϕ, according to Tw,ϕ := MwCϕ. Often we will denote this operator by Tfor short. Depending on ϕ, the operator Cϕcan be compact or not. As we already said, passing from Cϕto MwCϕcan only improve this compactness, or the behavior of singular numbers, thanks to the ideal property of both notions. It is the purpose of this paper to investigate the question more closely. 2.2 A simple general upper bound Throughout this section and the rest of this paper, we use the notation A.B(resp. A&B) to indicate that A≤λB (resp. A≥λB) where λ is a uniform positive constant, “uniform” being clear from the context. Let ϕbe a symbol continuous on D, fixing 1, and w0∈H∞a weight. We set γ(t) = ϕ(eit). We assume that ϕ(D)has no other contact points than 1with Tand more precisely that: (2.5) |t| ≤ π⇒1−|γ(t)| ≥ ω(|t|), where ω: [0, π]→R+is an increasing function with ω(0) = 0. We also set (2.6) δw0(h) = sup |t|≤ω−1(h)|w0(γ(t))|. We then have an upper bound for a special class of weights related to ϕ: 6 Theorem 2.2. Let ϕbe a symbol satisfying (2.5), w0a weight, w=w0◦ϕ, T=MwCϕ, and an=an(T). Then: (2.7) an.inf 0<h<1he−nh +δw0(h)i=: ρn. Proof. The proof is close to that of [26, Thm. 5.1]. Let f=zng∈znH2=: E, a subspace of H2of codimension n < n + 1. Assume that kfk= 1, so that kgk= 1. We see that, given 0< h < 1: kT(f)k2=ZT|w0(ϕ(u))|2|ϕ(u)|2n|g(ϕ(u))|2dm(u) =ZD|w0(z)|2|z|2n|g(z)|2dmϕ(z) =Z(1−h)D|w0(z)|2|z|2n|g(z)|2dmϕ(z) + ZD|z|2n|g(z)|2dµh(z) .(1 −h)2n+kµhkC.e−2nh +kµhkC, where µhdenotes the restriction (trace) of the measure |w0|2dmϕto the annulus Ah={z: 1 −h < |z|<1}. It remains to estimate kµhkC, which we do through Carleson’s embedding theorem. Since µhis carried by Ah, we can consider only boxes S=S(ξ, r)with 0< r ≤h. Then Ir:= Z1S(u)dµh(u) = Z1S(u)|w0(u)|2dmϕ(u) =ZS∩ϕ(∂D)|w0(u)|2dmϕ(u)≤sup S∩ϕ(∂D)|w0(u)|2mϕ(S) since mϕis carried by ϕ(∂D). Now, if u=ϕ(eit)∈Sand |t| ≤ π, then ω(|t|)≤1−|γ(t)| ≤ |ξ−γ(t)| ≤ r, that is |t| ≤ ω−1(r). Therefore, Ir≤δ2 w0(r)ρϕ(r).rδ2 w0(h), and by Carleson’s theorem: kµhkC.sup 0<r<h Ir r.δ2 w0(h), giving an+1 .ρnin view of the alternative definition (2.3) of approximation numbers, and ending the proof after a change of n+ 1 to n. 7 2.3 From non-compactness to compactness It is known that twisting a non-compact composition operator Cϕwith a multiplication operator Mwcan result in MwCϕbeing compact (see, for example, [19]). We now give a first application of Theorem 2.2, in which this effect is demonstrated in terms of explicit estimates on approximation numbers, which seems to be new. Note that the compactness result of part ii)of the following theorem also follows by applying [19, Thm. 2.8] to the specified symbol ϕand weight w. Theorem 2.3. Let ϕ(z) = 1+z 2and w(z) = (1 −z)α, α > 0. Then: i) Cϕis non-compact and indeed kCϕke=kCϕk=√2. ii) T=MwCϕis compact and its approximation numbers verify an(T).log n nα/2. In particular, a weighted composition operator T=MwCϕcan be compact while its “compositional symbol” ϕhas no fixed point inside D. Proof. Recall that kCϕke=: limn→∞ an(Cϕ)is the essential norm of Cϕ. The first item i) is well-known ([34], see also [13]). For upper bounds, we may use Theorem 2.2 since w=w0◦ϕwhere w0(z) = 2α(1 −z)α. Next, we observe that, for |t| ≤ π:1−|γ(t)|= 1 −cos(t/2) = 2 sin2(t/4) ≥ δt2, so that, up to absolute constants, we are allowed to take ω(h) = h2and ω−1(h) = √hin Theorem 2.2. Since |w0(γ(t))|.|1−γ(t)|α≤ |t|α, this implies that δw0(h).hα/2, and subsequently that an(T).inf 0<h<1he−nh +δw0(h)i.inf 0<h<1he−nh +hα/2i.log n nα/2 by taking h=Clog n nwhere Cis a large numerical constant. This gives the claimed upper bound. Finally, the fixed point of ϕis 1and 1/∈D. Remark: The simple estimates given here are not sharp (Theorem 4.1 will give sharper results) and are just intended to show that multiplication by w can improve the decay of approximation numbers. In particular, the logarithmic factor can be dropped in the example of Thm. 2.3, and whereas the estimate in Theorem 2.3 ii) give membership in the Hilbert-Schmidt class for α > 1, one actually has the following stronger statement. 8 Proposition 2.4. The following are equivalent in the previous example: i) MwCϕis Hilbert-Schmidt. ii) α > 1/2. Proof. If T=MwCϕand if (en)is the canonical basis of H2, we find that ∞ X n=0 kT(en)k2= 2 ZT |w(eiθ)|2 1−cos θdθ ≈Zπ 0 θ2α−2dθ and the latter integral is finite iff α > 1/2. Alternatively, we could use kT(en)k ≈ n−α/2−1/4. 2.4 Maximal general possible decay It was proved in [26] that singular numbers of composition operators never have a superexponential decay. The same holds for weighted composition operators. Theorem 2.5. Let T=MwCϕbe a weighted composition operator. Then β−(T)>0, that is, there exist positive constants δand ρsuch that for any integer n≥1: an(T)≥δρn. Proof. We first recall that the interpolation constant Izof a (finite or not) sequence z= (zj)of distinct points of Dis the smallest constant Ksuch that, for any bounded sequence c= (cj), one can find h∈H∞such that h(zj) = cj∀jand khk∞≤Ksup j|cj|. The connection between interpolation constants and reproducing kernels is given by the well-known two-sided inequality [31, p. 302-303], valid for all scalars λj: (2.8) I−2 zX j|λj|2kKzjk2≤  X j λjKzj   2≤I2 zX j|λj|2kKzjk2. We shall now rely on the following lemma: 9 This shows that Sis anti unitary on D. ii) This is clear because ∆it commutes with ∆1/2. iii) For ψ∈ D, the H-valued function z7→ ∆−izψis holomorphic on the strip 0<Im(z)<1 2, continuous on its closure, and norm-constant in the real direction, k∆−i(x+iy)ψk=k∆yψk. Thus the three lines theorem [15] implies, 0≤y≤1 2, k∆yψk1/2≤ kψk1/2−y·k∆1/2ψky, from which we read off k∆yψk ≤ kψk1−2y·k∆1/2ψk2y≤qkψk2+k∆1/2ψk2=kψk∆. iv) This follows from, ψ, ϕ ∈dom (∆1/2), hJψ, Jϕi∆=hJψ, Jϕi+h∆1/2Jψ, ∆1/2Jϕi =hψ, ϕi+h∆−1/2ψ, ∆−1/2ϕi=hJψ, Jϕi∆−1. To draw the connection to Hardy spaces, we recall a theorem of Borchers [6] on the commutation relation of the modular unitaries ∆it,t∈R, and the translation unitaries U(x),x∈R2: If (N, U, Ω) is a Borchers triple, then there holds ∆itU(x)∆−it =U(Λ(t)x), JU(x)J=U(−x), x ∈R2,(3.4) where (Λ(t)x)±=e∓2πtx±. The commutation relations (3.4) imply that in the presence of a Borchers triple, the operators U(x),∆it, and Jgenerate a (anti-)unitary strongly continuous representation of the proper Poincaré group P+=SO(1,1) ⋊ R2. We will denote this extended representation by the same letter U. As the basic building blocks, we are interested in the irreducible subrepresentations of U. We call a representation degenerate if there exists a non-zero vector Ψwhich is invariant under U(x)for all x∈R2, i.e. Ψ∈ker P+∩ker P−. In physical models, this vector can usually only be a multiple of Ω, so that we may restrict to non-degenerate representations of P+. The non-degenerate, irreducible, unitary, strongly continuous positive energy representations of P+can be classified up to unitary equivalence according to the joint spectrum of the generators P±, denoted Sp (U|R2). There are equivalence classes of three types: 16 m) Sp (U|R2) = {p∈R2:p±>0, p+·p−=m2}for some m > 0, 0+) Sp (U|R2) = {p∈R2:p+≥0, p−= 0}, 0−) Sp (U|R2) = {p∈R2:p−≥0, p+= 0}. The parameter mhas the physical interpretation of a mass. We therefore refer to representations of type m > 0as “massive”, and to representations of type 0±as “massless”. For a concise notation, we will adopt the convention to label objects by a single label m, which can either take positive values, referring to type m), or the two special values m= 0±, referring to type 0±). The irreducible representation Umof type mcan be conveniently realized on the Hilbert space H=L2(R, dθ). In fact, we have for all types the same modular unitaries and conjugation [23] H=L2(R, dθ),(∆itψ)(θ) = ψ(θ−2πt),(Jψ)(θ) = ψ(θ).(3.5) The translation operators are multiplication operators depending on the type, namely (Um(x)ψ)(θ) = wm,x(θ)·ψ(θ),(3.6) w0±,x(θ) = eix±e±θ, wm,x(θ) = eim(x+eθ+x−e−θ).(3.7) The functions wm,x will later serve as the weights of our weighted composition operators. Considering the representation (3.5), it becomes clear that the domain D ⊂ L2(R)of ∆1/2consists of functions that have an analytic continuation to the strip region S0,π, a special case of the more general strip Sa,b := {ζ∈C:a < Im ζ < b}, a < b ,(3.8) and satisfy certain bounds on this strip. Before we make this precise, let us recall some properties of functions analytic in a strip, and corresponding function spaces. Given f∈Hol(Sa,b)(the holomorphic functions Sa,b →C), we write fλ, a < λ < b, for its restriction to the line R+iλ. Denoting the usual norm of L2(R)by k·k2, we consider the norm |||f|||Sa,b := sup a<λ<b kfλk2∈[0,+∞](3.9) and set H2 B(Sa,b) := {f∈Hol(Sa,b) : |||f|||Sa,b <∞}.(3.10) We recall the following facts [37]: 17 i) (H2 B(Sa,b),|||·|||Sa,b )is a Banach space. ii) Any f∈H2 B(Sa,b)has L2-boundary values on the two boundaries R+ia and R+ib, i.e. fa+εand fb−εconverge in L2(R)as εց0. By a slight abuse of notation, we will denote these boundary values as fa, fb∈L2(R). iii) As an expression of the maximum principle, the function (a, b)∋λ7→ kfλk2is logarithmically convex for f∈H2 B(S). In particular, |||f||| = max{kfak2,kfbk2}. We will refer to H2 B(Sa,b)as Hardy Banach space to distinguish it from a Hardy Hilbert space on Sa,b to be introduced next. Indeed, as the strip is an unbounded region, there exist two different types of Hardy Hilbert spaces for this domain: The conformally invariant Hardy space, defined in terms of harmonic majorants, and the not conformally invariant Hardy space, defined in terms of L2-integrals over a sequence of Jordan curves tending to the boundary of the strip [17, Ch. 10]. For our purposes, only the latter space will be relevant. It will be convenient to characterize it in terms of a Riemann map τ:D→S. (To lighten our notation, we write Sinstead of Sa,b when the boundaries of the strip are arbitrary.) Namely, we define H2(S) := {f∈Hol(S) : √τ′·(f◦τ)∈H2(D)}.(3.11) This is a Hilbert space with scalar product hf, giS:= h√τ′·(f◦τ),√τ′·(g◦τ)iD,(3.12) and (H2(S),h·,·iD)depends on the choice of τonly up to changing the norm kfkS:= hf, fi1/2 Sto an equivalent Hilbert norm. We may therefore fix τ, and make the choice τ:D→Sa,b , τ(z) := 2(b−a) πarctanh(z) + i 2(a+b).(3.13) This is a biholomorphic mapping τ:D→Sa,b, and elementary calculations show that it has inverse and derivative, ζ∈Sa,b,z∈D, τ−1(ζ) = tanh π 2(b−a)ζ−iπ 4 b+a b−a, τ′(z) = 2(b−a) π 1 1−z2,(3.14) (τ−1)′(ζ) = π 2(b−a) 1 cosh2π 2(b−a)ζ−iπ 4 b+a b−a.(3.15) 18 Proposition 3.3. i) H2 B(S)and H2(S)coincide as linear spaces. ii) The two norms ||| · ||| and k · kSa,b are equivalent: For any f∈Hol(S), there holds 1 √2π|||f||| ≤ kfkS≤1 √π|||f|||.(3.16) iii) The scalar product of H2(Sa,b)can be written as hf, giSa,b =1 2πhfa, gai2+hfb, gbi2, f, g ∈H2(Sa,b).(3.17) Proof. We first work on the special strip Sgiven by a=−1,b= 1, and introduce for f∈Hol(S)the notation s(f) := 1 2sup0≤y<1(kfyk2 2+kf−yk2 2)∈ [0,+∞]. It was shown in [3, Thm. 2.1 & 2.2] that for f∈Hol(S), one has f◦τ∈ H2(D)if and only if s(w·f)<∞, and in this case, kf◦τk2 D=s(w·f), with the weight w(ζ) = (2 cosh πζ 4)−1. The condition that some f∈Hol(S)lies in H2(S), i.e. that √τ′(f◦τ) = (f/p(τ−1)′)◦τ∈H2(D), is therefore equivalent to s(w/p(τ−1)′·f)<∞. But in view of (3.15), w(ζ)/p(τ−1)′(ζ) = π−1/2. We thus have that f∈ Hol(S)lies in H2(S)if and only if s(f)<∞, and in this case, kfk2 S=k√τ′(f◦τ)k2 D=1 πs(f) = 1 2πsup 0≤y<1kfyk2 2+kf−yk2 2.(3.18) As the supremum on the right hand side clearly lies between |||f|||2and 2|||f|||2, the claimed equivalence of norms in ii)follows. This also implies i). To establish iii), we use that (−1,1) ∋y7→ kfyk2is logarithmically convex for f∈H2 B(S). Thus y7→ kfyk2 2+kf−yk2 2is convex, which implies that the supremum in (3.18) is taken for the boundary values at y= 1, i.e. kfk2 S=1 2πkf1k2 2+kf−1k2 2=1 2πhf1, f1i2+hf−1, f−1i2. This implies iii). It remains to proceed from S−1,1to a general strip Sa,b by means of the variable transformation ϕ:S−1,1→Sa,b,ϕ(ζ) := b−a 2ζ+i 2(a+b). But by elementary substitutions, one finds that f7→ f◦ϕis a bijection H2 B(Sa,b)→H2 B(S−1,1), with k(f◦ϕ)yk2 2=2 b−akf(b−a)y/2k2 2. Since also kf◦ϕk2 S−1,1=2 b−akfk2 Sa,b , the properties i)–iii) follow from the special case a=−1,b= 1. 19 Given the action of the unitaries ∆it (3.5) in the irreducible representations Umarising from the modular data of our Borchers triple, the scalar product (3.17) of H2(0, π)is strongly reminiscent of the graph scalar product (3.3). To make this match exact, we will use a rescaled version of the graph scalar product, namely hψ, ϕi∆:= 1 2πhψ, ϕi+h∆1/2ψ, ∆1/2ϕi.(3.19) Clearly, Proposition 3.2 still holds with this equivalent scalar product. Moreover, we have the following concrete realization of D. Proposition 3.4. Consider the modular data (3.5), and denote by Dthe complex Hilbert space dom ∆1/2with scalar product (3.19). i) D=H2(S0,π)as complex Hilbert spaces. ii) The Tomita operator Sacts on H2(S0,π)by “crossing symmetry”, i.e. (Sψ)(ζ) = ψ(iπ +¯ ζ), ζ ∈S0,π .(3.20) Proof. It was shown in [23, Lemma A.1] that the real standard subspace K= ker(1 −J∆1/2)is given by K={ψ∈H2 B(S0,π) : ψπ(θ) = ψ0(θ)a.e.}. In view of the analyticity properties of ψ, this implies that Kconsists exactly of those functions ψ∈H2 B(S0,π)that satisfy ψ(iπ +ζ) = ψ(ζ), ζ ∈S0,π .(3.21) Clearly any f∈H2 B(S0,π)can be written as f=ψ+iϕ with ψ, ϕ ∈K, so that we see H2 B(S0,π) = K+iK. But as Sis an antilinear involution, with domain D=K+iK, and H2 B(S0,π) = H2(S0,π), it follows that D=H2(S0,π)as linear spaces. Also the graph scalar product (3.19) coincides with the scalar product of H2(S0,π)by Prop. 3.3 iii). This shows i). The Tomita operator Sis uniquely fixed by being an antilinear involution and Sk =kfor all k∈K. But in view of the characterization (3.21) of K, it is clear that the antilinear involution defined in (3.20) leaves Kpointwise invariant. This shows ii). Having established the connection between modular data and Hardy spaces, we now explain how composition operators appear in this setting. 20 Any Borchers triple defines a quantum field theory on R2[10], which makes this concept interesting in the context of constructing models. However, the quantum field theories arising from Borchers triples might be pathological in the sense of containing no strictly local observables, a situation that arises when the inclusions U(x)NU(x)−1⊂ N,x∈W, have trivial relative commutants. These pathological situations can however be ruled out [9, 22] when the so-called modular nuclearity condition [8, 7] holds. This condition requires that the maps Ξx,µ :N → H,Ξx,µN:= ∆µU(x)NΩ, x ∈W, 0< µ < 1 2,(3.22) are nuclear2as linear maps between the two Banach spaces (N,k · kB(H)) and H. Whereas Ξx,µ is always bounded by modular theory, it is in general not compact, so that nuclearity of (3.22) is a non-trivial requirement. To investigate the approximation numbers of Ξx,µ, we split this map as Ξx,µ :NY −→ D U(x) −→ D ∆µ −→ H,(3.23) where the first operator, defined as Y(N) := NΩ, is bounded: For any N∈ N, we have kY(N)k2 ∆=kNΩk2+h∆1/2NΩ,∆1/2NΩi=kNΩk2+hJN∗Ω, JN∗Ωi ≤2kNk2, because kJk= 1 and kN∗k=kNk. The last operator in (3.23), ∆µ, is bounded as an operator D → H (see Prop. 3.2 iii)). Lemma 3.5. Let (N, U, Ω) be a Borchers triple. Then the translations U(x), x∈W, are isometries as maps on the Hilbert space D. Proof. It is known that the commutation relations (3.4) imply that the operator ∆1/2U(x)∆−1/2,x∈W, is defined on dom ∆−1/2, and coincides there with JU(x)J=U(−x)[30, Thm. 2.3.1 f)]. We therefore find for ψ∈ D and x∈Wthe equation ∆1/2U(x)ψ= ∆1/2U(x)∆−1/2∆1/2ψ=U(−x)∆1/2ψ, and consequently kU(x)ψk2 ∆=kU(x)ψk2+k∆1/2U(x)ψk2 =kU(x)ψk2+kU(−x)∆1/2ψk2=kψk2 ∆, 2The condition that a linear map Xbetween two Banach spaces is nuclear is slightly weaker than Xhaving summable approximation numbers [32]. 21 where we have used that U(x)is unitary on H. This shows that U(x)is an isometry on D. Note that U(x)is (except for trivial cases) not unitary because it does not have full range. The product of the last two operators in the split (3.23), Dx,µ :D → H, Dx,µ = ∆µU(x),(3.24) can however be compact (and even have approximation number that go to zero quite fast), analogously to the situation encountered in Thm 2.3. To obtain estimates on the approximation numbers an(Dx,µ), one splits Hand Dinto irreducible subspaces of U. Then each subspace takes the form H=L2(R, dθ),D=H2(S0,π), and Um(x) : D → D acts by multiplication with the (analytic continuation of the) weight wm,x (3.6), depending on the representation type m. (Note that the analytically continued weight functions wm,x ∈H∞(S0,π)(3.6) are bounded and inner, for any mand x.) Explicitly, the operator D(m) x,µ =Dx,µ then takes the concrete form of a “weighted restriction operator”, 0< µ < π,x∈W, D(m) x,µ :H2(S0,π)→L2(R),(3.25) (D(m) x,µ ψ)(θ) := wm,x(θ+iµ)·ψ(θ+iµ).(3.26) This observation warrants a more systematic analysis of weighted restriction operators on Hardy spaces on strips. Before we enter into this analysis in the next section, let us comment on the relation between the operators D(m) x,µ and Ξx,µ (3.22). Estimates on the approximation numbers of D(m) x,µ do not imply corresponding estimates on the maps Ξx,µ (3.22). To establish bounds on the an(Ξx,µ), one also has to take into account the multiplicities occurring in the decomposition of Uinto irreducibles. But for typical examples of Borchers triples, this analysis involves basically only (symmetrized) tensor powers of operators of the form D(m) x,µ [1]. For this reason, it is of interest to determine strong decay properties of the approximation numbers of D(m) x,µ . 3.2 Weighted restriction operators on H2(S) As a slight generalization of what appeared before, we consider here the following setting: Let S⊂Cbe a strip domain, which will be fixed in the following. To define the operators we want to study, we take a narrower strip ˜ S⊂S, such that the closure of the smaller strip is contained in the 22 larger one, and a weight function w∈H∞(S). We are then interested in the mappings f7→ (w·f)|˜ S, considered as operators H2(S)→H2(˜ S). For simplicity, we will always assume that both Sand ˜ Sare symmetric around the real axis, i.e. S=S−b,b for some b > 0and ˜ S=λSfor some 0< λ < 1. We then define Rw,λ :H2(S)→H2(λS), Rw,λf:= (w·f)|λS.(3.27) As a limiting case as λ→0, we also define Rw,0:H2(S)→L2(R), Rw,0:= (w·f)0 (3.28) as the restriction of wf to the real line. It is clear from Prop. 3.3 i)and the form of the norm (3.9) that the restriction maps R1,λ,0≤λ < 1, with trivial weight w= 1 are bounded. Since restriction of f∈H2(S)to λ′Sis the same as first restricting fto λS,λ > λ′, and then to λ′S, we find that there is a constant csuch that an(Rw,λ′)≤c an(Rw,λ),0≤λ′≤λ≤1, n ∈N.(3.29) In particular, the operators Rw,0mapping to L2(R)(3.28) can be estimated in terms of the Rw,λ,λ > 0. The latter operators map between Hardy spaces and can be reformulated as composition operators as follows. Let 0< λ < 1and Lλ:H2(λS)→H2(S),(Lλf)(z) := √λf(λz).(3.30) Taking into account that the Riemann maps τλfor λSand τfor Sare related by τλ=λτ, it follows that Lλis unitary. Furthermore, the product λ−1/2LλR1,λ is easily seen to be the composition operator Cλon H2(S)with linear symbol z7→ λz. This shows that the restriction operators R1,λ are unitarily similar to composition operators. Furthermore, we note that Cλ (and thus R1,λ) are not compact. This is so because on H2(S)there exist no compact composition operators at all [36]. The weighted operators Rw,λ can however be compact, depending on the weight w, similar to the example in Thm. 2.3. For the following analysis of the approximation numbers of Rw,λ, we recall that H2(S)is a reproducing kernel Hilbert space. Its kernel function KSis related to the well-known Szegö kernel [35] KD(z, z′) = (1 −zz′)−1of H2(D)by KS(ζ, ζ′) = q(τ−1)′(ζ)·KD(τ−1(ζ), τ−1(ζ′)) ·p(τ−1)′(ζ′).(3.31) 23 To compute this explicitly for the strip S=S−b,b, we insert (3.14) and get KS(ζ, ζ′) = π 4b 1 cosh π(¯ ζ−ζ′) 4b .(3.32) We also recall that given any orthonormal basis {ψn}nof H2(S), we have Pnψn(ζ)ψn(ζ′) = KS(ζ, ζ′). So, in particular, X n|ψn(ζ)|2=π 4b 1 cos πIm(ζ) 2b .(3.33) Proposition 3.6. Let w∈H∞(S)and 0≤λ < 1. i) If w|λS∈H2(λS)(for λ > 0) or w0∈L2(R)(for λ= 0), then Rw,λ is Hilbert-Schmidt, with Hilbert-Schmidt norm kRw,λk2=sπ 4bcos πλ 2·kwkλS, λ > 0,kRw,0k2=rπ 4b·kwk2. (3.34) ii) If wis non-vanishing and rapidly decreasing in the sense that for any k∈N, sup θ∈R −λb≤µ≤λb |w(θ+iµ)|(1 + θ2)k<∞,(3.35) then the approximation numbers of Rw,λ′satisfy for any N∈N sup n∈NnNan(Rw,λ)<∞.(3.36) iii) If w(θ)→c,c6= 0, as θ→ ∞ or θ→ −∞, then Rw,λ is not compact. Proof. i) Let {ψn}nbe some orthonormal basis of H2(S), and 0< λ < 1. Then, using (3.33), X nkRw,λψnk2 λS=1 2πX nZR dθ |w−λb(θ)|2|ψn,−λb(θ)|2+|wλb(θ)|2|ψn,λb(θ)|2 =1 8bZR dθ |w(θ−iλb)|2 cos πλ 2 +|w(θ+iλb)|2 cos πλ 2! =π 4bcos πλ 2·kwk2 λS. 24 Since kRw,λk2 2= Tr(R∗ w,λRw,λ), this finishes the proof for λ > 0. The argument for λ= 0 is analogous. ii) By (3.29), it is sufficient to show the claim for λ > 0. Let k∈N. We find intermediate strip regions S⊃λ1S⊃λ2S2⊃... ⊃λk−1S⊃λSsuch that in each inclusion, the closure of the smaller strip is contained in the larger strip. In view of the assumption on w, we may furthermore write our weight as a product w=w1·w2···wk, with w1, ..., wk∈H2(S)(We can take wj:= w1/k for j= 1, ..., k.) Thus our operator can be written as Rw,λ =Rλ/λk−1,wk·Rλk−1/λk−2,wk−1···Rλ1,w1. By part i), each of the kfactors is Hilbert-Schmidt. That is, Rw,λ can be written as a product of an arbitrary number of Hilbert-Schmidt operators. This implies the claim by standard estimates on approximation numbers [32]. iii) We consider the case that w(θ)→c6= 0 as θ→+∞, the opposite limit is analogous. By (3.29), it is sufficient to consider the case λ= 0. For non-zero f∈H2(S), we consider the sequence fn(ζ) := f(ζ−n). To show that Rw,0is not compact, we show that {Rw,0fn}nhas no convergent subsequence. In fact, by dominated convergence we have kRw,0fnk2 2=ZR dθ |w(θ)|2|f(θ−n)|2→c2kf0k2 2. Since c6= 0 and f6= 0, this limit is non-zero, i.e. kRw,0fnk2≥c0>0for sufficiently large n. On the other hand, we have |hRw,0fn1, Rw,0fn2i2| ≤ kwk2 ∞ZR dθ |f(θ)|·|f(θ+n1−n2)|, and this converges to 0for n1−n2→ ∞ by the falloff properties of Hardy space functions. Thus for large n1,n2,|n1−n2|, the vectors Rw,0fn1,Rw,0fn1 have approximately identical non-zero length and are approximately orthogonal to each other. Thus {Rw,0fn}ncan have no convergent subsequence. The situations described in item ii) and iii) of this proposition fit to the weights appearing in the massive and massless irreducible Poincaré representations, introduced in the previous section. To see this, we need to translate the weights wm,x ∈H∞(S0,π)(3.6) to the symmetric strip S−π/2,π/2by shifting their argument ζ→ζ+iπ 2. In the massive case m > 0, this results in the weight (denoted by the same symbol) wm,x(ζ) = em(−x+eζ+x−e−ζ), ζ ∈S−π/2,π/2.(3.37) 25 Since γλγλ=γλ2, we get γλgϕ =γλ2g, and an explicit formula for wis (4.6) w(z) = exp h−1 + z 1−zλ2iexp h−1−z 1 + zλ2i=: w′ 1(z)w′ −1(z). We now apply Lemma 2.6. To that effect, we must make a good choice of the uj’s. As in [27] for lens maps, we choose uj= 1 −e−jε where ε > 0 has to be adjusted, and vj=ϕ(uj). We know from [26, Lemma 6.5] that Iv≤exp(C/ε), and we have q1−u2 j≥c e−nε. Moreover, since 1 + uj 1−ujλ2 ≤2 1−ujλ2 , we see that inf 1≤j≤n|w′ 1(uj)| ≥ exp(−Cenε). And clearly inf1≤j≤n|w′ −1(uj)| ≥ e−1. 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