On truncations of Hankel and Toeplitz operators

Publicacions Matem`atiques, Vol 43 (1999), 235–250. ON TRUNCATIONS OF HANKEL AND TOEPLITZ OPERATORS Aline Bonami and Joaquim Bruna Abstract We study the boundedness properties of truncation operators acting on bounded Hankel (or Toeplitz) infinite matrices. A relation with the Lacey-Thiele theorem on the bilinear Hilbert transform is established. We also study the behaviour of the truncation operators when restricted to Hankel matrices in the Schatten classes. 1. Statement of results In this note we will be dealing with infinite matrices B=(bm,n)m,n≥0, bm,n ∈C, which we identify with linear operators on l2(N). More precisely we assume that sup n m |bm,n|2<∞, and consider Bas defined on almost finite sequences. Also we identify l2(N) with the Hardy space H2(D) of the unit disc and consider Bas a linear map from the space of polynomials to H2(D). One property we will be considering is whether Bextends to a bounded operator on the whole space, that is  m  n anbm,n 2 ≤B2 n |an|2. The matrix Bis called of Toeplitz type if bm,n =bm−nfor some sequence b∈l2(Z), and we write B=Tb. We identify bwith the L2 function b(ζ)=+∞ j=−∞ bjζjon T=∂D, called the symbol. In this case it is immediate to see that Tbcan be realized as the integral operator from H2(D) to the space of holomorphic functions on Dgiven by 236 A. Bonami, J. Bruna Tbf=C(bf), where Cis the Cauchy integral giving the projection from L2(T)toH2(D): if f=nanzn, Tb(f)(z)= 1 2πT b(ζ)f(ζ) 1−ζz dσ(ζ)=  m≥0   n≥0 anbm−n zm,z∈D. It is well known that Tbis a bounded operator on H2(D) if and only if b∈L∞(T), with the same norms. The matrix Bis called of Hankel type if bm,n =bm+nfor some sequence b∈l2(N), and we write B=Hb. In this case the symbol is the H2function b(ζ)=+∞ j=0 bjζjand Hbcan be realized as the integral operator Hbf=C(bg), where gdenotes the antiholomorphic function g(ζ)=f(ζ), (1) Hb(f)(z)= 1 2πT b(ζ)f(ζ) 1−ζz dσ(ζ)=  m≥0   n≥0 anbm+n zm,z∈D. It is well known that Hbis a bounded operator on H2(D) if and only if b∈BMOA(T), again with comparable norms, and compact if and only if b∈VMOA(T). In these cases, bis the holomorphic part of some bounded (resp. continuous) function and since the antiholomorphic part gives zero contribution in the above integral, one can assume that Hbis given by the above integral with b∈L∞(T) (resp. b∈C(T)). We consider truncations of matrices defined as follows. For real β,γ and a matrix Bas before, we let Πβ,γ(B) the matrix whose entry at position m, n is bm,n when m≥βn +γand zero otherwise. For β=1, γ= 0 we use the notation Π. For β=0,Π 0,γ(B) is just Bfollowed by a projection, and hence we will only consider β=0. For general B, boundedness of Bdoes not imply boundedness of Πβ,γ(B), for β>0. For instance, if B=Tbwith bbounded then Π(B) is also a Toeplitz operator whose symbol is Cb, the holomorphic projection of b. Since Cb is not necessarily bounded, Π(B)maybean unbounded operator. For β<0, B−Πβ,γ(B) has finite rank and hence Πβ,γ(B) is always a bounded operator. In this case the natural question is whether the operator norms Πβ,γ(B)are bounded independently of γ. This is again false, in this case a Hankel operator gives a counterexample. Indeed, if B=Hbwith b∈BMOA(T), then Π−1,N (B) are again Hankel operators with symbols SNb. Here SNb(ζ)=N j=0 bjζj, and a counterexample is given by b∈BMOA such that the SNbare not uniformly bounded in BMO. On truncations of Hankel and Toeplitz operators 237 Within special classes of operators, the truncations Πβ,γ haveagood behaviour. For instance, it is obvious that they preserve the HilbertSchmidt class S2, defined by the condition m,n |bm,n|2<∞, and more generally it can be shown that they preserve the Schatten classes Spfor p>1; a proof is included in the last section. It is a well known fact for triangular truncations, as one can see in [DS]or[GK2]. For p= 1 and the trace class S1this is no longer true (this will be shown in the last section too). It is customary to present the above situation as analogous to the behaviour of the Hilbert transform or the Cauchy projection C in the range of Lp-spaces: Π plays the role of Hilbert transform on matrices and the classes Spare, by definition ([GK1], [Po]), Lpclasses of operators. Our first result establishes that with the exception of the counterexamples above, the truncations Πβ,γ behave well when restricted to bounded Hankel or Toeplitz operators: Theorem 1. (a) If Bis a bounded (resp. compact) Hankel operator, and β=−1,Πβ,γ(B)are bounded (resp. compact) operators. Moreover their norms are uniformly bounded in γ, and locally uniformly bounded in βaway from 0and −1. (b) If Bis a bounded Toeplitz operator, and β=1,Πβ,γ(B)are bounded operators. Moreover their norms are uniformly bounded in γ, and locally uniformly bounded in βaway from 0and 1. Our proof consists in exhibiting an integral operator realizing Πβ,γ(B), closely related to the bilinear Hilbert transform, and applying the a priori estimates of the Lacey-Thiele theorem ([LT1], [LT2], [LT3]). We do not know of any “operator theory proof” of the result. In the last section we consider Hankel operators in the trace class S1. A theorem of Peller ([Pe]) describes the corresponding symbols as those in the Besov space B1. In contrast with the result above, we show that, for such B,Π(B) is not in general in the trace class, and we give a sufficient condition on the symbol for this to be the case. Let us mention that the truncation operators have specially been under consideration when β=±1. See [KP] for considerations on the norms of Π−1,γ, and [ACN] for precise estimates of the norm of Π. Let us also mention that the theorem gives a positive answer to a conjecture attributed to V. Peller. 2. Proof of the theorem We consider first the case of Hankel operators. 238 A. Bonami, J. Bruna Assume first that B=Hbis bounded, and let b∈L∞(T) as in (1), with holomorphic part j≥0bjζj.Forβ=−1 we must see that, independently of γ, (2)  m≥0  m≥nβ+γ bm+nan 2 ≤C(β)b2 ∞ n≥0 |an|2. It is enough to prove that this holds for rational β,γwith a constant C(β) which is locally bounded away from −1 and 0. Without loss of generality we may assume that β=k lwith k,l∈Z,k≥1, k=−l, and lγ ∈Z. Moreover, we may assume that both b,fare trigonometric polynomials; this is clear for f, and for bit follows from the existence of a sequence (bN) of trigonometric polynomials, uniformly bounded by b∞, whose Fourier coefficients ck(bN) equal ck(b) for |k|≤N(for instance using the de la Vall´ee-Poussin kernel). Now, for ba trigonometric polynomial, we consider an operator Ab acting on periodic functions as follows. For f∈L2(T), Abf(x) = p.v. 1 2ππ −π b(kx +lt)f(t)e−itlγ dt tan x−t 2 = p.v. 1 2ππ −π b((k+l)x−lt)f(x−t)e−iγl(x−t)dt tan t 2 . We will make use of the following result by Lacey-Thiele in [LT3]: Theorem 2. Let α∈R,α=0,=−1, and let 1<p,q≤+∞with 1 r=1 p+1 q<3 2. Then there exists a constant C(α, p, q)such that for f,gin the Schwartz class S(R)the bilinear Hilbert transform Hα(g,f)(x)=p.v. +∞ −∞ g(x+αt)f(x−t)dt t satisfies the inequality H(g,f)r≤C(α, p, q)fpgq. Moreover the constant C(α, p, q)is locally bounded as a function of α. On truncations of Hankel and Toeplitz operators 239 If C(α, p, q) denotes the best constant satisfying the above inequality, note that C(α, p, q)=C(α−1,q,p). Also note that in the limiting cases, by the boundedness properties of the Hilbert transform, C(0,p,q) is finite if 1 <p<∞,1≤q≤+∞, and C(−1,p,q) is finite if 1 <r<∞. Let us transport the Lacey-Thiele result to the periodic situation. For a couple of integers k,l,k=l, we define the bilinear conjugate transform by Hk,l(g,f)(x) = p.v. 1 2ππ −π g(kx +lt)f(t)dt tan x−t 2 , with f,g∈C∞(T). Note that Hk,l(g,f) is again periodic. We claim that there exists a constant C(k, l, p, q), with the same conditions on p,q, such that (3) Hk,l(g,f)r≤C(k, l, p, q)fpgq. To show (3), we first observe that, writing the periodic function f as a sum of three functions which are periodizations of functions whose support is contained in an interval of length πand using translation invariance, we may assume that fis supported in [−π/2,π/2]. Consider now Hk,l(g,f)(x) for |x|≤π. Then |x−t|≤3π 2, and (2 tan x−t 2)−1can be written (x−t)−1+a(x−t) where ais a smooth function. The term with agives in Hk,l a term which is pointwise bounded by π/2 −π/2 |g(kx +lt)||f(t)|dt, and which, by the continuous Minkowski inequality, has finite Lqnorm bounded by f1gq. Since r≤q, this term satisfies the required estimate in (3). It remains to consider p.v. π/2 −π/2 g(kx +lt)f(t)1 x−tdt. For |x|≤π, this equals p.v. +∞ −∞ G(kx +lt)F(t)1 x−tdt, 240 A. Bonami, J. Bruna where F(t)=f(t)in|t|≤π/2 and 0 elsewhere, and G=gφ,φbeing some cutoff function which is 1 for |ξ|≤|k|π+|l|π 2. But this last expression is just Hα(˜ G, F)(x), with α=−l k+land ˜ G(ξ)=G((k+l)ξ), and hence (3) follows with C(k, l, p, q)≤C0+C0|k+l|−1 qC(−l l+k,p,q) for some constant C0. Applying (3) with g=b,q=∞,p=r=2,we thus obtain Abf2≤C(β)b∞f2, for f,gtrigonometric polynomials. Let us now check how Abworks in the usual basis of exponentials. It is enough to consider b(t)=eijt and f(t)=eiht for i,h∈Z. Using that p.v. 1 2ππ −π eijt dt tan x−t 2 = sign(j)eijx, we find in this case Abf(x) = sign(jl +h−lγ)ei(jk+jl+h−lγ)x. Hence for b=bjeijt,f(t)=eiht we obtain Abf(x)= j bjsign(jl +h−lγ)ei(jk+jl+h−lγ)x. To obtain the desired truncation we consider only frequencies of type h= −n(l+k), n∈N, and make the substitution j=m+n.Thuson f(t)=nane−i(l+k)nt, we find that Abf(x)eilγx = n∈N m∈Z anbm+nsign(l(m−nβ −γ))eim(k+l)x. Finally we consider the projection on the closed subspace spanned by eim(k+l)xfor m∈N. It follows that the matrix whose entries are bm+nsign(m−nβ −γ) acts boundedly on l2(N), with constant dominated by C(β)b∞, proving (2). We now show that a Hankel operator Bis bounded if Π(B) is. Indeed, if Π(B) is bounded so is Π(B), hence its adjoint (Π(B))∗, and then so is Bbecause it differs from Π(B) + (Π(B))∗by a diagonal matrix with bounded entries b2n. On truncations of Hankel and Toeplitz operators 241 For Toeplitz operators we argue in a similar way. As before, without loss of generality we may assume that β=k lwith k,l∈Z,k≥1, k=l, and lγ ∈Z. The auxiliary operator Abis defined now by Abf(x)= 1 2πp.v. π −π b(−kx +lt)f(t)e−itlγ dt tan x−t 2 =1 2πp.v. π −π b((l−k)x−lt)f(t−x))eiγl(t−x)dt tan t 2 , and we use (3) with α=l k−l. We make act Abon f(t)=ei(l−k)nt,n∈N, obtaining now Abf(x)eilγx = m∈Z bm−nsign(l(m−nβ −γ))eim(l−k)x, and the proof is finished as before. If B=Hbis compact then bcan be assumed in C(T). If bnare polynomials converging uniformly to b, the estimate above implies that Πβ,γ(B) is the limit of the finite rank operators Πβ,γ(Hbn) and hence it is compact. It is worth mentioning, though, that parts (a) and (b) are equivalent. This can be seen as follows. Let PN(x0,...,x N,...)=(x0,... ,x N,0,...) the standard coordinate projection, and let J(x0,... ,x N)=(xN,... ,x 0); note that supNPNBPN=Bfor every matrix Band that for a N×Nmatrix B,A=BJ is Toeplitz (Hankel) if and only if Bis Hankel (Toeplitz). Also note that Πβ,γ(B)J=Π β,γ(A), with β=−β, γ=γ−Nβ. Assume that (a) holds, let Tbe a bounded Toeplitz operator with symbol φ, and let TN=PNTPN. Then TNJis a Hankel matrix which equals PNΓNPN, where ΓNis the Hankel operator on H2having ζNφas a symbol. Hence, Πβ,γ(ΓN)≤c(β)ζNφ∞= c(β)φ∞for every γand so PNΠβ,γ(ΓN)PNJ=PNΠβ,γ(ΓN)PN≤c(β)φ∞. But PNΠβ,γ(ΓN)PNJ=Π β,γ(PNΓNPN)J=Π β,γ(TNJ)J=Π β,γ(TN), with β=−β,γ=γ−Nβ. Since γis arbitrary, so is γ, and letting N→+∞we conclude that Πβ,γ(T)≤c(β)φ∞,β=−βfor all γ, proving (b) (we thank the referee for this observation). The following is an easy corollary of the theorem: Corollary 3. Let Bbe a compact Hankel operator. Then Bis the limit in the operator norm of its upper triangular truncations Bn= B−Πβ,n(B)if β=−1. 242 A. Bonami, J. Bruna We have seen that the theorem of Lacey-Thiele implies its periodic version and also Theorem 1. We shall show that some converse is also true. We shall also show that the consideration of the operators Πβ,γ gives some indication on the norm of the bilinear Hilbert transform. The first point to note is that, conversely, Lacey-Thiele theorem follows from its periodic version. Indeed, in proving the estimate of Lacey-Thiele theorem, it is enough to consider g,fsupported in [−/, /] (otherwise take f(Nx), g(Nx)). We want to bound ∞ −∞  p.v. ∞ −∞ g((k+l)x−lt)f(x−t)dt t r dx. The principal value is supported in |x|≤(1+|l|) |k+l|=η. We call F,Gthe periodized of f,g. We claim that for |x|≤ηsmall enough, p.v. ∞ −∞ g((k+l)x−lt)f(x−t)dt t= p.v. π −π G((k+l)x−lt)F(x−t)dt t. Indeed, the last expression is nonzero only when there exists tsuch that (k+l)x−lt ∈2mπ +[−/, /], x−t∈2nπ +[−/, /] for some integers m,n, which is equivalent to x∈2π(m−ln)/(k+l)+[−η,η]. If η≤π 2|k+l|, only m=n= 0 contribute, establishing the claim. Now, changing 1/t into 1/(2 tan t 2) as before shows that Lacey-Thiele theorem follows from (3). Assume we know that Π is a bounded operator when restricted to Hankel operators. Clearly Π1,−1is also a bounded operator when restricted to Hankel operators. We claim that we have an a priori estimate for the periodic bilinear Hilbert transform H(b, f) with band ftrigonometric polynomials. More precisely, we will prove that H(b, f)2≤Cb∞f2, with a constant Cwhich only depends on the norm of Π. Indeed,from the above computation, it follows that, whenever f(t)=nane−2int, H(b, f)(x)= n∈N m∈Z anbm+nsign(m−n)e2imx. So H(b, f)(x)=−b(2x)f(x)+Π(Hb)f(x). When f(t)=nane−i(2n+1)t, this formula has to be changed into H(b, f)(x)=−b(2x)f(x)e2ix + Π1,−1(Hb)f(x). We find the required inequality for H(b, f)2when fis a Taylor polynomial, cutting it into even and odd frequencies. On truncations of Hankel and Toeplitz operators 243 We conclude for fa trigonometric polynomial taking real and imaginary parts. So the boundedness of Π implies the existence of the constant C(−1 2,2,∞), or, by duality (as shown in [LT3]), the existence of the constant C(1,2,2), the original Calder´on conjecture. Finally, for some values of p,q, we give some information on the behavior of the constant C(α, p, q) when αtends to 0, −1, or ∞.Ifwe consider Toeplitz operators and take l=k−1, we see that C(k,2,∞) cannot remain bounded when ktends to +∞, otherwise Π(Tb) would be bounded for Tbbounded. The same is valid for ktending to −∞.By duality, we find that C(α, 2,2) cannot remain bounded when αtends to −1. These properties could have been directly obtained from the consideration of the bilinear Hilbert transform. Using Hankel operators we find a bound below for these constants. Lemma 4. C(±N,2,∞)≥cln N. Let us take k=−l+ 1, so that β=−1+1 land α=−l. Then it is possible to find γso that Π−1,k =Π β,γ. Using the fact that the operator giving the partial sum of order Nof the Fourier series of a function in BMO has norm equivalent to ln N, we get that the norm of Π−1,k is equivalent to ln k. Since it is bounded by C(α, 2,∞), we get the estimate of the lemma for negative integers. The choice k=−l−1 gives the other case. We also point out that in an analogous way, the Lacey-Thiele theorem can be used to obtain a version of our theorem for “continuous” Hankel operators, i.e. the bounded integral operators defined in L2(0,+∞)of type Hbf(x)=∞ 0 b(x+y)f(y)dy. For instance, if Hbis bounded in L2(0,∞), so is Π(Hb)f(x)=∞ 0 b(x+y) sign(x−y)f(y)dy. We finish this section by posing two open problems suggested by the proof. The first concerns finite N×NHankel or Toeplitz matrices; for fixed β,γ, let cNdenote the norm of the truncation operator Πβ,γ when restricted to N×NHankel matrices. Which is the behaviour of cNas N→∞? Note that our result does not apply to such Hankel matrices. 250 A. Bonami, J. Bruna [Po] S. C. Power,“Hankel operators on Hilbert space,” Research Notes in Mathematics 64, Pitman, London, 1982. [Ru] J. L. Rubio de Francia, Martingales and integral transforms of Banach space valued functions, in “Probability and Banach spaces,” (J. Bastero & M. San Miguel, eds.), Lecture Notes in Math. 1221, Springer Verlag, Berlin-New York, 1986, pp. 195–222. Aline Bonami: Universit´e d’Orl´eans B.P. 6759 45067 Orl´eans Cedex 2 FRANCE e-mail: [email protected] Joaquim Bruna: Departament de Matem`atiques Universitat Aut`onoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN e-mail: [email protected] Primera versi´o rebuda el 2 de juny de 1998, darrera versi´o rebuda el 8 de mar¸c de 1999