Two Distinguished Subspaces of Product Bmo and the Nehari–aak Theory for Hankel Operators on the Torus

In this paper we show that the theory of Hankel operators in the torus T , for d > 1, presents striking differences with that on the circle T, starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbols L∞(T) by BMOr(T), a subspace of product BMO, and the singular numbers of Hankel operators by so-called sigma numbers. This leads to versions of the Nehari–AAK and Kronecker theorems, and provides conditions for the existence of solutions of product Pick problems through finite Pick-type matrices. We give geometric and duality characterizations of BMOr, and of a subspace of it, bmo, closely linked with A2 weights. This completes some aspects of the theory of BMO in product spaces. Introduction This paper deals with the extension of the classical theory of bounded Hankel operators in the circle T to (big) Hankel operators in the torus T, for d > 1. Some crucial results in the one-variable theory, involving the notions of L symbols and the singular numbers of the operators, cannot have, as stated, meaningful extensions to the torus. This difficulty can be overcome by introducing so-called BMOr symbols and sigma numbers of Hankel operators. To explain what changes are to be made in dimension d > 1, we recall some basic features of the theory in T. Each function φ ∈ L(T) gives rise to a Hankel operator Γφ, and φ is called a symbol for the operator. In the case d = 1, these operators are closely related to the space BMO, since, by the Nehari theorem [N], a Hankel operator Γ is bounded if and only if Γ1 ∈ BMO, and if and only if Γ = Γφ with φ ∈ L, while φ ∈ BMO implies Γφ bounded with ‖Γφ‖ = ‖φ‖BMO. In turn, the Helson–Szegő theorem [HS] relates BMO to the boundedness of the Hilbert transform in L(μ), for μ a given measure on the circle T. 1991 Mathematics Subject Classification. 47B35, 42B20. Sadosky was partially supported by NSF grants DMS-9205926, INT-9204043 and GER-9550373, and her visit to MSRI is supported by NSF grant DMS-9022140 to MSRI. 1 2 MISCHA COTLAR AND CORA SADOSKY The Nehari theorem gives the distance of a bounded function φ to the space H(T) as the norm of the Hankel operator Γφ, and the theorem of Adamjan, Arov and Krein (AAK) refines this by giving its distance to H(T) + Rn (where Rn is the space of rational functions with n poles in the disk) as the singular number sn of the operator, or, equivalently, as the distance of the operator to those Hankel operators of finite rank n [AAK]. From the Beurling characterization of the invariant subspaces of H(T) of finite codimension, it follows that a Hankel operator Γ is of finite rank n if and only if Γ = Γφ with φ = b̄h, where h ∈ H and b is a Blaschke product with n zeros at z1, . . . , zn. If this is the case, the operator Γφ is closely related to a model operator in a finite subspace of H, so that its norm ‖Γφ‖ equals that of a finite n× n matrix explicitly given in terms of the zk’s and φ(zk)’s: the Pick matrix. One of the main applications of the Nehari theorem is that it provides a condition for the existence of solutions of the Pick interpolation problems in terms of the norm of an associated Hankel operator Γφ of finite rank, thus yielding the classical Pick condition in terms of Pick matrices. The basic properties of BMO(T) can be deduced in a unified way [ACS] through a generalized Bochner theorem, which includes also the results of Nehari and Helson– Szegő. The extension of this theorem to several dimensions led in [CS2] and [CS3] to an extension of the Nehari theorem to T, for d > 1, in terms of a class of symbols that we called BMOr (for “restricted” BMO). The extension of the Helson–Szegő theorem to several dimensions was given in [CS1], in terms of a subspace of product BMO = BMO(T) (defined in [ChF1]), that here we call bmo (for “small” BMO). Section 1 gives some basic properties of these subspaces of product BMO, starting with the continuous proper inclusions L(T) ⊂ bmo ⊂ BMOr ⊂ BMO(T). The preduals of bmo and BMOr are determined, providing counterparts of the duality result of Chang and Fefferman in product domains [ChF2]. As a corollary of the duality result for BMOr, in Section 2 it is shown that, when d > 1, there are bounded Hankel operators without bounded symbols (Theorem 2.1). This indicates that L symbols are not enough to characterize bounded Hankel operators, and that BMOr is the right class of symbols in product domains [CS3]. For d > 1 it is known [Am] that the positivity of the Pick matrix is necessary but not sufficient for the existence of a solution of the Pick problem. Necessary and sufficient conditions involving Pick matrices have been given by Agler for d = 2 [Ag], and by Cole, Lewis and Wermer for all d > 1 [CLW]. However, their conditions are not verifiable in practice, and the relation with Hankel operators is lost in their approach. In Section 3 we return to the consideration of analogues to the Pick problem with NEHARI–AAK THEORY FOR HANKEL OPERATORS ON THE TORUS 3 BMOr-norm control initiated in [CS3], and give necessary and sufficient conditions for the existence of solutions of a coordinate-wise Pick problem in terms of either the boundedness of a Hankel operator with symbol specified by the data, or the positiveness of d associated n× n Pick matrices. In the case d > 1, all singular numbers of a Hankel operator are bounded below by d times its norm (Theorem B), so that all Hankel operators of finite rank are zero [CS2]. This abrupt change from the one-dimensional case is closely related to the failure of the Beurling characterization of invariant subspaces to hold in the polydisk [AhC], and shows that an AAK theory cannot be meaningful in T, for d > 1. To recover the main features of the Nehari–AAK theory we need to introduce, not only BMOr symbols, but sigma numbers to replace the singular numbers, and a notion of operators of finite type, to replace that of finite rank. In Section 4 we rely on a version of Beurling’s characterization in the polydisk given in [CS4] to characterize the symbols of Hankel operators of finite type in terms of tensor products of finite Blaschke products, and to extend the AAK result mentioned above in terms of the sigma numbers of the Hankel operators. In Section 1 it is shown that, when passing from T to T, for d > 1, the different equivalent characterizations of BMO(T) give rise to distinct spaces. Similarly, the different characterizations of Carleson measures in D give rise to different notions in D, for d > 1. One such characterization is that a measure in D is Carleson if and only if a canonically associated function is in BMO. In Section 5 we extend this canonical association to d > 1, by defining Carleson–Nikolskii measures, and proving that a measure is of this type if and only if a canonically associated function is in BMOr. In the circle, the norms of Hankel operators of finite rank coincide with the norms of multipliers acting in finite-dimensional model subspaces, which in turn are determined by finite Pick matrices. In Section 6 we prove that the norms of Hankel operators of finite type coincide with those of multipliers acting in corresponding model subspaces, which now are not finite-dimensional but of bi-finite type, like those appearing in Sections 4 and 5. This significantly reduces the number of steps required to verify norm boundedness. Acknowledgements. We want to thank Chandler Davis for extensive discussions with the second author on duality, and Nikolai Nikolskii for helpful comments. The last version of this paper was written while the second author was a Research Professor of the Mathematical Sciences Research Institute at Berkeley, and we are happy to acknowledge the hospitality received there by both of us. Basic Notations. The following notations will be used throughout the paper. For d ≥ 1, P = P(T) is the class of trigonometric polynomials; f̂ represents the Fourier 4 MISCHA COTLAR AND CORA SADOSKY transform of f ; H(T) = {f ∈ L(T) : f̂(n1, . . . , nd) = 0 if nk < 0 for some k = 1, . . . , d}; H xk = {f ∈ L (T) = f̂(n) = 0 for nk < 0}; H(T) = L(T)⊖H(T); and the orthogonal projector P : L → H(T) is called the analytic projector. The d shifts Sk = Sxk , in L (T), where k = 1, . . . , d, are defined by Skf(x) = Skf(x1, . . . , xd) := exp(ixk)f(x). In the case d = 2, we write (x, y) for (x1, x2) and (m,n) for (n1, n2), and consider the subspaces of H(T) given by H x = {f ∈ L(T) : f̂(m,n) = 0 for m < 0}, H −x = L ⊖H x, and H y = {f ∈ L(T) : f̂(m,n) = 0 for n < 0}, H −y = L ⊖H y , as well as the projectors Px : L 2 → H x, P−x := (I − Px) : L → H −x and Py : L 2 → H y , P−y := (I − Py). The two shifts S1 and S2 in L (T) satisfy S1f(x, y) = e f(x), S2f(x, y) = e f(y). Observe that H(T) = H −x +H 2 −y = H 2 −x ∔ (H 2 −y ∩H x) = H −y ∔ (H −x ∩H y ). 1. Two Distinguished Subspaces of Product BMO An integrable function in T is of bounded mean oscillation if 1 |I| ∫ I |f(x)− fI | dx ≤ C for all intervals I, (1.1) where fI = |I|−1 ∫ I f(x) dx. The class BMO of functions of bounded mean oscillation is important in analysis. It is closely related to the Carleson measures and to the Ap weights, as well as to bounded Hankel operators. A function φ is in BMO = BMO(T) if and only if a canonically associated measure μ in D is Carleson, and a measure μ in D is Carleson if and only if Γ1 ∈ BMO for a canonically associated Hankel operator Γ. (See definitions below.) As there are different characterizations for the elements of BMO in T, the same is true for Carleson measures in D. NEHARI–AAK THEORY FOR HANKEL OPERATORS ON THE TORUS 5 BMO coincides with the space L+HL, where H is the Hilbert transform. This characterization follows from Charles Fefferman’s famous duality result, asserting that BMO is the (real) dual of the Hardy space H. Another way to prove BMO = L +HL (1.2) is through the characterizations of the weights w for which H is bounded in L(w) given by the A2 condition and by the Helson–Szegő theorem [HS]. In passing from T to T, for d > 1, the extension of the BMO theory to product domains presents various difficulties [ChF2]. S.-Y. Alice Chang and Robert Fefferman were able to introduce a notion of product BMO = BMO(T), dual to the space H(D), and for which an analogue of (1.2) is retained [ChF1]. In fact, φ ∈ BMO(T) ⇐⇒ φ = f1 +Hx1f2 + · · ·+Hxdfd+1 +Hx1Hx2fd+2 + · · ·+Hx1Hx2 . . .Hxdf2d for f1, . . . , f2d ∈ L(T), (1.3) where Hxj is the Hilbert transform with respect to the variable xj , for j = 1, . . . , d, and BMO is a complete normed space with respect to ‖φ‖BMO := inf{max j ‖fj‖∞ : all decompositions (1.3)}. But for product BMO the geometric characterizations by mean oscillation and by associated Carleson measures become considerably more complicated (they do not correspond to bounded mean oscillation with respect to rectangles), and, furthermore, the connections with weights and Hankel operators are lost. In previous work ([CS1], [CS3]), we gave results in product spaces analogous to those linking BMO to weights and to Hankel operators in one variable, in terms of classes of functions that are properly contained in product BMO. In this section we clarify the relation of these classes with product BMO, give some of their basic properties, and characterize their preduals. Definition 1 (small BMO). A function φ ∈ L(T), for d ≥ 1, is in bmo(T) if there exist f1, . . . , fd, g1, . . . , gd ∈ L(T) such that φ = f1 +Hx1g1 = · · · = fd +Hxdgd (1.4) and ‖φ‖bmo := inf{max 1≤j≤d {‖fj‖∞, ‖gj‖∞} : all decompositions (1.4)}. Observe that ‖φ‖bmo = 0 if and only if φ is constant, and bmo/C is a complete normed space with respect to ‖ · ‖bmo. 6 MISCHA COTLAR AND CORA SADOSKY Definition 2 (restricted BMO). A function φ ∈ L(T), for d ≥ 1, is in BMOr if there exist φ0, φ1, . . . , φd ∈ L(T) such that { (I − Pxj)φ = (I − Pxj)φj for j = 1, . . . , d, and Px1Px2 . . . Pxdφ = Px1Px2 . . . Pxdφ0, (1.5) where Pxj : L 2 → H xj is the analytic projector in xj , for j = 1, . . . , d. Moreover, ‖φ‖BMOr := inf{max 0≤j≤d ‖φj‖∞ : all decompositions (1.5)} = max{max 1≤j≤d {inf{‖φ− hxj‖∞ : hxj ∈ H xj}}, inf{‖φ− h ‖∞ : h ∈ H}}. Observe that BMOr is a complete normed space with respect to ‖ · ‖BMOr, and coincides with the space restricted BMO introduced in [CS3]. The two definitions given above are justified by the following results. Theorem (Helson–Szegő theorem in T, for d ≥ 1). [CS1] A weight 0 ≤ w ∈ L(T) satisfies