ON TRUNCATIONS OF HANKEL AND TOEPLITZ OPERATORS

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 l(N). More precisely we assume that sup n ∑


Statement of results
In this note we will be dealing with infinite matrices B = (b m,n ) m,n≥0 , b m,n ∈ C, which we identify with linear operators on l 2 (N).More precisely we assume that sup and consider B as defined on almost finite sequences.Also we identify l 2 (N) with the Hardy space H 2 (D) of the unit disc and consider B as a linear map from the space of polynomials to H 2 (D).One property we will be considering is whether B extends to a bounded operator on the whole space, that is (1) It is well known that H b is a bounded operator on H 2 (D) if and only if b ∈ BMOA(T), again with comparable norms, and compact if and only if b ∈ VMOA(T).In these cases, b is 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 H b is 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 B as before, we let Π β,γ (B) the matrix whose entry at position m, n is b m,n when m ≥ βn + γ and zero otherwise.For β = 1, γ = 0 we use the notation Π.For β = 0, Π 0,γ (B) is just B followed by a projection, and hence we will only consider β = 0.
For general B, boundedness of B does not imply boundedness of Π β,γ (B), for β > 0. For instance, if B = T b with b bounded 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) may be an unbounded operator.For β < 0, B − Π β,γ (B) has finite rank and hence Π β,γ (B) is always a bounded operator.In this case the natural question is whether the operator norms Π β,γ (B) are bounded independently of γ.This is again false, in this case a Hankel operator gives a counterexample.Within special classes of operators, the truncations Π β,γ have a good behaviour.For instance, it is obvious that they preserve the Hilbert-Schmidt class S 2 , defined by the condition m,n |b m,n | 2 < ∞, and more generally it can be shown that they preserve the Schatten classes S p for 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 S 1 this 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 L p -spaces: Π plays the role of Hilbert transform on matrices and the classes S p are, by definition ( [GK1], [Po]), L p classes of operators.
Our first result establishes that with the exception of the counterexamples above, the truncations Π β,γ behave well when restricted to bounded Hankel or Toeplitz operators: Our proof consists in exhibiting an integral operator realizing Π β,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 S 1 .A theorem of Peller ([Pe]) describes the corresponding symbols as those in the Besov space B 1 .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.

Proof of the theorem
We consider first the case of Hankel operators.
Assume first that B = H b is bounded, and let b ∈ L ∞ (T) as in (1), with holomorphic part j≥0 b j ζ j .For β = −1 we must see that, independently of γ, (2) It is enough to prove that this holds for rational β, γ with a constant C(β) which is locally bounded away from −1 and 0. Without loss of generality we may assume that β = k l with k, l ∈ Z, k ≥ 1, k = −l, and lγ ∈ Z.Moreover, we may assume that both b, f are trigonometric polynomials; this is clear for f , and for b it follows from the existence of a sequence (b N ) of trigonometric polynomials, uniformly bounded by b ∞ , whose Fourier coefficients c k (b N ) equal c k (b) for |k| ≤ N (for instance using the de la Vallée-Poussin kernel).Now, for b a trigonometric polynomial, we consider an operator A b acting on periodic functions as follows.For f ∈ L 2 (T), We will make use of the following result by Lacey-Thiele in [LT3]: Then there exists a constant C(α, p, q) such that for f , g in the Schwartz class S(R) the bilinear Hilbert transform Moreover the constant C(α, p, q) is locally bounded as a function of α.
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 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 ) is again periodic.We claim that there exists a constant C(k, l, p, q), with the same conditions on p, q, such that 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 π and using translation invariance, we may assume that f is supported in where a is a smooth function.The term with a gives in H k,l a term which is pointwise bounded by and which, by the continuous Minkowski inequality, has finite L q norm bounded by f 1 g q .Since r ≤ q, this term satisfies the required estimate in (3).It remains to consider where F (t) = f (t) in |t| ≤ π/2 and 0 elsewhere, and G = gφ, φ being some cutoff function which is 1 for |ξ| ≤ |k|π + |l| π 2 .But this last expression is just H α ( G, F )(x), with α = − l k+l and G(ξ) = G((k + l)ξ), and hence (3) follows with for some constant C 0 .Applying (3) with g = b, q = ∞, p = r = 2, we thus obtain for f , g trigonometric polynomials.Let us now check how A b works in the usual basis of exponentials.It is enough to consider b(t) = e ijt and f (t) = e iht for i, h ∈ Z.Using that we find in this case To obtain the desired truncation we consider only frequencies of type h = −n(l + k), n ∈ N, and make the substitution j = m + n.Thus on f (t) = n a n e −i(l+k)nt , we find that k+l)x .
Finally we consider the projection on the closed subspace spanned by e im (k+l)x for m ∈ N. It follows that the matrix whose entries are b m+n sign(m − nβ − γ) acts boundedly on l 2 (N), with constant dominated by C(β) b ∞ , proving (2).
We now show that a Hankel operator B is bounded if Π(B) is.Indeed, if Π(B) is bounded so is Π(B), hence its adjoint (Π(B)) * , and then so is B because it differs from Π(B) + (Π(B)) * by a diagonal matrix with bounded entries b 2n .
For Toeplitz operators we argue in a similar way.As before, without loss of generality we may assume that β = k l with k, l ∈ Z, k ≥ 1, k = l, and lγ ∈ Z.The auxiliary operator A b is defined now by and we use (3) with α = l k−l .We make act and the proof is finished as before.
If B = H b is compact then b can be assumed in C(T).If b n are polynomials converging uniformly to b, the estimate above implies that Π β,γ (B) is the limit of the finite rank operators Π β,γ (H bn ) and hence it is compact.
It is worth mentioning, though, that parts (a) and (b) are equivalent.This can be seen as follows.Let P N (x 0 , . .., x N , . ..)=(x 0 , . . ., x N , 0, . . . ) the standard coordinate projection, and let J(x 0 , . . ., x N )=(x N , . . ., x 0 ); note that sup N P N BP N = B for every matrix B and that for a Assume that (a) holds, let T be a bounded Toeplitz operator with symbol φ, and let T N = P N T P N .Then T N J is a Hankel matrix which equals P N Γ N P N , where Γ N is the Hankel operator on for every γ and so Since γ is arbitrary, so is γ , and letting N → +∞ we conclude that Π β ,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 B be a compact Hankel operator.Then B is the limit in the operator norm of its upper triangular truncations 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 Π β,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, f supported in [− , ] (otherwise take f (Nx), g(Nx)).We want to bound The principal value is supported in |x| ≤ (1+|l|) |k+l| = η.We call F , G the periodized of f , g.We claim that for |x| ≤ η small enough, p.v.
Indeed, the last expression is nonzero only when there exists t such that (k 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 Π 1,−1 is 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 b and f trigonometric polynomials.More precisely, we will prove that with a constant C which only depends on the norm of Π. Indeed,from the above computation, it follows that, whenever . We find the required inequality for H(b, f ) 2 when f is a Taylor polynomial, cutting it into even and odd frequencies.
We conclude for f a 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 Calderón conjecture.
Finally, for some values of p, q, we give some information on the behavior of the constant C(α, p, q) when α tends to 0, −1, or ∞.If we consider Toeplitz operators and take l = k − 1, we see that C(k, 2, ∞) cannot remain bounded when k tends to +∞, otherwise Π(T b ) would be bounded for T b bounded.The same is valid for k tending to −∞.By duality, we find that C(α, 2, 2) cannot remain bounded when α tends 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.
Let us take k = −l + 1, so that β = −1 + 1 l and α = −l.Then it is possible to find γ so that Π −1,k = Π β,γ .Using the fact that the operator giving the partial sum of order N of 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 L 2 (0, +∞) of type We finish this section by posing two open problems suggested by the proof.
The first concerns finite N × N Hankel or Toeplitz matrices; for fixed β, γ, let c N denote the norm of the truncation operator Π β,γ when restricted to N × N Hankel matrices.Which is the behaviour of c N as N → ∞?Note that our result does not apply to such Hankel matrices.
The second concerns weighted Cauchy integral operators: let b(ζ, z) bounded and consider For which symbols b is T b bounded in H 2 (D)? (in our proof we used symbols of type b(

Hankel operators in the trace class
We shall first prove that the Schatten classes S p are globally preserved by the truncation operators Π β,γ when 1 < p < ∞, as said in the introduction.Our proof follows from well known arguments.We give it for completeness.
Remember that, for p > 0, given a compact operator T on a Hilbert space, we say that T belongs to the Schatten class S p if j s p j < ∞, where s j = {inf T − E : rank E ≤ j}.
We will make use of two facts.The first one is that the Schatten classes are closed ideals in the algebra of bounded operators (see [GK1]).The second one is that the Hilbert transform maps also continuously L p into itself, for 1 < p < ∞, when extended to S p -valued functions (see [Ru]).
For t ∈ T, let us define R t as the diagonal matrix with diagonal entries given by e int , n = 0, 1, 2, . . . .Then an elementary computation shows that, when ms+nt) .Let us now go back to the truncation operators Π β,γ , for which we shall prove that they are uniformly bounded in the Schatten classes S p .Without loss of generality we may assume as before that β = k l with k, l ∈ Z, k ≥ 1, and lγ ∈ Z.For B ∈ S p , let us consider the S p -valued function whose entries are given by b m,n e i(ml−kn−γ)t .Then the Cauchy projection of B(t), extended to S p -valued functions, is equal to We know that this extension of the Cauchy projection is uniformly bounded in L p .Moreover, it is easy to see that the norm in S p of B(t) does not depend of t, and is equal to B Sp .The same is valid for CB(t), for which we find Π β,γ (B) Sp .The conclusion follows at once.
Let us now consider trace class operators, that is the case p = 1.
Theorem 5.The operator Π is not bounded in S 1 , even when restricted to Hankel operators.
Proof: Let us consider the Hankel operators H r with symbol b r (z) = 1−r 2 1−rz .The definition shows that H r f (z) = f (r)b r (z), that is, H r is the projection on the subspace generated by the function b r .In particular H r is of rank 1, and of norm 1.Since We set, for 0 < r < 1, Since the quotient |b r 2 /b r | is bounded above and below and S 1 is an ideal, the norms in the trace class of Π(H r ) and U r 2 are equivalent.So the unboundedness of Π for Hankel operators in the trace class follows from the following lemma.
Lemma 6.The norm of U r in the trace class S 1 is equivalent to ln 1 1−r .
Proof: For the lower bound we adapt a method which is employed to find necessary conditions on symbols of Hankel operators.L 2 (dV ) denotes the ordinary Lebesgue space on the unit disc D endowed with Lebesgue measure dV .We shall use the fact that for A a bounded operator from H 2 into L 2 (dV ), since S 1 is an ideal, the norm in the trace class of the operator AU r A * is bounded by A 2 U r S1 .We combine this with the fact that the trace class norm is given by ( 4) the supremum being taken over all orthonormal systems.Hence, if (e k ) K k=1 is an orthonormal system in L 2 (dV ), then For k = 1, 2, . . ., K, let us set w k = re iθ k , with θ k = k(1 − r), and K the biggest integer such that K(1 − r) ≤ π/4.It is clear that K (1 − r) −1 .Now, let A be the map given by where χ D k denotes the characteristic function of the disc D k which has center w k and radius η(1 − r).For 0 < η < 1/2 fixed, small enough, the discs D k are disjoint.It follows from this fact and from the mean value inequality that So A is a bounded operator from H 2 into L 2 (dV ), with norm bounded independently of r.We set 1−w k z .It follows from this choice that there is some constant C, which does not depend on r, such that We now use the reproducing property of the Szegö kernel to replace this last inequality by We observe now that |1 − rw k | (1 − r 2 )k.This proves the bound from below.Now we will prove the upper bound.Let us denote by M r the operator of multiplication by the function ϕ , and P r the operator defined on H 2 by P r f (z) = f (rz).We claim that U r and M r P r M r have equivalent norms in the trace class.Indeed, and multiplication by a function which is bounded below and above preserves the norm, up to a constant.Let us consider the operator V r , defined in the whole L 2 (T) by the same expression M r P r M r , where M r denotes again multiplication by the same function ϕ, restricted to the unit circle, and now P r denotes convolution with the Poisson kernel.
Clearly the operator V r is an extension of U r and so it is sufficient to prove that V r S1 ≤ C ln 1 1 − r as an operator in L 2 (T).Now the adjoint of M r as an operator on , which has the same modulus as ϕ(e it ).It follows that V r and M * r P r M r have same norm.This last operator is self adjoint and positive since P r is.Now, recall that for positive operators the trace class norm is of course given by the trace, and this can be computed in any orthormal basis (this follows from (4) too).Hence, where f k (t) = e −ikt .Now, M r f k (t) = ϕ(t)e −ikt = l φ(l)e i(l−k)t , P r M r f k (t) = l φ(l)r |l| e i(l−k)t and therefore by Plancherel formula, Summing first in k, and using Plancherel formula again, we get This finishes the proof of the lemma and of Theorem 5.
Let us recall that the operator H b is in the Schatten class S 1 if and only if the symbol b belongs to the Besov space B 1 , where B 1 is the space of holomorphic functions b in the unit disc such that D |b (z)| dV (z) < ∞.We give now a sufficient condition for Π(H b ) to be in S 1 which is of the same type, but stronger.Proof: For b(z) = n≥1 a n z 3 n , Π(H b ) is clearly in the trace class if the sum of the absolute values of the entries of its matrix is bounded, that is n 3 n |a n | < ∞.It is easy to find sequences (a n ) for which this quantity is finite but, for a given ρ, the integral above is not.
The matrix B is called of Toeplitz type if b m,n = b m−n for some sequence b ∈ l 2 (Z), and we write B = T b .We identify b with the L 2 function b(ζ) = +∞ j=−∞ b j ζ j on T = ∂D, called the symbol.In this case it is immediate to see that T b can be realized as the integral operator from H 2 (D) to the space of holomorphic functions on D given by T b f = C(bf ), where C is the Cauchy integral giving the projection from L 2 (T) to H 2 (D): if f = n a n z n , m , z ∈ D. It is well known that T b is a bounded operator on H 2 (D) if and only if b ∈ L ∞ (T), with the same norms.The matrix B is called of Hankel type if b m,n = b m+n for some sequence b ∈ l 2 (N), and we write B = H b .In this case the symbol is the H 2 function b(ζ) = +∞ j=0 b j ζ j and H b can be realized as the integral operator H b f = C(bg), where g denotes the antiholomorphic function g(ζ) = f (ζ), are again Hankel operators with symbols S N b.Here S N b(ζ) = N j=0 b j ζ j , and a counterexample is given by b ∈ BMOA such that the S N b are not uniformly bounded in BMO.