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ZERO PRODUCTS OF TOEPLITZ OPERATORS ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ Abstract. We prove that the product of finitely many Toeplitz operators on the Hardy space is zero if and only if at least one of the operators is zero. We use some new vector-valued techniques which not only lead to a vector-valued version of this result but also appear to be needed in the scalar case. Introduction A Toeplitz operator Tu with the symbol u is defined as the operator of multiplication by u followed by an appropriate projection. Toeplitz operators acting on Hardy spaces have very simple and natural matrix representations via infinite Toeplitz matrices, i.e., they have constant entries on the diagonals parallel to the main one. These operators also have many natural relationships with other relevant mathematical objects such as sequence spaces, reproducing kernels, maximal ideals in Banach algebras, Wiener-Hopf processes, the commutant of the shift operator, etc. The algebra generated by Toeplitz operators is of particular importance and has generated a lot of interest in the recent years. It is easy to check that a Toeplitz operator is the zero operator if and only if its symbol is zero. Likewise, the product of two such operators is zero if and only if the symbol of one of them is zero [3]. The following natural and basic conjecture about finite products of Toeplitz operators has been well-known for a long time: If a product of n Toeplitz operators is the zero operator then at least one of these operators must be zero. This was shown to hold for three operators by Axler in the 1970’s (unpublished) but the method used in [3] becomes quite complicated for handling more operators. Although the question has received some Date: 29 January, 2009. 2000 Mathematics Subject Classification. 47B35, 30H05. The second author was supported by MICINN grant MTM2006-14449-C02-02 and partially by MTM2007-30904-E and CSD2006-00032 (i-MATH), Spain, as well as by the European ESF Network HCAA (“Harmonic and Complex Analysis and Applications”). 1 2 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ attention, only recently has the conjecture been verified for n = 5 and n = 6 by Guo [7] and Gu [6] respectively. In the present paper we show that the above conjecture is true in general and actually holds in the appropriate sense in the vector-valued case as well. In fact, the vector-valued approach has not been considered here merely for the purpose of greater generality but is crucial in our proof even in the scalar case. In order to state our result, we first fix the notation. Denote by dσ = dθ/(2π) the normalized arc length measure on the unit circle T and by H p the usual Hardy space of the circle. The space (H p )m is the set of all vector-valued functions f : T → Cm , written as a column matrix f = (f 1 , f 2 , . . . f m )t , where each f j ∈ H p . An r × m Toeplitz operator matrix is an r × m operator-matrix from (H p )m into (H p )r whose entries are Toeplitz operators. Such an operator is defined via an r ×m matrix-valued function u with entries in L∞ (T) called symbol . As in the scalar case, the corresponding Toeplitz operator-matrix Tu is defined via the Szegő projection: Z (uf )(z) Tu f (ζ) = P+ (uf )(ζ) = dσ(z), f ∈ (H p )m , T 1 − z̄ζ where the product uf is to be understood in the standard matrix P product sense; that is, the i-th component is given by (Tu f )i = j Tuij f j . The values f (z) above denote, as is usual, the nontangential boundary values of f which exist a.e. on T and belong to Lp (T, Cm ). Throughout the paper we assume that p > 1, so that the Toeplitz operator-matrices considered above are bounded on (H p )m . We shall use the standard notations H ∞ (Mr×m (C)), N (Mr×m (C)) for the matrices whose entries are bounded analytic functions, or meromorphic functions in the Nevanlinna class on D. Similarly L∞ (T, Mr×m (C)) denotes the matrices with essentially bounded measurable entries. We can now state our main result. Theorem A. Let u1 , . . . , un ∈ L∞ (T, Mm×m (C)). If Tu1 · · · Tun = 0 then at least one symbol uk , 1 ≤ k ≤ n, satisfies det uk = 0 a.e.. This is a natural generalization of the conjecture for the vector-valued case since there are even constant non-zero matrices whose product is the zero matrix. The theorem easily extends to the finite-rank products of finitely many Toeplitz operators: in order for the rank of the product to be finite, again at least the symbol of one of the factors must be singular a.e.. ZERO PRODUCTS OF TOEPLITZ OPERATORS 3 Here is an outline of our method, presented in the scalar case m = 1. We exploit the close relationship of kernels of Toeplitz operators to the so-called nearly invariant subspaces [10]. These are subspaces of H p with the property that each zero of an element in the space which is not a common zero can be divided out without leaving the space. It is wellknown that the kernel of a Toeplitz operator is nearly invariant (see also Lemma 2.1 below). Almost conversely, every nearly invariant subspace is contained in a Toeplitz kernel. However, the situation is more complicated for the kernels of finite products of Toeplitz operators. Our starting point is the observation that if u1 , . . . , un ∈ L∞ (T) then (see Lemma 2.2 below) the subspace N = {(f, Tun f, . . . , Tu2 · · · Tun f )t : f ∈ ker Tu1 · · · Tun } is a closed nearly invariant subspace of (H p )n , which brings into the picture the vector-valued case in a natural way. With this idea in mind, we consider nearly invariant subspaces in the vector-valued case and prove in Section 1 that such subspaces are always contained in the kernel of an appropriate Toeplitz operator-matrix which we denote here by V . If we assume that Tu1 Tu2 · · · Tun = 0, the equality TV N = {0} leads to TV (f, Tun f, . . . , Tu2 · · · Tun f )t = 0 , f ∈ Hp . Now we can use an approximate identity argument, Lemma 2.3 (the result goes back to Douglas [5], see also [2]) to show that V (f, un f, . . . , u2 · · · un f )t = 0 a.e. , f ∈ Hp . The conjecture would now follow if, under the assumption that none of the symbols uk is zero, we could choose V to be an n × n matrixvalued function invertible on a set of positive measure on T. Even if we cannot conclude the proof at this early stage, the analysis of this type of situation is essentially involved in our argument. It requires several steps which will be briefly discussed below. They are based on two kernel inclusion theorems for products of Toeplitz operator-matrices proved in Section 3. The reasoning described here can be applied to the partial products u1 · · · uk , k < n and we begin with the case when k = 2. It can be easily verified that under our assumption we must have u1 u2 6= 0 (this is actually not needed in our proof). Then the simplest variant of our Theorem 3.1 asserts that there exists a bounded measurable 2 × 2 matrix-valued function V2 , invertible a.e. on T, such that N2 = {(f, Tu2 f )t : f ∈ ker Tu1 Tu2 } ⊂ ker TV2 , 4 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ or equivalently, {(f, f )t : f ∈ ker Tu1 Tu2 } ⊂ ker TV2 TU2 , where µ U2 (z) = 1 0 0 u2 (z) ¶ . If n ≥ 3 this means that {(f, f )t : f ∈ ker Tu1 Tu2 Tu3 } ⊂ ker TV2 TU2 TU3 , where µ U3 (z) = u3 (z) 0 0 u3 (z) ¶ . The point is that Theorem 3.1 applies to ker TV2 TU2 TU3 as well, and if we assume without loss of generality that Tu2 · · · Tun 6= 0 we can start an inductive process which is described in Theorem 3.2. The outcome of this process is that the diagonal in (H p )(n−1)! = (ker Tu1 · · · Tun )(n−1)! satisfies the inclusion (n−1)! (H p )d ⊂ ker TVn TW1 · · · TWn−1 , where Vn is a bounded measurable (n − 1)! × (n − 1)! matrix-valued function, invertible a.e. on T, and the product W1 · · · Wn−1 has the form µ ¶ J(z) 0 W1 (z) · · · Wn−1 (z) = , A(z) B(z) where A, B, J are bounded measurable matrix-valued functions with J ∈ L∞ (M(n−2)!×(n−2)! (C)) diagonal and non-zero. We then use again the approximate identity argument mentioned above to obtain Vn (z)W1 (z) · · · Wn−1 (z)(f (z), f (z), . . . f (z))t = 0, a.e. on T , for all f ∈ H p , which leads to the contradiction J = 0 and concludes the proof. In the vector-valued case we actually obtain a somewhat stronger result than Theorem A. In Section 4 we prove that if each of the symbols uk , 1 ≤ k ≤ n, is invertible on a set of positive measure on T then the product Tu1 · · · Tun satisfies dim{Tu1 · · · Tun f (λ) : f ∈ (H p )m } = m for all λ ∈ D except possibly a discrete subset. Acknowledgments. We would like to thank N. K. Nikolskiı̆ for helpful discussions, S. Shimorin for pointing out some initial errors in the first draft, and especially D. Sarason for a careful reading of the first draft and many useful comments and suggestions. Part of this work ZERO PRODUCTS OF TOEPLITZ OPERATORS 5 was done while the first author was visiting the Department of Mathematics at the Universidad Autonóma de Madrid. He wants to thank the department for its hospitality and a stimulating working environment. 1. Nearly invariant subspaces of vector-valued functions Throughout the paper we shall denote by ζ the identity function on D, ζ(z) = z. Following Sarason [10] we call a closed subspace N of H p nearly invariant if for every f ∈ N and every λ ∈ D which is a f zero of f but not a common zero of N we have that ζ−λ ∈ N . Nearly invariant subspaces generalize Toeplitz kernels and play an important role in many other problems. The main result about such subspaces is the characterization obtained by Hitt [8] and Sarason [10] for the case when p = 2. Their theorem asserts that a non-zero nearly invariant subspace N of H 2 is either invariant for the unilateral shift, or it has the form (1.1) N = θN F Kθ where θN is the common inner divisor of N , F is an outer function, θ is inner, and, as usual Kθ = H 2 ª θH 2 . In addition, multiplication by θN F is an isometry from Kθ onto N . The parameters F, θ are not uniquely determined by N . However F can be chosen to be the −1 extremal function of θN N for the origin, that is, the unique solution of the extremal problem −1 F (0) = sup{Ref (0), f ∈ θN N , kf k = 1}. An immediate consequence of this result is that a nontrivial nearly invariant subspace N = θN F Kθ is contained in the kernel of the nonzero Toeplitz operator with symbol ζθθN F /F . Indeed, any function f is such a space N can be written as f = θN F g for some g ∈ H 2 which is orthogonal to θH 2 ; hence for all n ≥ 0 we get hTζθθN F /F f, ζ n i = hP (ζF θg), ζ n i = hζF θg, ζ n i = hg, θF ζ n+1 i = 0, which implies that TζθθN F /F f = 0. Thus, we can state the following result which is the starting point for the present section. Proposition 1.1. A nontrivial nearly invariant subspace of H 2 is either invariant for the unilateral shift Mζ on H 2 , Mζ f = ζf , or it is contained in the kernel of a non-zero Toeplitz operator. We should point out here that the above proposition can be proved directly without use of the powerful theorem of Hitt and Sarason (see also Theorem 1.1 below). The purpose of this section is to extend this 6 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ result to the vector-valued case. To this end, fix a positive integer m and p > 1. If M is a subspace of (H p )m and λ ∈ D we define the fiber dimension of M at λ by dimM (λ) = dim{f (λ) : f ∈ M}. It is a simple exercise to show that for all α ∈ D except possibly a discrete subset we have (1.2) dimM (α) = max dimM (λ). λ∈D Moreover, if max dimM (λ) = k λ∈D then we can find f1 , . . . , fk ∈ M such that the matrix with columns f1 (λ), . . . , fk (λ) has rank k. Since the minors are analytic functions of λ, the claim follows. The same argument shows that there is a relatively open dense subset of k-tuples (f1 , . . . , fk ) ∈ Mk with _ dim {f1 (λ), . . . , fk (λ)} = k for all λ ∈ D \ Λ, where Λ is a discrete subset of D which may depend on f1 , . . . , fk . For further reference, let us record the following simple observation regarding the maximal fiber dimension. As is usual, by the kernel of a matrix we shall mean the kernel of the induced linear operator on Cm . Proposition 1.2. Let M be a subspace of (H p )m with k = max dimM (λ) < m. λ∈D ∞ Then there exists u ∈ H (Mm×m (C)) and a discrete set Λ ⊂ D with dim ker u(λ) = k for all λ ∈ D \ Λ, such that M ⊂ ker Tu . Proof. Let λ0 ∈ D and (f1 , . . . , fk ) ∈ Mk with _ dim {f1 (λ0 ), . . . , fk (λ0 )} = k, or, equivalently, rank(fjl (λ0 )) 1≤j≤k = k, 1≤l≤m l where f denotes the l-th component of f . Assume without loss of generality that the first k rows of this matrix are linearly independent. Then there exists a discrete set Λ ⊂ D such that det(fjs (λ))1≤j,s≤k 6= 0 ZERO PRODUCTS OF TOEPLITZ OPERATORS 7 for all λ ∈ D \ Λ, since the above determinant is easily seen to be an analytic function of λ ∈ D. On the other hand, for every f ∈ M, λ ∈ D and l > k we have l f (λ) f1l (λ) . . . fkl (λ) f 1 (λ) f11 (λ) . . . fk1 (λ) =0 det .. .. ... . ... . f k (λ) f1k (λ) . . . fkk (λ) which implies that for each such l we have (1.3) l f (λ) + k X as (λ)f s (λ) = 0, s=1 where the coefficients as (·) are independent of f and belong to the Nevanlinna class on D (each of them being a quotient of two H p/k functions, by generalized Hölder’s inequality). This can easily be seen from the expansion by the first column. There exists a function F ∈ H ∞ such that F as ∈ H ∞ , 1 ≤ s ≤ k, and which vanishes at most on a discrete exceptional set which does not depend on f . Indeed,Qwriting each as = gs /hs with gs , hs ∈ H ∞ it suffices to choose F = ks=1 hs . Then the lower-triangular block matrix u = (uls )1≤l,s≤m with F as , if 1 ≤ s ≤ k < l F, if l = s > k uls = 0, otherwise satisfies the conditions in the statement. Indeed, it is obvious that Tu f = uf since u ∈ H ∞ (Mm×m (C)) and one easily verifies that uf is the zero vector in view of (1.3). Again by (1.3) and the non-vanishing property of F , it is also quite evident that for each λ ∈ D \ Λ we have dim ker u(λ) = k since this kernel is formed precisely by those vectors f = (f 1 (λ), f 2 (λ), . . . , f m (λ)) ∈ Cm for which f l (λ) = 0 whenever k < l ≤ m. ¤ Let us now turn to nearly invariant subspaces of vector-valued functions. Definition 1.1. A closed subspace N of (H p )m is called nearly invariant if for every f ∈ N and every λ ∈ D with f (λ) = 0 and 1 f ∈ N. dimN (λ) = maxα∈D dimN (α) we have that ζ−λ As pointed out above, we want to extend Proposition 1.1 to this context. The only complication which occurs in the vector-valued case is 8 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ to find the appropriate interpretation for the dichotomy given by that proposition. This appears naturally, since there are nearly invariant subspaces which are shift-invariant in certain components but not in others. In the proof of the next result we shall make use of the well known jump theorem for Cauchy transforms (see, for example, [4, § 2.4]). Given h ∈ L1 (T, Cm ) its Cauchy transform is defined by Z h(z) Ch(λ) = dσ(z), λ ∈ C \ T. T z −λ If Ci h(λ), Ce h(λ) denote the nontangential limits of Ch from the inside and outside of the unit disc respectively, then these limits exist σ-a.e. and the jump theorem asserts that Ci h(λ) − Ce h(λ) = λ̄h(λ) a.e. on T. In what follows we shall use the notation N ⊥ for the annihilator of the closed subspace N of H p . When p = 2, we mean by this the usual orthogonal complement of N in H 2 while for p 6= 2 it will denote the annihilator in the dual space of N , a subspace of the dual space of (H p )m . That is, if we denote by x · y the scalar product of the vectors x, y ∈ Cm then saying that g ∈ N ⊥ means that Z g(z) · h(z)dσ(z) = 0 T for all h ∈ N . See [9, Chapter 7] for a description of the dual of H p when m = 1; the description is similar in the vector-valued case. Proposition 1.3. Let N be a nontrivial nearly invariant subspace of (H p )m . (i) For every λ ∈ D with dimN (λ) = maxα∈D dimN (α) we have dimN (λ) = dim N /(N ∩ (ζ − λ)N ) . (ii) If α ∈ C \ D then dim N /(N ∩ (ζ − α)N ) ≤ max dimN (λ) . λ∈D Proof. (i) Let λ be a value for which the maximum fiber dimension is attained. It follows from the definition of a nearly invariant subspace that a function f ∈ N vanishes at such λ if and only it can be written as f = (ζ − λ)g for some g ∈ N , hence N ∩ (ζ − λ)N = {f ∈ N : f (λ) = ZERO PRODUCTS OF TOEPLITZ OPERATORS 9 0}. The functional of point evaluation at λ is linear, hence by standard linear algebra dim N /{f ∈ N : f (λ) = 0} = dim{f (λ) : f ∈ N } and therefore dim N /(N ∩ (ζ − λ)N ) = dimN (λ) . (ii) Let k = maxλ∈D dimNW (λ), let λ0 ∈ D with dimN (λ0 ) = k, and let f1 , . . . , fk ∈ N with dim {f1 (λ0 ), . . . , fk (λ0 )} = k. Then there is a discrete, possibly void, subset Λ of D and a fixed minor of order k of the determinant with column vectors f1 (λ), . . . , fk (λ) which does not vanish on D \ Λ. If λ ∈ D \ Λ and g ∈ N we deduce, as above, that there exist scalars cj (λ, g), 1 ≤ j ≤ k such that (1.4) g(λ) = k X cj (λ, g)fj (λ) j=1 and consequently by near-invariance, à ! k X 1 (1.5) Rλ g = g− cj (λ, g)fj ∈ N . ζ −λ j=1 The coefficients cj (λ, g) can be calculated from (1.4) with help of the inverse of the matrix corresponding to the minor considered above. It is important to note that the entries of this matrix are functions in H p , hence the entries of its inverse are meromorphic functions in the Nevanlinna class on the unit disc. Thus by inserting zero column vectors in this matrix if necessary, we obtain that there exists a k × mmatrix A(λ), λ ∈ D \ Λ of rank k such that (c1 (λ, g), . . . , ck (λ, g))t = A(λ)g(λ), λ ∈ D \ Λ, where the entries of A(λ) extend to meromorphic functions in the Nevanlinna class on D. In particular, cj (·, g) , 1 ≤ j ≤ k are meromorphic functions in the Nevanlinna class on D and (1.4) continues to hold a.e. on T. Now let α ∈ C \ D and let h ∈ (N ∩ (ζ − α)N )⊥ . Note that (ζ − α)Rλ g = g − k X cj (λ, g)fj + (λ − α)Rλ g ∈ N ∩ (ζ − α)N j=1 hence Z (z − α)Rλ g(z) · h(z)dσ(z) = 0, T λ ∈ D \ Λ, 10 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ or equivalently, (1.6) C((ζ − α)g · h)(λ) = k X cj (λ, g)C((ζ − α)fj · h), λ ∈ D \ Λ. j=1 By taking nontangential limits we then have i (1.7) C ((ζ − α)g · h)(λ) = k X cj (λ, g)Ci ((ζ − α)fj · h) , j=1 a.e. on T. On the other hand, the jump theorem gives Ce ((ζ − α)g · h)(λ) = Ci ((ζ − α)g · h)(λ) − λ̄(λ − α)g · h(λ) (1.8) a.e. on T. From (1.6), (1.7) and (1.8) it follows that if g1 , . . . , gl ∈ N and h ∈ (N ∩ (ζ − α)N )⊥ then (Ce ((ζ − α)g1 · h)(λ), . . . , Ce ((ζ − α)gl · h)(λ))t = Γkl (λ)x(λ) , where Γkl (λ) = (cj (λ, gi )) 1≤j≤k , 1≤i≤l and x(λ) is the k-dimensional vector with components xj (λ) = Ci ((ζ − α)fj · h)(λ) − λ̄(λ − α)fj · h(λ) , e e 1 ≤ j ≤ k. t Thus all vectors (C (g1 · h)(λ), . . . , C (gl · h)(λ)) lie in the range of the linear map induced by Γkl (λ). Note that this matrix does not depend on the choice of h ∈ (N ∩ (ζ − α)N )⊥ and clearly, at any point λ ∈ T where it is defined, its rank is at most k. This means that if l > k, g1 , . . . , gl ∈ N and h1 , . . . , hl ∈ (N ∩ (ζ − α)N )⊥ the vectors (Ce ((ζ − α)g1 · hi )(λ), . . . , Ce ((ζ − α)gl · hi )(λ))t , 1≤i≤l are linearly dependent, or equivalently, det(Ce ((ζ − α)gj · hi )(λ))1≤i,j≤l = 0 a.e. on T. Then these determinants vanish identically on C \ D since they belong to the Nevanlinna class of C \ D. In particular, µZ ¶ det gj · hi dσ = det(Ce ((ζ − α)gj · hi )(α))1≤i,j≤l = 0 , T 1≤i,j≤l and we obtain a contradiction which completes the proof. ¤ Given a nearly invariant subspace of (H p )m , the ordered pair (max dimN (λ), max dim N /(N ∩ (ζ − λ)N )) λ∈D λ∈C\D will be called the index of N . It is easy to see that N is invariant for the shift operator Mζ if and only if it has index (k, 0) for some k ≤ m. ZERO PRODUCTS OF TOEPLITZ OPERATORS 11 The aim of this section is to give a vector-valued version of Proposition 1.1. Of course, for a nearly invariant subspace N 6= {0} of (H p )m with index (k, l), where k < m, Proposition 1.2 immediately provides a choice of a Toeplitz operator-matrix Tu with N ⊂ ker Tu . In this case we can choose u ∈ H ∞ (Mm×m (C) with rank u(λ) = m − k a.e. on T. The remaining case k = m is the most interesting one and the rest of this section is devoted to this case (see next Theorem). For further purposes let us record the definition and some simple properties of the generalized backward shifts used in this proof. Given fj ∈ (H p )m , 1 ≤ j ≤ k we shall use throughout the notation f = (f1 , . . . , fk ) ∈ [(H p )m ]k Since the dimension of f is always clear from the context it will not be specified in the notation. Also, we shall frequently identify [(H p )m ]k with (H p )mk . Let N be a nearly invariant subspace of (H p )m with index (k, l), let λ0 ∈ D with dimN (λ0 ) = k and let fj ∈ N , 1 ≤ j ≤ k such that _ dim {f1 (λ0 ), . . . , fk (λ0 )} = k. As we have seen in the proof of Proposition 1.3 (formula (1.4)), if λ ∈ D with _ dim {f1 (λ), . . . , fk (λ)} = k then for every f ∈ N we can write f (λ) = k X cj (λ, f )fj (λ) , j=1 where the coefficients cj (λ, f ) are given by (1.9) (c1 (λ, f ), . . . , ck (λ, f ))t = A(λ)f (λ) with A(·) ∈ N (Mk×m (C)), rank(A(λ)) = k for all λ ∈ D outside of a discrete exceptional set. The matrix-valued function A is uniquely determined by (1.9) and satisfies A(λ)fj (λ) = (δ1j , . . . , δkj )t , 1 ≤ j ≤ k. For such λ ∈ D the linear operator defined by P f − kj=1 cj (λ, f )fj , (1.10) Rλ,f f = ζ −λ 12 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ maps N into itself. By the closed graph theorem Rλ,f is bounded. For our purposes, the crucial property of these operators is that for every α∈C (1.11) (ζ −α)Rλ,f f = f − k X cj (λ, f )fj +(λ−α)Rλ,f f ∈ N ∩(ζ −α)N . j=1 We also note that if α, λ ∈ D are such that _ _ dim {f1 (α), . . . , fk (α)} = dim {f1 (λ), . . . , fk (λ)} = k then 1 A(λ)(Rα,f f )(λ) = λ−α à A(λ)f (λ) − k X ! cj (α, f )A(λ)fj (λ) , j=1 and from above we obtain that cj (λ, f ) − cj (α, f ) (1.12) cj (λ, Rα,f f ) = , 1 ≤ j ≤ k. λ−α We can now turn to the vector-valued version of Proposition 1.1. Theorem 1.1. Let N be a nontrivial nearly invariant subspace of (H p )m with index (k, l), where l > 0. Then there exists a non-zero l × m-Toeplitz operator-matrix Tu with N ⊂ ker Tu . The symbol matrix u can be chosen to have the form u(λ) = B(λ)A(λ) a.e. on T, where A ∈ N (Mk×m (C)) has rank k a.e. on T and B ∈ H ∞ (Ml×k (C)) has rank l a.e. on T. Proof. Let λ0 ∈ D with dimN (λ0 ) = k and let λ1 ∈ C \ D with dim N /N ∩ (ζ − λ1 )N = l . Recall that by Proposition 1.3 (ii) we have l ≤ k. Denote by [g] = g + (ζ − λ1 )N ∩ N the coset of g ∈ N in (ζ −λ1 )N ∩N and use again the simple argument mentioned at the beginning of the section to conclude that both sets of tuples n o _ (f1 , . . . , fk ) ∈ N k : dim {f1 (λ0 ), . . . , fk (λ0 )} = k and n (g1 , . . . , gk ) ∈ N k : dim _ o {[g1 ], . . . , [gl ]} = l ZERO PRODUCTS OF TOEPLITZ OPERATORS 13 are open and dense in N k . Then we can choose f1 , . . . , fk ∈ N such that _ _ dim {f1 (λ0 ), . . . , fk (λ0 )} = k, dim {[f1 ], . . . , [fl ]} = l . Clearly, there exist h1 , . . . , hl ∈ (N ∩ (ζ − λ1 )N )⊥ such that µZ ¶ det fj · hr dσ 6= 0 . T 1≤r,j≤l Let f = (f1 , . . . , fk ) and let Rλ,f be the operator defined in (1.10). By (1.11) we have Z (ζ − λ1 )Rλ,f f · hr dσ = 0 , 1 ≤ r ≤ l , T for all λ ∈ D \ Λ, where Λ is a discrete subset of D. This gives C((ζ − λ1 )f · hr )(λ) = k X cj (λ, f )C((ζ − λ1 )fj · hr ), 1 ≤ r ≤ l, j=1 and after taking nontangential limits i C ((ζ − λ1 )f · hr )(λ) = k X cj (λ, f )Ci ((ζ − λ1 )fj · hr ), 1 ≤ r ≤ l, j=1 a.e. on T. Recall also that cj (., f ) are Nevanlinna functions with k X cj (λ, f )fj (λ) = f (λ) , j=1 a.e. on T. Then by an application of the jump theorem we obtain e C ((ζ − λ1 )f · hr )(λ) = k X cj (λ, f )Ce ((ζ − λ1 )fj · hr )(λ), 1 ≤ r ≤ l, j=1 a.e. on T. If A is the k × m matrix given by (1.9) and B1 (λ) = (Ce ((ζ − λ1 )fj · hr )(λ)) 1≤j≤k 1≤r≤l then the last equality can be rewritten as (Ce ((ζ − λ1 )f · h1 )(λ), . . . , Ce ((ζ − λ1 )f · hl )(λ))t = B1 (λ)A(λ)f (λ) a.e. on T. Finally, by an argument similar to the ones used before, we can find a function F ∈ H ∞ such that F B1 ∈ H ∞ (Ml×k (C)) and F B1 A ∈ L∞ (T, Mk×m (C)). Set u = F B1 A to obtain the desired result. ¤ 14 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ 2. Near-invariance and kernels of products of Toeplitz operator-matrices The starting point and motivation for the work in this section is the following observation essentially related to the commutator of a Toeplitz operator-matrix with a backward shift. For simplicity we shall consider throughout the case of quadratic m × m-matrices Lemma 2.1. If u ∈ L∞ (T, Mm×m (C)) and f ∈ (H p )m has the zero λ ∈ D then f Tu f (z) − Tu f (λ) Tu (z) = . ζ −λ z−λ Proof. We have Z f ζf Tu dσ (z) = ζ −λ T (1 − ζλ)(1 − ζz) Z Z 1 1 Tu f (z) − Tu f (λ) f f = dσ − dσ = z − λ T 1 − ζz z − λ T 1 − ζλ z−λ and the result follows. ¤ As an immediate application of the lemma we obtain that the kernel of such an operator is a nearly invariant subspace of (H p )m . Then by Theorem 1.1 we obtain that ker Tu ⊂ ker Tv where v is a matrix as given in Theorem 1.1. The above result has an immediate generalization for kernels of products of Toeplitz operators which will be given below. Given u = (u1 , . . . , un ) ∈ (L∞ (T, Mm×m (C)))n , we define the operator Tu : (H p )m → (H p )mn by (2.13) Tu f = (f t , (Tun f )t , . . . (Tu2 · · · Tun f )t )t , f ∈ (H p )m . Note that the operator Tu1 does not appear in this definition. Lemma 2.2. If u = (u1 , . . . , un ) ∈ (L∞ (T, Mm×m (C)))n then Tu ker Tu1 · · · Tun is a closed nearly invariant subspace of (H p )mn . Proof. The fact that the subspace is closed is obvious by the continuity of Toeplitz operators. To see the near-invariance, let f ∈ ker Tu1 · · · Tun and λ ∈ D with f (λ) = Tun f (λ) = . . . = Tu2 · · · Tun f (λ) = 0 . ZERO PRODUCTS OF TOEPLITZ OPERATORS 15 Then using repeatedly Lemma 2.1 we get Tu · · · Tun f f Tu f f = n , . . . , Tuj · · · Tun = j ζ −λ ζ −λ ζ −λ ζ −λ for all j ≥ 2. Also, Tun Tu1 · · · Tun Thus, f Tu · · · Tun f − Tu1 · · · Tun f (λ) = 1 = 0. ζ −λ ζ −λ Tu f f = Tu ζ −λ ζ −λ and f ∈ ker Tu1 · · · Tun . ζ −λ which completes the proof. ¤ The main result of this section will be a refinement of Lemma 2.2. In order to state our theorem we need the following notations. We shall use throughout multi-indices i = (i1 , . . . , il ) where the integers i1 , . . . , il satisfy i1 < i2 < . . . < il . For s ≥ l we denote by Isi = (aij )1≤i,j≤s the s × s-matrix with entries aij = 1 if i = j = iν , 1 ≤ ν ≤ l, and aij = 0 otherwise. We shall keep the usual notation for the identity matrix, that is, if s = l and i0 = (1, . . . , l) we write Isi0 = Is . Next, if R is a subspace of Cm , a set J ⊂ {1, . . . , m} will be called a coordinate basis for R if the functionals of evaluation of the j-th coordinate, j ∈ J , form a basis in the algebraic dual of R. Finally, if M is a closed subspace of (H p )m and z ∈ T, consider the vector space Mz ⊂ Cm consisting of all nontangential limits f (z), f ∈ M (whenever these limits exist). It is not difficult to verify that dim Mz = max dimM (λ) λ∈D a.e. on T. Indeed, let λ0 ∈ D with dimM (λ0 ) = max dimM (λ) = r . λ∈D Then there exist f1 , . . . , fr ∈ M with f1 (λ0 ), . . . , fr (λ0 ) linearly independent. As we have seen before, for every λ ∈ D except possibly for a discrete set, f1 (λ), . . . , fr (λ) are linearly independent and again since determinants formed with the components of these vectors are functions in the Nevanlinna class of the unit disc it follows that f1 ,. . . ,fr are linearly independent a.e. on T. Moreover, if f ∈ M then f can be 16 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ expressed as a linear combination of the functions fj , 1 ≤ j ≤ m with coefficients in the Nevanlinna class, which shows that _ Mz = {f1 (z), . . . , fr (z)} a.e. . The main result of this section is the following theorem. Recall that σ denotes the normalized Lebesgue measure on the unit circle. Theorem 2.1. Let u = (u1 , . . . , un ) ∈ (L∞ (T, Mm×m (C)))n , R a fixed subspace of Cm such that M = {f ∈ ker Tu1 · · · Tun : f (λ) ∈ R, λ ∈ D} 6= {0} , and let γ ≥ 0 be an integer with σ({z ∈ T : dim ker((u1 · · · un )(z)|Mz ) ≤ γ}) > 0 . Let i = (i1 , . . . , iM ), M ≤ mn, be a multi-index with the following properties: (i) {i1 , . . . , iM } ∩ {1, . . . , m} is a coordinate basis for R; (ii) maxλ∈D dimImn i T M (λ) = maxλ∈D dimTu M (λ) . u i Then Imn Tu M is a closed nearly invariant subspace of (H p )mn with index (k, k ∗ ), where k = maxλ∈D dimTu M (λ) and k − γ ≤ k∗ ≤ k . At this stage, the use of the subspace R may appear as an unnecessary complication, but this more general form of the theorem will be used in full strength later. The proof is based on the following lemma which is a quantitative version of a result on the symbol homomorphism on the Toeplitz algebra; for m = 1 and p = 2 this goes back to Douglas (see [5] or [2]). We have included a proof for the sake of completeness. For a general Cm -valued function h on D we denote by nt − limλ→z h(λ) the nontangential limit of h at z ∈ T whenever this limit exists. The nontangential lim sup is defined in a similar way. Lemma 2.3. If u1 , . . . , un ∈ L∞ (T, Mm×m (C)) and f ∈ (H p )m then ¶ Z µ f dσ(ξ) 2 nt − lim (1 − |λ| ) Tu1 . . . Tun (ξ) λ→z 1 − λζ 1 − λξ T = u1 (z) . . . un (z)f (z) a.e. on T. Proof. We proceed by induction. The case when n = 1 is a direct consequence of Fatou’s theorem. We actually have a stronger estimate which ZERO PRODUCTS OF TOEPLITZ OPERATORS 17 will be used in the sequel. If P+ denotes the usual Szegő projection then for almost every z ∈ T we have µ ¶ µ ¶ f 1 u1 f 1 Tu1 (ξ)− u1 (z)f (z) = P+ − u1 (z)f (z) (ξ) 1 − λζ 1 − λξ 1 − λζ 1 − λζ a.e. on T. Then by the M. Riesz theorem we obtain (2.14) 2 p−1 (1 − |λ| ) °p µ ¶ Z ° ° ° f 1 ° dσ(ξ) °Tu1 (ξ) − u (z)f (z) 1 ° m ° 1 − λζ 1 − λξ T C Z dσ(ξ) ≤ Cp (1 − |λ|2 )p−1 ku1 (ξ)f (ξ) − u1 (z)f (z)kpCm , |1 − λξ|p T where Cp > 0 depends only on p. It is not difficult to see that the right hand side converges nontangentially to zero as λ → z for almost every z ∈ T. This follows by standard estimates from [11, p. 57] adapted to the disk instead of the half-plane. Namely, it is shown there that Fatou’s theorem holds for a wide class of kernels bounded by a constant multiple of a certain integrable function; our kernel actually equals such a function. Now assume that the statement holds for some n ≥ 1. For u1 , . . . , un+1 ∈ L∞ (T, Mm×m (C)), f ∈ (H p )m and x ∈ Cm write À ¶ µ Z ¿ f 1 Tu1 · · · Tun+1 x dσ(ξ) (ξ), 1 − λζ 1 − λξ Cm T ¶ À ¶ µ µ Z ¿ 1 f = x (ξ) dσ(ξ) Tu2 · · · Tun+1 (ξ), Tu∗1 1 − λζ 1 − λζ T Cm ¶ µ ¶ À µ Z ¿ 1 1 f ∗ = x (ξ) − u1 (z)x dσ(ξ) Tu2 · · · Tun+1 (ξ), Tu∗1 1 − λζ 1 − λζ 1 − λξ T Cm À µ ¶ Z ¿ 1 f ∗ + u1 (z)x dσ(ξ) Tu2 · · · Tun+1 (ξ), 1 − λζ 1 − λξ T Cm = J1 (λ, z) + J2 (λ, z) . Use Hölder’s inequality to conclude that ° ° ° f ° ° (1 − |λ| )|J1 (λ, z)| ≤ (1 − |λ| )kTu2 · · · Tun+1 k ° ° 1 − λζ ° p °q µ ¶1/q µZ ° ¶ ° ° 1 1 ∗ °Tu∗ ° dσ(ξ) , × (z)x x (ξ) − u 1 ° 1 1 − λζ ° m 1 − λξ T C 2 2 18 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ + 1q = 1. By the standard estimates mentioned above we have ° ° ° f ° 2 1/q ° ° <∞ nt − lim sup(1 − |λ| ) ° 1 − λζ °p λ→z a.e. on T. Using also (2.14) and the same estimates we obtain that °q µ ¶1/q µZ ° ¶ ° ° 1 1 2 1/p ∗ ° ° =0 nt−lim (1−|λ| ) °Tu∗1 1 − λζ x (ξ) − 1 − λξ u1 (z)x° m dσ(ξ) λ→z T C where that 1 p a.e. on T, i.e. nt − limλ→z (1 − |λ|2 )J1 (λ, z) = 0 a.e. on T. By the induction hypothesis we have that nt − lim (1 − |λ|2 )J2 (λ, z) = hu2 (z) · · · un+1 (z)f (z), u∗1 (z)xi λ→z a.e. on T and the result follows. ¤ We are now ready to prove our theorem. Proof of Theorem 2.1. Note that by assumption (i) on the multi-index i, if (2.15) {i1 , . . . , iM } ∩ {1, . . . , m} = {i1 , . . . , il } then every coordinate of a vector f ∈ M can be written as a linear combination of the coordinates f i1 , . . . , f il with constant coefficients independent of f . This, together with the continuity of Toeplitz i operator-matrices, implies that Imn Tu M is closed. i To see that Imn Tu M is nearly invariant let λ0 ∈ D be such that dimImn i T M (λ0 ) = max dimI i T M (λ) = max dimTu M (λ) u mn u λ∈D λ∈D (here we are using assumption (ii)) and let f ∈ M be such that i Imn Tu f (λ0 ) = 0. Assumption (i) on i implies that Tu f (λ0 ) = 0 hence by Lemma 2.2 and its proof we have Tu f f = Tu ∈ Tu ker Tu1 · · · Tun . ζ − λ0 ζ − λ0 The fact that f (α) ∈ R , α ∈ D \ {λ0 } and f 0 (λ0 ) ∈ R α − λ0 is obvious, hence f ζ−λ0 ∈ M. Thus, i f Tu f Imn i i Tu Tu M , = Imn ∈ Imn ζ − λ0 ζ − λ0 i Tu M is nearly invariant. which shows that Imn ZERO PRODUCTS OF TOEPLITZ OPERATORS 19 Let us now estimate k ∗ . Let λ0 ∈ D \ {0} such that i i i dim Imn Tu M/(Imn Tu M ∩ (1 − λ0 ζ)Imn Tu M) = k ∗ and consider a basis of cosets [g1 ], . . . , [gk∗ ] in this quotient space. Then there exists a discrete subset Λ of D such that [g1 ], . . . , [gk∗ ] is a basis i i i Tu M) for every λ ∈ D \ Λ. Thus in Imn Tu M/(Imn Tu M ∩ (1 − λζ)Imn i if λ ∈ D \ Λ and g ∈ Imn Tu M there exist scalars a1 (λ), . . . ak∗ (λ) such that ! à k∗ X 1 i (2.16) Gλ = aν (λ)gν ∈ Imn Tu M . g+ 1 − λζ ν=1 It is easily seen that aν extend to meromorphic functions in the Nevanlinna class on D. To verify this we use (2.16) to obtain i i λ(1−λ0 ζ)Gλ = (λ−λ0 )Gλ +λ0 (1−λζ)Gλ ∈ Imn Tu M∩(1−λ0 ζ)Imn Tu M that is, Z λ (1 − λ0 ζ)Gλ · hdσ = 0 , T i i whenever h ∈ (Imn Tu M ∩ (1 − λ0 ζ)Imn Tu M)⊥ and λ ∈ D. If we now choose functions i i hµ ∈ (Imn Tu M ∩ (1 − λ0 ζ)Imn Tu M)⊥ , such that 1 ≤ µ ≤ k∗ Z gν · hµ dσ = δµν , T this leads to the linear system of equations B(λ)(a1 (λ), . . . ak∗ (λ))t = (x1 (λ), . . . , xk∗ (λ))t , λ ∈ D, where the matrix B(λ) = (bµν (λ))1≤µ,ν≤k∗ is given by Z gν · hµ dσ , 1 ≤ µ, ν ≤ k ∗ , λ ∈ D , bµν (λ) = λ (1 − λ0 ζ) 1 − λζ T and Z g · hµ xµ (λ) = −λ (1 − λ0 ζ) dσ, λ ∈ D, 1 ≤ µ ≤ k ∗ . 1 − λζ T We have B(λ0 ) = λ0 Ik∗ , and the entries bµν (λ) of B(λ) are complex conjugates of analytic functions of λ in the Nevanlinna class on D. This implies that there exists a discrete subset Λ1 of D such that det B(λ) 6= 0 for all λ ∈ D \ Λ1 . Since the components xµ , 1 ≤ µ ≤ k ∗ , are also complex conjugates of analytic functions of λ in the Nevanlinna class on D, the claim follows immediately from the equality (a1 (λ), . . . ak∗ (λ))t = B −1 (λ)(x1 (λ), . . . , xk∗ (λ))t , λ ∈ D \ Λ1 . 20 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ Denote by Ts h, m < s ≤ mn, the corresponding components of Tu h. Each Ts is a 1 × m operator-matrix whose entries are finite sums of products of Toeplitz operators of the form Tsµ (s) X NY = j Tvijµ , 1 ≤ µ ≤ m. i=1 Also, let Tes (z) be the 1 × m matrix-valued functions with entries Tesµ (z) = (s) X NY j vijµ (z) , 1 ≤ µ ≤ m. i=1 i i If g, g1 , . . . , gk∗ are as above, write g = Imn Tu f and gν = Imn Tu fν with ∗ f, fν ∈ M, 1 ≤ ν ≤ k , and let ∗ fλ = f + k X aν (λ)fν . ν=1 Then (2.16) can be written as 1 i i Imn Tu fλ = Imn Tu F λ , 1 − λζ fλ (z) with Fλ ∈ M. Note that for each z ∈ D we have 1−λz , Fλ (z) ∈ R and since the first dim R components of i form a coordinate basis of R we conclude that fλ (z) = Fλ (z) , z ∈ D . 1 − λz Thus for a.e. ξ ∈ T we have µ ¶ fλ (2.17) 0 = Tu1 · · · Tun (ξ) , 1 − λζ and if iM > m then (2.18) 1 (Ts fλ )(ξ) = 1 − λξ µ fλ Ts 1 − λζ ¶ (ξ) , whenever s is an entry of i with s > m. 2 Multiply these equalities by 1−|λ| , integrate on T with respect to σ 1−λξ and let λ approach z ∈ T nontangentially. The limit of the left-hand side of (2.17) is, of course, zero. From the definition of fλ and the fact that the coefficients aν have nontangential limits a.e. on T, on the left ZERO PRODUCTS OF TOEPLITZ OPERATORS 21 hand side of the equations contained in (2.18), in the limit we obtain by Poisson’s formula Z k∗ X 1 − |λ|2 nt − lim (Ts fλ )(ξ)dσ(ξ) = Ts f (z) + aν (z)Ts fν (z) λ→z T |1 − λξ|2 ν=1 a.e. on T. On the right-hand sides of (2.17) and (2.18) we use Lemma 2.3 and the above remark on the coefficients aν to obtain for a.e. z ∈ T µ ¶ Z fλ 1 − |λ|2 nt − lim Tu1 · · · Tun (ξ)dσ(ξ) λ→z T 1 − λξ 1 − λζ à ! k∗ X = (u1 · · · un )(z) f (z) + aν (z)fν (z) ν=1 in (2.17), and µ ¶ Z k∗ X 1 − |λ|2 fλ nt−lim Ts (ξ)dσ(ξ) = Tes (z)f (z)+ aν (z)Tes (z)fν (z) λ→z T 1 − λξ 1 − λζ ν=1 in (2.18). Assume first that iM > m and let j = (1, . . . , m, il+1 , . . .), where l is given by (2.15). Our argument applied to the equations (2.17) and (2.18) yields à ! k∗ X j j (2.19) W (z) Imn Tu f (z) + aν (z)Imn Tu fν (z) = 0 , a.e. on T , ν=1 where W (z) = u1 · · · un (z) 0 0 . . . −Teil+1 (z) 1 0 . . . .. . ... e −TiM (z) 0 0 . . . 0 0 .. . . 1 Here the zeros in the first row are 1 × m zero matrices, while the rest of zeros and ones are scalars. Let j K = {Imn Tu f : f ∈ M} and note that maxλ∈D dimK (λ) = k. As was done in the text preceding the statement of Theorem 2.1, define the vector space Kz ⊂ Cm of all nontangential limits g(z), g ∈ K (whenever these limits exist). Then (2.19) shows that dim W (z)Kz ≤ k ∗ . On the other hand, from the special form of the matrix W (z) we see j Tu fα (z)} is a basis for ker(W (z)|Kz ) then {fα (z)} is linearly that if {Imn 22 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ independent in ker((u1 · · · un )(z)|Mz ), hence dim ker(W (z)|Kz ) ≤ dim ker((u1 · · · un )(z)|Mz ) . By assumption, there exists a set of positive measure E ⊂ T such that dim ker(u1 · · · un (z)|Mz ) ≤ γ on E. If we choose z ∈ E so that dim Kz = k we obtain that k ∗ ≥ dim W (z)Kz ≥ k − γ and the result follows. The case when iM ≤ m is easier. The above argument applies with j = (1, . . . , m) and W (z) = u1 · · · un (z). The proof is now complete. ¤ 3. Two Kernel Inclusion Theorems We want to apply Theorem 1.1 to the nearly invariant subspaces considered in the previous section. This will lead to two kernel inclusion theorems for products of finitely many Toeplitz operator-matrices. Our first result is essentially a direct application of both Theorem 1.1 and Theorem 2.1. We make use of the following notation. Given u1 , . . . , un ∈ L∞ (T, Mm×m (C)) we consider for j ≥ 2 the mn × mn diagonal block-matrices Uj = diag(Im , . . . , Im , uj , . . . , uj ) , where the block Im occurs n − j + 1 times. Note that the matrix-valued function U1 which can, of course, be defined similarly, is not needed in the sequel. Theorem 3.1. Let u = (u1 , . . . , un ) ∈ (L∞ (T, Mm×m (C)))n , let R be a fixed subspace of Cm such that M = {f ∈ ker Tu1 · · · Tun : f (λ) ∈ R, λ ∈ D} 6= {0} , γ ≥ 0 an integer with σ({z ∈ T : dim ker((u1 · · · un )(z)|Mz ) ≤ γ}) > 0 , and let r = dim R, m = maxλ∈D dimM (λ). Then there exists a multi-index i with the properties stated in Theorem 2.1 and V ∈ L∞ (T, Mmn×mn (C)) with rank(V ) ≥ mn − r + m − γ a.e. on T such that Mnd ⊂ ker TV Imn TU2 · · · TUn , i n where Md stands for the diagonal in Mn . ZERO PRODUCTS OF TOEPLITZ OPERATORS 23 Proof. Let k = maxλ∈D dimTu M (λ) We will show that r + k − m ≤ mn and construct the multi-index i = (i1 , . . . , ir+k−m ) as follows. We start with λ0 ∈ D such that dimM (λ0 ) = m , max g1 ,...,gmn ∈M rank(Tu gj (λ0 ))1≤j≤mn = k . Let f1 , . . . fm ∈ M be such that f1 (λ0 ), . . . , fm (λ0 ) are linearly independent and let the indices 1 ≤ i1 < . . . < im ≤ m be chosen so that det(fsil (λ0 ))1≤s,l≤m 6= 0. We can find f1 , . . . fm ∈ M as above and fm+1 , . . . fmn such that rank(Tu fj (λ0 ))1≤j≤mn = k . This is because the sets of tuples (f1 , . . . , fm ) and (g1 , . . . , gmn ) as above are easily seen to be open and dense in Mm and Mmn respectively. If m < k, there exist j1 , . . . , jk−m which satisfy j1 < . . . < jk−m ≤ mn and the corresponding rows of (Tu fj (λ0 ))1≤j≤mn are linearly independent. Obviously, j1 > m, which shows that mn − m > k − m, hence r + k − m ≤ mn (whenever m < k). If m = k, the inequality is obvious. Finally, note that the functionals of evaluation of the il -th coordinate, 1 ≤ l ≤ m, on R must be linearly independent because of the linear independence of f1 (λ0 ), . . . fm (λ0 ). Then by a straightforward inductive procedure, it is easy to see that there exist im+1 , . . . , ir such that im < im+1 < . . . < ir ≤ m and {i1 , . . . , ir } is a coordinate basis for R. Now define (i1 , . . . , ir+k−m ) ½ (i1 , . . . , ir , j1 , . . . , jk−m ) , if m < k i= (i1 , . . . , ir ) , if m = k which obviously also satisfies conditions (i) and (ii) of Theorem 2.1. i We now apply Theorem 2.1 to obtain that Imn Tu M is a closed nearly p mn ∗ invariant subspace of (H ) with index (k, k ), where k − γ ≤ k ∗ ≤ i k. Suppress the zero entries of the vectors in the subspace Imn Tu M p k+r−m . By and identify it with a nearly invariant subspace N of (H ) Theorem 1.1 applied to N there exists v ∈ L∞ (T, Mk∗ ×(k+r−m) (C)), rank(v) = k ∗ , such that N ⊆ ker Tv . We then add mn − k ∗ zero rows to the matrix v in arbitrary positions. If mn − k − r + m = 0, we have obtained the desired matrix V . If mn − k − r + m > 0, add mn − k − r + m zero columns so as to make it a square mn × mn matrix w. After permuting some columns in w if necessary, we may assume that if the l-th zero column in w is one of the newly added zero columns, then i is zero. Note that the entry in the place (l, l) on the diagonal of Imn 24 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ mn−k ≥ mn−k−r+m. Finally, in these newly added zero columns we place exactly one entry = 1 in each one of the newly added rows off the diagonal. Note that this is possible because mn − k ∗ ≥ mn − k − r + m. Let V the matrix obtained in this way. Note that in either case rank (V ) = k ∗ +mn−k−r+m ≥ mn−r+m−γ. i i Also, it is easy to verify that V Imn = wImn . Finally, by construction we f Tun f TV Imn i .. . i have Tw Imn Tu M = {0} hence Tu2 · · · Tun f I 0 ... 0 0 I 0 ... 0 0 .. 0 I . . . 0 0 ... . . . . = TV Imn i .. × ... ... . 0 0 . . . Tu3 0 0 0 . . . 0 Tu2 0 0 . . . 0 Tu3 I 0 ... 0 0 f 0 Tun . . . 0 0 f ··· × × . , .. ... ... . .. 0 0 . . . 0 Tun × f which is what we had to prove. ¤ We now turn to our second kernel inclusion theorem. Although it looks rather technical, it is a crucial step towards proving our main theorem. Theorem 3.2. Let v1 , . . . , vs+1 ∈ L∞ (T, Mm×m (C)) with N = ker Tv1 · · · Tvs+1 6= {0}. Suppose that v1 6= 0 a.e. on T and that for 1 ≤ l ≤ s + 1 there exists a set of positive measure El ⊂ T such that vl |El is invertible. Also assume that for 3 ≤ j ≤ s + 1 (3.20) max dimTvj ···Tvs+1 N (λ) = m. λ∈D If γ0 < m is an integer with dim ker v1 (ξ) ≤ γ0 a.e. on T, then the diagonal Nds! in N s! satisfies (3.21) Nds! ⊂ ker TV Is!m TW1 · · · TWs , i where V ∈ L∞ (T, Ms!m×s!m (C)) with rank(V ) ≥ s!m − γ0 , ZERO PRODUCTS OF TOEPLITZ OPERATORS 25 a.e. on T, i is a multi-index with at least m entries in {1, . . . , (s−1)!m} which form a coordinate basis for the diagonal in (C m )s! , Ws = diag(vs+1 , . . . , vs+1 ) ∈ L∞ (T, Ms!m×s!m (C)), and W1 , . . . , Ws−1 are block diagonal matrix-valued functions of the form Wj (z) = diag(I(s−1)!m , Wj1 (z), . . . Wjs−1 (z)) with Wjl ∈ L∞ (T, M(s−1)!m×(s−1)!m (C)), 1 ≤ j, l ≤ s − 1. Proof. We proceed by induction. For s = 2 note that trivially ker Tv1 Tv2 ⊃ Tv3 N and, by assumption, maxλ∈D dimTv3 N (λ) = m. Thus, we can apply Theorem 3.1 to the product Tv1 Tv2 with R = Cm and m = r = m, γ = γ0 (since v2 is invertible on a set of positive measure) and obtain (Tv3 N )2d ⊂ ker TV I2m i TU with V , U and i as in Theorem 3.1. This clearly yields the statement in the case s = 2. Assume the result holds true for some integer s ≥ 2 and let v1 , . . . , vs+2 be as in the statement. Since Tvs+2 N ⊂ ker Tv1 · · · Tvs+1 it follows easily that v1 , . . . , vs+1 satisfy (3.20), hence by the induction hypothesis, (3.21) holds for v1 , . . . vs+1 with matrix-valued functions V, W1 , . . . Ws and multi-index i. We want to apply Theorem 3.1 to the product TW1 · · · TWs T = TV Is!m i of s + 1 Toeplitz operator-matrices on (H p )s!m , with R being the diagonal in (Cm )s! . Clearly, r = dim R = m. Moreover, if M = { (f t , f t , . . . , f t )t ∈ ker T } , | {z } s! times then from the fact that (Tvs+2 f t , . . . , Tvs+2 f t )t ∈ M whenever f ∈ ker Tv1 · · · Tvs+2 , we deduce that m = max dimM (λ) = m. λ∈D In order to find an integer γ as in Theorem 3.1 recall that vs+1 is defined and invertible on a set of positive measure Es+1 ⊂ T. Then for every z ∈ Es+1 , Ws (z) is invertible and if Mz is defined then Ws (z)Mz is 26 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ contained in (actually equal to) the diagonal in (Cm )s! . But by the i conditions on i and W1 , . . . Ws−1 , it follows that Is!m W1 (z) · · · Ws−1 (z) i is injective on Mz , hence Is!m W1 (z) · · · Ws (z)|Mz is injective. Thus by hypothesis i dim ker V (z)Is!m W1 (z) · · · Ws (z)|Mz ≤ γ0 a.e. on Es+1 and we can apply Theorem 3.1 with the parameters given above and γ = γ0 . We obtain that (s+1)! (ker Tv1 · · · Tvs+1 )d ⊂ Ms+1 ⊂ ker TV1 I i1 d (s+1)! TU2 . . . TUs+1 , where V1 , Uj and the multi-index i1 are as in the theorem. In particular, they satisfy the requirements in the statement. This means that for all f ∈ ker Tv1 · · · Tvs+2 we have TV1 I i1 (s+1)!m TU2 . . . TUs+1 (Tvs+2 f t , . . . , Tvs+2 f t )t = 0, i.e. (3.21) holds for v1 , . . . vs+2 with V = V1 , Wν = Uν+1 , 1 ≤ ν ≤ s, i = i1 , and the result follows. ¤ 4. Main result We can now prove the main result announced in the introduction. This will be deduced from the following stronger theorem. Theorem 4.1. Let u1 , . . . , un ∈ L∞ (T, Mm×m (C)) and assume that for 1 ≤ j ≤ n there exists a set of positive measure Ej ⊂ T such that uj |Ej is invertible. Then max dimTu1 ···Tun (H p )m (λ) = m . λ∈D Proof. Assume the contrary, i.e. max dimTu1 ···Tun (H p )m (λ) < m. λ∈D Then there exists k ∈ {1, . . . , n} such that max dimTul ···Tun (H p )m (λ) = m λ∈D if l > n − k + 1 and γ0 = max dimTun−k+1 ···Tun (H p )m (λ) < m . λ∈D By Proposition 1.2 there exists u0 ∈ H ∞ (Mm×m (C)) with dim ker u0 (z) = γ0 a.e. on T such that Tu0 Tun−k+1 · · · Tun = 0 . ZERO PRODUCTS OF TOEPLITZ OPERATORS 27 Then, if k > 1, we can apply Theorem 3.2 to the product Tu0 Tun−k+1 · · · Tun to obtain that [(H p )m ]k! ⊂ ker TV Ik!m TW1 · · · TWk i with V, W1 , . . . , Wk and the multi-index i as in that statement. Thus for f ∈ H p and λ ∈ D we have ¶ Z µ 1 − |λ|2 t 1 t t TV Ik!m TW1 · · · TWk (f , . . . , f ) (ξ) dσ(ξ) = 0 . i 1 − λζ 1 − λξ T Let λ approach nontangentially to z ∈ T and apply Lemma 2.3 to obtain for a.e. z ∈ T and every f ∈ (H p )m i V (z)Ik!m W1 (z) · · · Wk (z)(f (z)t , . . . , f (z)t )t = 0. Since dim ker V (z) ≤ γ0 a.e. this implies i rank(Ik!m W1 (z) · · · Wk (z)) ≤ γ0 < m a.e. on T. On the other hand, i has at least m entries in {1, . . . , k!m} which form a coordinate basis for the diagonal in (Cm )k! and for 1 ≤ j ≤ k, Wj (z) has the form Wj (z) = diag(Ik!m , Wj1 (z), . . . Wjn (z)), so that the rank in question is at least m. This is a contradiction which concludes the proof. In the case k = 1, we can still apply Theorem 3.2 to Tu0 Tun TIm = Tu0 Tun . The argument is completely analogous to the above so we omit it. ¤ Corollary 4.1. Let u1 , . . . , un ∈ L∞ (T, Mm×m (C)) and assume that for 1 ≤ j ≤ n there exists a set of positive measure Ej ⊂ T such that uj |Ej is invertible. Then Tu1 · · · Tun cannot have finite rank. Proof. Suppose the operator S = Tu1 Tu2 . . . Tun is of rank N . Then there exist N non-zero Toeplitz operators: Tv1 ,. . . ,TvN such that (4.22) W TvN . . . Tv1 S = 0 . Indeed, if S((H p )m ) = {f1 , . . . , fN } with f1 ,. . . ,fN ∈ (H p )m , choose v1 to be the diagonal m × m matrix whose diagonal entries are ( l ζf1 , if f1l 6= 0 f1l ull = 1, otherwise Then Tv1 f1 = P (ζf1 ) = 0 28 ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ and therefore Tv1 S((H p )m ) ⊂ _ {Tv1 f2 , . . . , Tv1 fN } . Hence dim Tv1 S((H p )m ) ≤ N − 1, so we can proceed inductively to obtain (4.22). Now Theorem 4.1 implies that one of the operators Tv1 ,. . . ,TvN , Tu1 , . . . , Tun is the zero operator. Since none of the operators Tv1 ,. . . ,TvN is zero by construction, it follows that some Tuν = 0, which contradicts the assumptions of the statement. This ends the proof. ¤ One may ask if analogous results hold for Toeplitz operators acting on Bergman spaces. In the case n = 2, m = 1, p = 2 this has been proved for harmonic symbols by Ahern and Čučković in [1]. The question remains open for n ≥ 3; however, we have not been able to adapt our methods to that context. References [1] P. Ahern and Ž. Čučković, A theorem of Brown-Halmos type for Bergman space Toeplitz operators, J. Funct. Anal. 187 (2001), 200– 210. [2] J. Barrı́a and P. Halmos, Asymptotic Toeplitz operators, Trans. Amer. Math. Soc. 273 (1982), no. 2, 621–630. [3] A. Brown and P. Halmos, Algebraic properties of Toeplitz operators, J. reine angew. Math. 213 (1963), 89–102. [4] J. Cima, A. Matheson, and W. T. Ross, The Cauchy Transform, Mathematical Surevys and Monongraphs 125, AMS, Providence, RI, 2006. [5] R. G. Douglas, Banach algebra techniques in the theory of Toeplitz operators, Expository Lectures from the CBMS Regional Conference (Athens, GA 1972.) Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 15, American Mathematical Society, Providence, R.I., 1973. [6] C. Gu, Products of several Toeplitz operators, J. Funct. Anal. 171 (2000), no. 2, 483–527. [7] K. Y. Guo, A problem on products of Toeplitz operators, Proc. Amer. Math. Soc. 124 (1996), no. 3, 869–871. [8] D. Hitt, Invariant subspaces of H 2 of an annulus, Pacific J. Math. 134 (1988), no. 1, 101-120. [9] P. Koosis, Introduction to Hp spaces, Second edition, Cambridge University Press, Cambridge 1998. [10] D. Sarason, Nearly invariant subspaces of the backward shift, in: Contributions to operator theory and its applications, (Mesa, AZ, 1987), 481-493, Oper. Theory Adv. Appl. 35, Birkhäuser, Basel, 1988. ZERO PRODUCTS OF TOEPLITZ OPERATORS 29 [11] E. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ 1993. Alexandru Aleman, Department of Mathematics, Lund University, P.O. Box 118, S-221 00 Lund, Sweden E-mail address: [email protected] Dragan Vukotić, Departamento de Matemáticas & ICMAT, Módulo C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain E-mail address: [email protected]