THE MULTIPLICATIVE HILBERT MATRIX

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1 THE MULTIPLICATIVE HILBERT MATRIX OLE FREDRIK BREVIG, KARL-MIKAEL PERFEKT, KRISTIAN SEIP, ARISTOMENIS G. SISKAKIS, AND DRAGAN VUKOTIĆ arxiv: v1 [math.fa] 26 Nov 214 ABSTRACT. It is observed that the infinite matrix with entries mn logmn)) 1 for m,n 2 appears as the matrix of the integral operator Hf s) := + f w)ζw+ s) 1)d w with respect to the basis n s ) n 2 ; here ζs) is the Riemann zeta function and H is defined on the Hilbert space H 2 of Dirichlet series vanishing at + and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson s terminology or, alternatively, it can be viewed as a bona fide small) Hankel operator on the infinite-dimensional torus T. By analogy with the standard integral representation of the classical Hilbert matrix, this matrix is referred to as the multiplicative Hilbert matrix. It is shown that its norm equals π and that it has a purely continuous spectrum which is the interval [,π]; these results are in agreement with known facts about the classical Hilbert matrix. It is shown that the matrix m 1/p n p 1)/p logmn)) 1 has norm π/sinπ/p) when acting on l p for 1< p <. However, the multiplicative Hilbert matrix fails to define a bounded operator on H p for p 2, where H p are H p spaces of Dirichlet series. It remains an interesting problem to decide whether the analytic symbol n 2logn) 1 n s of the multiplicative Hilbert matrix arises as the Riesz projection of a bounded function on the infinite-dimensional torus T. 1. INTRODUCTION The classical Hilbert matrix ) 1 A := m+ n+ 1 m,n is the prime example of an infinite Hankel matrix, i.e., a matrix whose entries a m,n only depend on the sum m+n. The Hilbert matrix can be viewed as the matrix of the integral operator 1) H a f z) := 1 f t)1 zt) 1 d t with respect to the standard basis z n ) n for the Hardy space H 2 D). This representation was first used by Magnus [13] who found that the Hilbert matrix has no eigenvalues and that its continuous spectrum is [, π]. The purpose of this paper is to identify and study a multiplicative analogue of A. This means that we seek an infinite matrix with entries a m,n that depend only on the product mn 21 Mathematics Subject Classification. 11M99, 42B3, 47B35, 47G1. The research of the first and the third author is supported by Grant of the Research Council of Norway. The fifth author s work was supported by MINECO grant MTM C2-2, Spain. This research was initiated while the third author served as a one month visitor in the Postgraduate Excellence Program at the Department of Mathematics at Universidad Autónoma de Madrid. 1

2 2 O. F. BREVIG, K.-M. PERFEKT, K. SEIP, A. SISKAKIS, AND D. VUKOTIĆ and with properties that parallel those of A. Our starting point is the multiplicative counterpart to 1) which we have found to be the integral operator 2) Hf s) := + f w)ζw+ s) 1)d w acting on Dirichlet series f s) = a n n s. Here ζs) denotes the Riemann zeta function, and we assume that f is in H 2, which means that f 2 := a H 2 n 2 <. By the Cauchy Schwarz inequality, every f in H 2 represents an analytic function in the halfplane σ=re s>. As is readily seen, the reproducing kernel K w of H 2 is K ws)=ζs+w) 1. This implies that 3) Hf, g H 2 = f w)g w)d w when f and g are Dirichlet polynomials. Since arc length measure on the half-line, + ) is a Carleson measure for H 2 see [15, Theorem 4]), we get that 3) in fact holds for arbitrary functions f and g in H 2, and hence H is well defined and bounded on H 2. Taking into account that every f in H 2 is analytic when σ>, we find that f, Hf H 2 = if and only if f. Hence 3) also implies that H is a strictly positive operator. Now an explicit computation of the integral on the right-hand side of 2) shows that the matrix of H with respect to the orthonormal basis n s ) n 2 is ) 1 M :=. mn logmn) m,n 2 We will refer to this matrix as the multiplicative Hilbert matrix. We will be interested in understanding M as an operator on l 2 = l 2 N \ {1}), which means that, equivalently, we will be concerned with the properties of the integral operator H. Our main result reads as follows. Theorem 1. The operator H is a bounded and strictly positive operator on H 2 has no eigenvalues, and its continuous spectrum is [,π]. with H =π. It This theorem, which is in agreement with what is known about the classical Hilbert matrix, should be seen as an outgrowth of Helson s last two papers [11, 12]. In these works, a study of multiplicative Hankel matrices was initiated, mainly focused on the question of to which extent Nehari s theorem [14] extends to the multiplicative setting. We will return to this interesting question in the final section of this paper. At this point, we just wish to point out that the existence of a canonical operator like H, closely related to the Riemann zeta function, clearly demonstrates that multiplicative Hankel matrices may arise quite naturally. The computation of the norm of H is straightforward, by a simple adaption of the classical proof of [9, pp ]. In fact, this adaption leads us to consider an l p version of the

3 multiplicative Hilbert matrix M, namely ) 1 M p := m 1/p n p 1)/p logmn) THE MULTIPLICATIVE HILBERT MATRIX 3, m,n 2 where 1 < p <. We will see that M p has norm π/sinπ/p), viewed as an operator on l p, which is analogous to the classical fact that A has norm π/sinπ/p) when it acts on l p. We will explain this link in Section 2. The identification of the spectrum is the hardest part of the proof of Theorem 1. Inspired by Magnus s work [13], it is split into two main parts. First, in Section 3, we establish estimates near the singular point s = for the anticipated solutions f to equations of the form H λ)f = c ψ, where c is a constant and ψ is the analytic symbol of H. This means that ψ is the primitive of ζs+ ) 1) belonging to H 2. The point of this estimation is to show that f w) must be square integrable on, ). Here we make use of the fact that ζs) s 1) 1 is an entire function, which allows us to relate H to a classical operator studied by Carleman. This analysis requires a fair amount of classical-type computations involving Mellin transforms. In Section 4, we may then finish the proof by resorting to the following commutation relation, obtained by integration by parts, between H and the differentiation operator D: DHf s)= f )ζs+ ) 1) HDf s). After finishing the proof of Theorem 1, we turn to two questions related to Helson s viewpoint, namely that multiplicative Hankel operators are bona fide small) Hankel operators on the infinite-dimensional torus T. The first question is whether there is a counterpart to the result of [5, 6] saying that the norm of H a viewed as an operator on H p D) is again π/sinπ/p). We will show in Section 5 that the analogy with H a breaks down at this point, or, more precisely, that H does not extend to a bounded operator on the H p analogues of H 2, which by Bayart s work [1] can be associated with H p T ). This negative result is related to, though not a trivial consequence of, the fact that H p T ) is not complemented in L p T ) [7]. The final question to be discussed concerns the analytic symbol 4) ψs) := n s n logn of the multiplicative Hankel matrix. Since ψ is, up to a linear term, a primitive of the Riemann zeta function, it appears to be of interest to investigate it more closely. While it is known from [16] that Nehari s theorem does not hold in the multiplicative setting, it could still be true that ψ is the Riesz projection of a bounded function. In the final Section 6, we will explain the exact meaning of this statement and show how this question relates to a long-standing embedding problem for H p spaces of Dirichlet spaces. A word on notation: Throughout this paper, the notation U z) V z) or equivalently V z) U z)) means that there is a constant C such that U z) CV z) holds for all z in the set in question, which may be a space of functions or a set of numbers. We write U z) V z) to signify that both U z) V z) and V z) U z) hold.

4 4 O. F. BREVIG, K.-M. PERFEKT, K. SEIP, A. SISKAKIS, AND D. VUKOTIĆ 2. THE NORM OF THE MATRIX M p In this section, M p p will denote the norm of M p viewed as an operator on l p. Our aim is to prove the following theorem, which in particular shows that H =π. Theorem 2. We have M p p = π/sinπ/p) for 1<p <. Proof. The proof relies, as in [9, pp ], on the following homogeneity property of the kernel x+y) 1 : 5) x 1/p 1 1+ x d x = x p 1)/p 1 1+ x d x = π sinπ/p). The exact computation of the integral can be found in [19, p. 254, Example 4]. We prove first that M p p π/sinπ/p). We write q = p/p 1) and assume that a n ) is in l p and b n ) is in l q. By Hölder s inequality, we find that a m b n m 1/p n 1/q logmn)) 1 P Q, m,n 2 where 6) P := m m 2 b p ) logm 1/q n 1 1 n 2 logn logmn) ) 1/p and 7) Q := n n 2 a q ) ) logn 1/p 1/q m 1 1. m 2 logm logmn) By a change of variables argument, each of the inner sums is dominated by the integral in 5), and hence we obtain the desired bound by duality. To prove that the norm is bounded below by π/sinπ/p), we use the sequences for which we have a n = n 1/q logn) 1+ε)/q and b n = n 1/p logn) 1+ε)/p 8) a n ) q q = 1 ε +O1) and b n) p p = 1 ε +O1) when ε +. We see that a m b n m 1/p n 1/q 1 logmn) = logm) 1+ε)/p logn) 1+ε)/q m 1 n 1 1 logmn) m,n 2 m,n 2 log3 log3 x 1+ε)/q y 1+ε)/q 1 d xd y. x+y This iterated integral can computed as the corresponding integral in [9, 9.5.2) p. 233] so that we get a m b n m 1/p n 1/q = 1 ) π m,n 2 ε sinπ/p) + o1) when ε +. Combining this estimate with 8), we get the desired bound M p p π/sinπp).

5 THE MULTIPLICATIVE HILBERT MATRIX 5 It is of interest to observe that when we replace the inner sums in 6) and 7) by the respective integrals in 5), we get a strict inequality. In particular, we get that 9) Hf H 2 < π f H 2 for every nontrivial function f in H 2. This means that we have already shown that π is not an eigenvalue for H. Another observation is that the matrix M p fails to be bounded on l p when p p. This is most easily seen when p > p because we can find a sequence a in l p for which the entries in M p a become infinite. When p < p, we can apply the same argument to the conjugate exponents q and q and the matrix M q. 3. ESTIMATES FOR SOLUTIONS OF H λ)f = cψ In preparation for the characterization of the spectrum of H, we will in this section prove precise asymptotics as s for solutions f in H 2 of the equation H λ)f = cψ, where c is a constant and ψ is the analytic symbol of H defined by 4). The considerations to come are in fact of a rather general nature, providing a spectral decomposition of f in terms of generalized eigenvectors of the shifted) Carleman operator [3, p. 169] defined by f w) Cf s)= d w, s>. s+ w 1 We choose to focus on H for simplicity, but it will be clear from the proof of the next theorem that minor modifications yield similar results for other integral operators whose kernels are perturbations K s + w), K analytic, of the Carleman kernel. Theorem 3. Suppose that <λ<π, and let ψ denote the analytic symbol of H, that is 1 ψs)= n s, Re s >. n logn If f in H 2 satisfies H λ)f = cψ, then there exists a complex number d and polynomially bounded sequences of complex numbers c k ) k=1 and d k) such that f has the series representation k=1 1) f s)=cd+ s ) 2k c k s ) iθ + d k s ) iθ), < s < 3/2, k=1 where θ is a real number dependent on λ, namely θ= 1 ) π log π π ) 2 λ 1. λ Remark. Note that for each k, the functions s s ) 2k ±iθ are generalized eigenvectors of the Carleman operator C belonging to the eigenvalue λ, < λ < π; see Lemma 1. The constant function s cd is not such an eigenfunction, and its appearance in 1) will allow us to derive a contradiction in the case that c. It is also possible to treat the case λ=π with the methods below, although we choose not to since we do not need it. Carrying out the details, one obtains for λ=π a decomposition of

6 6 O. F. BREVIG, K.-M. PERFEKT, K. SEIP, A. SISKAKIS, AND D. VUKOTIĆ f in terms of the eigenfunctions s s ) 2k and s s ) 2k logs ) of the Carleman operator. To simplify the computations and to align our proof with the classical representation of the Carleman operator, we will in this section shift everything to R + =, ), and prove Theorem 3 on this ray. Shifting the representation back to, + ) will then give 1). This means that we consider H 2 the space of Dirichlet series f s)= with coefficients a n ) l 2, and the operator 11) Hf s)= a n n n s, f w)ζs+ w+ 1) 1) d s. We let {x} denote the fractional part of x, and use the well-known formula ζs+ 1) 1= 1 s s+ 1) {x}x s+1) d x x = 1 s+ 1) s 1 The function 1/s is the kernel of Carleman s operator, defined on L 2 R + ) as Cf s)= f w) s+ w d w. {e x }e s+1)x d x =: 1/s K s). We will let K denote the similarly defined integral operator with kernel s, w) K s+ w), so that H=C K. By the fact that the arc length measure on the ray,+ ) is a Carleson measure for H 2, we have for f H 2 that 12) f s) 2 d s n a 2 = f 2, H 2 and hence f is in L 2 R + ). For <λ<π and f in H 2, we consider the equation H λ)f = cψ, where ψ denotes 1 ψs)= n logn n s. Note that this function also differs by a shift from the actual symbol appearing in Theorem 3.) It is convenient to rewrite this equation in the form 13) C λ)f = Kf + cψ. To analyze the equation 13), we will use the Mellin transform, which is defined by 14) M f z)= s z f s) d s s. By the Cauchy Schwarz inequality and 12), taking into account the rapid decay near infinity, we obtain that if f is in H 2, then the integral 14) converges absolutely when Re z >. This means that the function M f z) is analytic in at least) Re z >. Our first goals are thus to compute M Cf and M Kf for f in H 2, as well as the special transform M ψ.

7 Lemma 1. Suppose that f is in H 2. Then THE MULTIPLICATIVE HILBERT MATRIX 7 π 15) M Cf )z)= M f )z), sinπz) when Re z > and z Z. Proof. When Re z < 1, z Z and w >, we have 16) s z 1 s+ w d s= π sinπz) w z 1, which is the same integral 5) which was used in the proof of Theorem 2. By this formula and Fubini s theorem, we obtain 15) in the strip <Re z < 1. However, the right hand side of 15) has a meromorphic continuation to the domain Re z >. Remark. Note that the choice of θ is such that π/ sinπi θ + )) = λ. This motivates the appearance of the functions s s 2k ±iθ in 1) as generalized eigenfunctions to the Carleman operator. Compare with the remark following Theorem 3. Lemma 2. Let f be a function in H 2. Then M Kf )z) has a meromorphic continuation to Re z < 1 with simple poles at the non-positive integers, and satisfies in the same half plane when Im z ε for some positive ε, the estimate 17) M Kf )z) f H 2 z 3/2 e π/2 Imz. Proof. We begin by computing 18) Kf s)= where α n s)= β n s)= A n x) se sx xd x, f w)k s+ w)d w = 1 A n x)= 1+ x/logn 2B n x)e sx xd x, B n x)= 1 2 a n n logn αn s)+β n s) ), {e x } x e x, 1 1+ x/logn) x/logn ) {e x } x e x. We will only need the estimates A n x), B n x) e x, which imply that Kf s) is analytic in Re s> 1, since a n / n logn)) is in l 1. We apply the Mellin transform of 18), initially with < Re z < 1, obtaining a n M Kf )z)= Γ1+ z) αn z)+γz) β n z) ), n logn where Γ denotes the Gamma function and α n z)= A n x) x 1 z d x x and βn z)= 2B n x) x 2 z d x x. When Re z < 1, we use the estimates A n x), B n x) e x along with the triangle inequality to obtain α n z) Γ1 Re z) and β n z) 2Γ2 Re z).

8 8 O. F. BREVIG, K.-M. PERFEKT, K. SEIP, A. SISKAKIS, AND D. VUKOTIĆ Hence M Kf has a meromorphic continuation to Re z < 1, with simple poles at the poles of Γz). Moreover, by the Cauchy Schwarz inequality, we obtain that M Kf )z) f H 2 Γ1+ z) Γ1 Re z)+2 Γz) Γ2 Re z) ). When Im z ε, we use the functional equation and reflection formula for the Gamma function, and estimate further that M Kf )z) f H 2 Γ1+ z) Γ1 Re z) ) = f H 2 π Γ1 Re z). sinπz) Γ z) Using the assumption that Re z < 1 along with the product formulas for Γz) and sinπz), we get Γ1 Re z) n z = Re z z 1 z n z Γ z) n= n Re z 1 Re z n 2+ z n Re z n 2+Re z = z 1 z n z 1 Re z = z 1 z sinπz) n Re z 1 Re z sinπre z), n= we therefore arrive at the desired bound Lemma 3. For Re z >, we have M Kf )z) f H 2 z 3/2 19) M ψz)= 1 z 2 + sinπz) f H 2 z 3/2 e π/2 Imz. n= b n z+ n + E ψz), where b n decays super-exponentially, and E ψ z) is an entire function that, for every real number R, is bounded in the half-plane Re z < R. Proof. Set hs) := ψs) log s. Since h s) = ζs+ 1) 1 1/s, hs) = n= b n s n is an entire function. Note now that for Re z > we have while 1 1 s z 1 log s d s= 1 z 2, s z 1 hs)d s= n= b n z+ n. We finish the proof by setting E ψ z) := 1 sz 1 ψs)d s. Proof of Theorem 3. Suppose that < λ < π. Transforming the equation 13) by the Mellin transform and solving for M f, we obtain ) π 1 2) M f z)= sinπz) λ ) M Kf z)+cm ψz), Re z > Initially this formula is only valid for < Re z < 1, but we note that we may continue the right-hand side meromorphically to Re z >.

9 21) The inverse Mellin transform is given by THE MULTIPLICATIVE HILBERT MATRIX 9 M 1 h ) s)= 1 2πi κ+i κ i s z hz)d z for a suitable κ. Our expressions for M Kf and M ψ show that the right-hand side of 2) is meromorphic in Re z < 1 with possible) simple poles at the solutions of sinπz) = π/λ as well as at z =. Note here that the factor in front of M Kf z)+cm ψz) has simple zeroes at the integers. Note also that there actually are no poles in Re z >, since M f z) is analytic there. Hence we are left with the pole z = if c ) and those given by 1 λ sinπz)= Re z z =±iθ+ 2k+ ), π where k =, 1, 2,... We now compute 21) for h = M f and κ=2/3 by the method of residues. Let J n = [θ]+n and form the rectangular contour J n with corners in 2/3±i J n and 2J n +3/2)±i J n, traversed counter-clockwise. Using 17) and 19), straightforward estimates show that for < s < 1 we have lim s z M f z)d z = 1 2/3+i s z M f z)d z. n J n 2πi 2/3 i Evaluating the left-hand side by residues, we obtain f s)=cd+ s 2k c k s iθ + d k s iθ), <s< 1, k= where cd, c k, and d k are obtained as the residues of the right-hand side of 2) at z =, z = iθ 2k+ and z = iθ 2k+, respectively. In fact, it is clear that c k and d k grow at most polynomially in k, as seen from the estimates of Lemma 2 and Lemma 3. It remains to show that c = d =, However, either of them assuming a non-zero value contradicts the fact that f is in L 2 R + ). Moving back to,+ ), we obtain 1). Note that in the excluded case λ=π one may use the same argument, but the representation of f is different because all poles of the right-hand side of 2) except z = are double. We also note that a more careful analysis would show that the sequences c k ) and d k ) are in fact bounded, but since we don t need this, we have not made an effort to optimize this part of the theorem. 4. THE SPECTRUM OF THE MULTIPLICATIVE HILBERT MATRIX In this section we establish that H has the purely continuous spectrum [,π] on H 2. Our argument is based on a commutation relation between H and the operator D of differentiation, Df s)= f s). To establish this relation, we observe that DHf s)= f w)dζw+ s) 1)d w, s>.

10 1 O. F. BREVIG, K.-M. PERFEKT, K. SEIP, A. SISKAKIS, AND D. VUKOTIĆ Supposing that f is integrable on the segment,1), we get that DHf s) = f )ζs+ ) 1) f w)ζw+ s) 1)d w = f )ζs+ ) 1) HDf s), s>, where we have defined f )= f 1) 1 f w)d w. Thus, D and H anti-commute up to an unbounded) rank-one term. This observation is crucial for the characterization of the spectrum of H. To demonstrate that H has the purely continuous spectrum [,π], it suffices to show that H has no eigenvalues and that H λ does not have full range for λ in,π). Indeed, H is a positive operator with norm π, and so it follows that its spectrum is [,π]. Since any λ in the spectrum of a self-adjoint operator must either be an eigenvalue or part of the continuous spectrum, we can conclude that H has purely continuous spectrum. With this in mind we now finish the proof of Theorem 1. Theorem 4. The operator H : H 2 H 2 has no point spectrum. Furthermore, if f in H 2 solves the equation H λ) f = cψ, where c is a complex number and 1 ψs)= n s, n logn then f = c =. In particular, the spectrum of H is [, π] and purely continuous. Proof. We have already proved that λ=, π are not eigenvalues, since we have shown in Section 2 that H is a strictly positive operator for which Hf H 2 < π f H 2, f. It is hence sufficient to verify the second part of Theorem 4, since it shows simultaneously that no λ in,π) is an eigenvalue, and that H λ does not have full range. Accordingly, we suppose that f in H 2 satisfies H λ)f = cψ. By Theorem 3, we have the series representation 1). In particular f is square-integrable on, ) and f )=cd. But noting that ψ s)=ζs+ ) 1 and using the commutation relation of H and D, we then get that H+λ)f cdζs+ ) 1)=cζs+ ) 1). Since f is L 2, ) while ζ has a pole of order 1 in s = 1, it follows that d = 1. Hence, we have obtained that 22) H+λ)f =. Note now that since f is in L 2, ), we have that Hf is in H 2 see 3)), and thus we actually obtain that f is H 2 by 22). But since H is a positive operator, applying 22) again, we find that f. 5. FAILURE OF BOUNDEDNESS OF H ON H p WHEN p 2 We follow [1] and define H p as the completion of the set of Dirichlet polynomials Ps)= Nn=1 a n n s with respect to the norm P H p := lim T 1 T T Pi t) p d t) 1/p.

11 THE MULTIPLICATIVE HILBERT MATRIX 11 The space H p is the subspace of H p consisting of Dirichlet series of the form a n n s, which means that series in H p vanish at+. Theorem 5. H does not act boundedly on H p for p 2. The proof of this theorem requires us to associate H p with H p T ). This means that we need to invoke the so-called Bohr lift, which we now recall see [1, 17] for further details). For every positive integer n, the fundamental theorem of arithmetic allows the prime factorization πn) n= j=1 which associates n to the finite non-negative multi-index κn)=κ 1, κ 2, κ 3,...). The Bohr lift of the Dirichlet series f s)= n=1 a n n s is the power series 23) B f z)= a n z κn), n=1 where z = z 1, z 2, z 3,...). Hence 23) is a power series in infinitely many variables, but each term contains only a finite number of these variables. Under the Bohr lift, H p corresponds to the Hardy space H p T ), which we view as a subspace of L p T ). This means that B is an isometric map from H p onto H p T ). We refer to [1, 1] for the details, mentioning only that the Haar measure of the compact abelian group T is simply the product of the normalized Lebesgue measures for each variable. In particular, H p D d ) is a natural subspace of H p T ). We denote orthogonal projection Riesz projection) from L 2 T ) onto H 2 T ) by P +. We require the following lemma which is established by direct computation. Here and in what follows, L p norm with respect to normalized Lebesgue measure on T or T ) is denoted by p. Lemma 4. Let λ be a real parameter and suppose that <ε1+ λ )<1, 1 p <. Then 1+εz+ λz) p p = 1+ p [ p 1)1+λ) λ) 2] ε 2 +Oε 3 ). 4 The norm is minimal when λ=2 p)/p: ) 1+ε 2 p) p z+ z = 1+p 1)ε 2 +Oε 3 ). p p Proof. We write z = e iθ so that we have p κ j j, 1+εz+ λz) p = 1+2ε1+λ)cosθ+ ε 2 1+λ) 2 cos 2 θ+ ε 2 1 λ) 2 sin 2 θ) p/2 Integrating, we get = 1+ pε1+λ)cosθ + p 2 ε2[ 1+2 p 2 1) ]1+λ) 2 cos 2 θ+ p 2 ε2 1 λ) 2 sin 2 θ+oε 3 ). 1+εz+ λz) p p = 1+ p 4 [ p 1)1+λ) λ) 2] ε 2 +Oε 3 ).

12 12 O. F. BREVIG, K.-M. PERFEKT, K. SEIP, A. SISKAKIS, AND D. VUKOTIĆ The point of the lemma is that p 2 /4>p 1 whenever p 2, so that one-dimensional) Riesz projection acts expansively on the vector 1 + εz + λz). Proof of Theorem 5. We will choose G of norm 1 in L q T ) with 1/p+ 1/q = 1. Then Hf H p f w)b 1 P + Gw)d w. More specifically, we set Gz)= j=1 where α> and c q = j 1+ 2 q p α j z q p α j z j + 2 q) q ) q z = 1+ q 2 q) z j ))/c j, q 4q 1) q 2 p 2α j +Op 3α j ) by Lemma 4. It follows that P + Gw)= 1+2/q)p α w j )/c j 2α 1) 4/pq) /α+ w 1) 2/q. We now choose j=1 which has f H p 1. We see that f s)= j=1 1+2/p)p α s j 1+1/p)p 2α j 1, Hf H p 2α 1) 4/pq2 )+1/p 1. Here the exponent is negative if p 2 since, in this case, pq > 4 so that H f H p when α. 6. SYMBOLS OF THE MULTIPLICATIVE HILBERT MATRIX To place our discussion in context, we begin with some general considerations concerning Hankel forms, i.e., the bilinear forms associated with additive or multiplicative) Hankel matrices. We recall that any function ψ in H 2 D) defines a Hankel form H ψ by the relation H ψ f, g )= f g,ψ L 2 T), which makes sense at least for polynomials f and g. Nehari s theorem [14] says that H ψ extends to a bounded form on H 2 D) H 2 D) if and only if ψ=p + ϕ for a bounded function ϕ in L T). Moreover, H ψ = ϕ if we choose ϕ to have minimal L norm. Since n 1 e i nθ = 2iP + arg1 e iθ ), n=1 Nehari s theorem gives an alternate proof of the fact that the Hilbert matrix has norm π. By the Hahn Banach theorem, Nehari s theorem is equivalent to the statement that c n ρ n H ψ f 1, n=

13 THE MULTIPLICATIVE HILBERT MATRIX 13 where f z)= n c n z n and ψz)= n ρ n z n. In the case of the Hilbert matrix, we have in fact the stronger inequality 24) c n n+ 1) 1 π f 1, n= which was proved by Hardy and Littlewood [8]. We turn next to what is known about multiplicative Hankel forms. Every sequence = 1, 2, 3,...) in l 2 defines in an obvious way a multiplicative Hankel matrix, and we associate with it the corresponding multiplicative Hankel form given by 25) a, b) = mn a m b n, m=1 n=1 which initially is defined at least for finitely supported sequences a and b in l 2. We will now explain, using the Bohr lift, that every multiplicative Hankel matrix can be uniquely associated with either a Hankel form on H 2 D ) H 2 D ) or equivalently a small) Hankel operator acting on H 2 D ). If f, g, and ϕ are Dirichlet series in H 2 with coefficients a n, b n, and n, respectively, a computation shows that f g,ϕ H 2 = a,b). A formal computation gives that B f Bg,Bϕ L 2 T )= f g,ϕ H 2, allowing us to compute the multiplicative Hankel form 25) on T. This means that we may equivalently study Hankel forms 26) H Φ FG)= FG,Φ L 2 T ), F,G H 2 T ). In our previous considerations we required that Φ be in H 2 T ), but there is nothing to prevent us from considering arbitrary symbols Φ from L 2 T ). Hence, each Φ in L 2 T ) induces by 26) a possibly unbounded) Hankel form H ϕ on H 2 T ) H 2 T ). Of course, this is not a real generalization. Each form H Φ is also induced by a symbol Ψ in H 2 T ); setting Ψ=P + Φ we have H Φ = H Ψ. On the polydisc, the Hankel form H Φ is naturally realized as a small) Hankel operator H Φ, which when bounded acts as an operator from H 2 T ) to the anti-analytic space H 2 D ). Letting P + denote the orthogonal projection of L 2 T ) onto H 2 D ), we have at least for polynomials F in H 2 T ) that H Φ F = P + ΦF ). We now come to the question of to which extent Nehari s theorem remains valid in the multiplicative setting. Note first that if Ψ is in L T ), then the corresponding multiplicative Hankel form is bounded, since H Ψ f g ) = f g,ψ f 2 g 2 Ψ. We say that H Φ has a bounded symbol if there exists a Ψ in L T ) such that H Φ = H Ψ. In [11], Helson proved that every Hankel form in the Hilbert Schmidt class S 2 has a bounded

14 14 O. F. BREVIG, K.-M. PERFEKT, K. SEIP, A. SISKAKIS, AND D. VUKOTIĆ symbol, but it was shown in [16] that there exist bounded multiplicative Hankel forms without bounded symbols. This result was strengthened in [2] which showed that there are Hankel forms in Schatten classes S p without bounded symbols whenever p > 1 logπ/log4) 1 = In the opposite direction, we have the following positive result about Hankel forms with bounded symbols. Theorem 6. If αn) is a completely multiplicative function, then the symbol B n=1 αn)n s is the Riesz projection of a bounded function on T if and only if αn) is in l 2 N). Proof. This result is just a reformulation of the fact that the set of bounded point evaluations for H 1 D ) is D l 2 [4]. Following [4, p. 122], corresponding bounded functions can be found explicitly: For every point z = z j ) on T, we set Φz)= 1 αp) 2 ) 1 1 αp j )z j. p 1 αp j )z j This is a bounded function on T because the latter product is the ratio between two nonvanishing functions in L 2 T ). One may check that B 1 P + Φs) = n=1 αn)n s by a direct computation or by checking that Φ represents the functional of point evaluation at αp j )). The symbol 4) of the multiplicative Hilbert matrix does not have completely multiplicative coefficients, and we know from Theorem 1 that it is not compact. This means that the preceding discussion gives no answer to the following question. Question. Does the multiplicative Hilbert matrix have a bounded symbol? Equivalently, we may ask whether we have 27) a a n 1+ f H 1 n logn when f s) = a n n s is in H 1. We could even ask if the analogue of the Hardy Littlewood inequality 24) is valid: Does 27) hold when we put absolute values on a n? We observe that the left-hand side of 27) can be written as an integral, so that another reformulation of the question is to ask if the linear functional defined by L f = j=1 f w)d w extends to a bounded linear functional on H 1. A Carleson measure argument see [15, Theorem 4]) shows that if there exists an absolute constant C such that 28) 1 P+it) d t C P H 1 holds for all Dirichlet polynomials, then we also have f w) d w C f H 1

15 THE MULTIPLICATIVE HILBERT MATRIX 15 for another absolute constant C and all f in H 1. Whether 28) holds has been an open problem for a number of years; we refer to [18] for a discussion of it. We conclude that a positive solution to this embedding problem would imply a positive answer to question 1), and conversely, a negative answer to question 1) would imply a negative solution to the embedding problem 28). REFERENCES [1] F. Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math ), [2] O. F. Brevig and K.-M. Perfekt, Failure of Nehari s theorem for multiplicative Hankel forms in Schatten classes, arxiv: ). [3] T. Carleman, Sur les équations intégrales singulièeres à noyau réel et symétrique, A. B. Lundequistska, Uppsala, [4] B. J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc ), [5] E. Diamantopoulos and A. G. Siskakis, Composition operators and the Hilbert matrix, Studia Math. 14 2), [6] M. Dostanić, M. Jevtić, and D. Vukotić, Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type, J. Funct. Anal ), [7] S. E. Ebenstein, Some H p -spaces which are uncomplemented in L p, Pacific. J. Math ), [8] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, Math. Ann ), [9] G. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, [1] H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L 2,1) Duke Math. J ), [11] H. Helson, Hankel forms and random variables, Studia Math ), [12] H. Helson, Hankel forms, Studia Math ), [13] W. Magnus, On the spectrum of Hilbert s matrix, Amer. J. Math ), [14] Z. Nehari, On bounded bilinear forms, Ann. of Math. 2) ), [15] J.-F. Olsen and E. Saksman, On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate, J. Reine Angew. Math ), [16] J. Ortega-Cerdà and K. Seip, A lower bound in Nehari s theorem on the polydisc, J. Anal. Math ), [17] H. Queffélec and M. Queffélec, Diophantine Approximation and Dirichlet series, HRI Lecture Notes Series - 2, Hindustan Book Agency, New Delhi, 213. [18] E. Saksman and K. Seip, Integral means and boundary limits of Dirichlet series, Bull London Math. Soc ), [19] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Reprint of the fourth 1927) edition, Cambridge University Press, 198.

16 16 O. F. BREVIG, K.-M. PERFEKT, K. SEIP, A. SISKAKIS, AND D. VUKOTIĆ OLE FREDRIK BREVIG, DEPARTMENT OF MATHEMATICAL SCIENCES, NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY, NO-7491 TRONDHEIM, NORWAY address: KARL-MIKAEL PERFEKT, DEPARTMENT OF MATHEMATICAL SCIENCES, NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY, NO-7491 TRONDHEIM, NORWAY address: KRISTIAN SEIP, DEPARTMENT OF MATHEMATICAL SCIENCES, NORWEGIAN UNIVERSITY OF SCIENCE AND TECH- NOLOGY, NO-7491 TRONDHEIM, NORWAY address: ARISTOMENIS G. SISKAKIS, DEPARTMENT OF MATHEMATICS, ARISTOTLE UNIVERSITY OF THESSALONIKI, 54124, THESSALONIKI, GREECE address: DRAGAN VUKOTIĆ, DEPARTAMENTO DE MATEMÁTICAS, MÓDULO 17, FACULTAD DE CIENCIAS, UNIVERSIDAD AUTÓNOMA DE MADRID, 2849 MADRID, SPAIN address: