Exterior powers and pointwise creation operators

We develop a theory of pointwise wedge products of vector-valued functions on the circle and the disc, and obtain results which give rise to a new approach to the analysis of the matricial Nehari problem. We investigate properties of pointwise creation operators and pointwise orthogonal complements in the context of operator theory and the study of vector-valued analytic functions on the unit disc.


Introduction
The wedge product of Hilbert spaces, though a long established theory, deserves in our view to be better exploited in operator theory than it has been hitherto. In this paper we put forward a new approach to some aspects of the analysis of E-valued functions on the unit disc D and on the unit circle T, where E is a separable Hilbert space. Our first application of this idea was to the problem of the superoptimal analytic approximation of a continuous matrix-valued function on the unit circle [2]. The computation of such approximants arises naturally in the context of the classical "Nehari problem", and also in its application to the "robust stabilization problem" in control engineering. In [2] we give a new algorithm for the construction of the unique superoptimal analytic approximant of a continuous matrix-valued function on the unit circle, making use of exterior powers of operators in preference to spectral or Wiener-Masani factorizations. This algorithm is parallel to the construction of [10], but has the advantage that it requires only the spectral factorisation of scalar functions on the circle, together with the calculation of singularvalue decompositions, and otherwise requires only rational arithmetic. In the algorithm we make use of pointwise creation operators and pointwise orthogonal complements.
We show how the Hilbert space geometry of the Hardy space H 2 (D, E), where E is a separable Hilbert space, interacts with the pointwise geometry of H 2 (D, E), that is, the geometry of E over each point of T or D. We work with the completion of the algebraic tensor product of Hilbert spaces, namely the Hilbert tensor product (see [4] and [6]), and, in particular, with its closed linear subspace of antisymmetric tensors. The space of all antisymmetric tensors ∧ p E, also called a wedge or exterior product, is a closed linear subspace of the p-fold Hilbert tensor product ⊗ p H E, see Section 2. In Section 3, we define the pointwise wedge product of maps that are defined either on the unit disc or on the unit circle and take values in Hilbert spaces. We illustrate the fact that some classical results from function theory extend to the pointwise wedge product. An exemplary result is Proposition 3.7 which asserts that, for any separable Hilbert space E, the pointwise wedge product of two functions in the Hardy space H 2 (D, E) is an element of H 1 (D, ∧ 2 E). We study pointwise creation operators, which are of the form where ξ : D → E is a bounded analytic function, and the symbol ξ∧f denotes the function (ξ∧f )(z) = ξ(z) ∧ f (z) for all z ∈ D.
Analogous notation applies when ξ, f are E-valued functions defined almost everywhere on T. We connect the kernel of C ξ to the pointwise linear span and pointwise orthogonal complement of ξ.
In Section 4 we prove, among other things, the following two statements.
Theorem 4.21 Let E be a separable Hilbert space. Let ξ ∈ H ∞ (D, E) be an inner function. Then, for any h ∈ H 2 (D, E), Thus C * ξ C ξ is the Toeplitz operator with symbol 1 − ξξ * . Finally, we show that although C ξ is an isometry from the pointwise orthogonal complement of ξ to H 2 (D, ∧ 2 E), a simple example shows that C ξ fails to be a partial isometry.

Exterior powers of Hilbert spaces and operators
In this Section we recall the established notion of the exterior power, or wedge product, of Hilbert spaces and we explore wedge products of bounded linear operators acting between Hilbert spaces. One can find definitions and properties of wedge products in [3], [5], [9] and [11]. Here we present a concise version of this theory which we need for the development of pointwise wedge products of vector-valued functions on the circle and the disc. We shall assume the notion of the algebraic tensor product of linear spaces as given, for example, in [6].
2.1. Exterior powers. In this subsection, for E a Hilbert space, we consider the completion of the p-fold algebraic tensor product, denoted by ⊗ p H E, with respect to the norm induced by the inner product (2.1). We define an action of permutation operators on tensor products and examine various properties. These permutation operators generate two different types of tensors, symmetric and antisymmetric, and we focus on the properties of the antisymmetric tensors. The space of all antisymmetric tensors, also called wedge or exterior products, in ⊗ p H E is denoted by ∧ p E and in Theorem 2.9 we prove it is a closed linear subspace of ⊗ p H E. In the following E denotes a Hilbert space.
Definition 2.2. An inner product on ⊗ p E is given on elementary tensors by for any x 1 , . . . , x p , y 1 , . . . , y p ∈ E, and is extended to ⊗ p E by sesqui-linearity.
Observe that the inner product (2.1), in contrast to the majority of the ones included in the bibliography, invokes a multiple of p! . The reason for this choice will be apparent in Theorem 2.11.
In order to introduce antisymmetric tensors, we need to consider the action of the following permutation operators on tensors.
Definition 2.4. Let S p denote the symmetric group on {1, . . . , p}, with the operation of composition. For σ ∈ S p , we define for any x i j ∈ E and λ i ∈ C.
Remark 2.5. (S p , •) is a group, and so, for every permutation σ ∈ S p , there exists where id ∈ S p is the identity map on {1, . . . , p}. Then, Elements of S p induce unitary operators on ⊗ p H E in an obvious way.
Proposition 2.6. Let E be a Hilbert space, and let p be a positive integer. Then, for any σ ∈ S p , S σ is a linear operator on the normed space (⊗ p E, · ), which extends to an isometry S σ on (⊗ p H E, · ). Furthermore, S σ is a unitary operator on ⊗ p H E. Proof. It is easy to check that S σ is linear. For any elementary tensors w = x 1 ⊗x 2 ⊗· · ·⊗x p , v = y 1 ⊗ y 2 ⊗ · · · y p by the definition of the inner product on ⊗ p E, Hence S * σ = S σ −1 , and therefore S * σ S σ = S σ −1 S σ = I, the identity operator on ⊗ p E, and therefore S σ is an isometric linear self-map of ⊗ p E. Likewise, S σ S * Theorem 2.9 implies that the projection onto ∧ p E is well-defined. Definition 2.10. Let E be a Hilbert space. For x 1 , . . . , x p ∈ E, define x 1 ∧ x 2 ∧ · · · ∧ x p to be the orthogonal projection of the elementary tensor x 1 ⊗ x 2 ⊗ · · · ⊗ x p onto ∧ p E, that is Theorem 2.11. For all u ∈ ⊗ p H E, Proof. Let u ∈ ⊗ p H E. Then, for any σ ∈ S p , u = ǫ σ S σ (u) + (u − ǫ σ S σ (u)), and so Since the first sum on the right hand side is clearly antisymmetric, it suffices to show that is orthogonal to the set of antisymmetric tensors, in other words, For every w = σ∈Sp ǫ σ S σ (u) ∈ ⊗ p H E and for every τ ∈ S p , we have Remark 2.12. If p > 1, then S p contains a transposition, for instance σ = (1 2), and Proposition 2.13. Let E be a Hilbert space and let p ≥ 2. The set of antisymmetric tensors and the set of symmetric tensors are orthogonal in ⊗ p H E. Proof. Suppose that u is a symmetric tensor, that is S σ u = u for all σ ∈ S p , and that v is an antisymmetric tensor, that is, S σ v = ǫ σ v for all σ ∈ S p . Since p ≥ 2 there exists a transposition σ ∈ S p , so that ǫ σ = −1, and therefore 14. Let E be a Hilbert space. The inner product in ∧ p E is given by for all x 1 , . . . , x p , y 1 , . . . , y p ∈ E.
Proof. By Theorem 2.11, we have Since we have already shown that ∧ p E is a closed linear subspace of the Hilbert space ⊗ p H E, the space (∧ p E, ·, · ∧ p E ) with inner product given by Proposition 2.14 is itself a Hilbert space.
By Proposition 2.14, By assumption, and hence If, for k = 1, · · · , j we multiply the k-th column of the determinant by u k , x E and subtract it from the (j + 1)-th column, we find that the latter equality by Pythagoras' theorem.
Moreover, we define a norm on Definition 2.17. Let E be a Hilbert space. We define the multilinear operator Proposition 2.18. [Hadamard's inequality, [7], p. 477] For any matrix Proposition 2.19. Let E be a Hilbert space. Then the multilinear mapping Observe that the matrix is Hermitian, and so its determinant is real. By Hadamard's inequality, Moreover, by the Cauchy-Schwartz inequality, Hence the p-linear operator Λ is bounded.

Wedge products of bounded linear operators.
Definition 2.20. Suppose H 1 , . . . , H p , K 1 , . . . , K p are Hilbert spaces and T i : H i → K i is a bounded linear operator for i = 1, . . . , p. We define the operator on algebraic tensor products
Alternative notations for T 1 ⊗ · · · ⊗ T p are ⊗ p i=1 T i , and, in the case that T i = T for each i, ⊗ p T . Proposition 2.21. Let H i and G i be Hilbert spaces, and let T i : H i → G i be bounded linear operators for i = 1, . . . , p. Then, the operator T 1 ⊗ · · · ⊗ T p of equation (2.2) has a continuous extension Proofs of this fact are given in [4, Chapitre 1, Section 2] and in [1, Section 3.5] Proposition 2.22. Let H, K be Hilbert spaces and let T : Definition 2.23. Let H, K be Hilbert spaces and T : H → K be a bounded linear operator. We define the operator Definition 2.24. Let E be a Hilbert space. We define the linear operator Proof.

Pointwise wedge products
In this section we introduce the notion of pointwise wedge product of vector-valued functions on the unit circle or in the unit disk and explore its features.  for almost all z ∈ T.
3.1. Pointwise wedge products on function spaces. For vector-valued L p spaces we use the terminology of [8]. (ii) H p (D, E) to be the normed space of analytic E-valued maps f : D → E such that is the space of essentially bounded measurable E-valued functions on the unit circle with essential supremum norm Proposition 3.5. Let E be a separable Hilbert space and let 1 p Then and Proof. By Proposition 2.14, for all z ∈ T, By Definition 3.4, Now by Hölder's inequality,  x 1∧ x 2∧ . . .∧x n : D → ∧ n E is also analytic on D and The proof is straightforward. It follows from Proposition 2.14 and Hadamard's inequalities.
Proof. By Proposition 3.6, x∧y is analytic on D. By Proposition 3.5, for Thus, Proposition 3. 10. Let E be a separable Hilbert space, and let ξ 0 , ξ 1 , · · · , ξ j ∈ L ∞ (T, E) be a pointwise orthonormal set on T, and let x ∈ L 2 (T, E). Then Proof. By Proposition 2.15, for almost all z ∈ T, Thus,   Then the radial limits lim r→1 (ξ(re iθ ) ∧ f (re iθ )) exist for almost all e iθ ∈ T and define functions in L 2 (T, ∧ 2 E).

Pointwise creation operators, orthogonal complements and linear spans
Proof. By Proposition 2.25, the bilinear operator Λ : E×E → ∧ 2 E is a continuous operator for the norms of E and ∧ 2 E. By Remark 4.2, the functions ξ ∈ H ∞ (D, E) and f ∈ H 2 (D, E) have radial limit functionsξ ∈ L ∞ (T, E) andf ∈ L 2 (T, E). Also, by Proposition 3.8, ξ∧f ∈ H 2 (D, ∧ 2 E). Hence lim r→1 ξ(re iθ ) ∧ f (re iθ ) −ξ(e iθ ) ∧f (e iθ ) ∧ 2 E = 0 almost everywhere on T and we conclude that This shows that the radial limits lim r→1 (ξ(re iθ ) ∧ f (re iθ )) exist almost everywhere on T and, by Lemma 3.5, define functions in L 2 (T, ∧ 2 E). Hence one can consider (C ξ f )(z) = (ξ∧f )(z) to be defined for either all z ∈ D or for almost all z ∈ T.   We define the pointwise orthogonal complement of X in F to be the set Our next aim is to show that POC(X, F ) is a closed subspace of F. We are going to need the following results.
Lemma 4.7. Let E be a Hilbert space and let x ∈ L 2 (T, E). The function ϕ : L 2 (T, E) → C given by Proof. Consider g 0 ∈ L 2 (T, E). For any ǫ > 0, we are looking for a δ > 0 such that Note that For each e iθ ∈ T, by the reverse triangle inequality, the integrand satisfies By the Cauchy-Schwarz inequality, For the given ǫ > 0, let δ be equal to ǫ x L 2 (T,E) + 1 , and let By equations (4.1) and (4.2), Hence ϕ is a continuous function.
Proof. V is a linear subspace of H 2 (D, E) since for λ, µ ∈ C, ψ, k ∈ V and for almost all z ∈ T, Now suppose that the sequence of functions (g n ) ∞ n=1 in V converges to a function g. We need to show that g ∈ V. Since g n ∈ V for all n ∈ N, we have g n (z), η(z) E = 0 for almost all z ∈ T. (4.3) Consider the function ϕ : H 2 (D, E) → C given by Thus | g(e iθ ), η(e iθ ) E | = 0 for almost all e iθ ∈ T, and, hence, g ∈ V. We have proved that V is a closed subspace of H 2 (D, E).
Our motivation for the next theorem is the following. In [2] spaces of the form play a crucial role in the superoptimal Nehari problem. They are the domains of Hankeltype operators whose norms are "superoptimal singular values" of error functions G − Q, where G is a given continuous approximand and Q is its analytic approximation.
Then, for all w ∈ H 2 (D, E) and for all z ∈ D, due to the pointwise linear dependence of ξ 0 and w, ξ 0 H 2 (D,E) ξ 0 on D. Note that By Proposition 4.9, Ξ 0 is a closed subspace of H 2 (D, E), hence and so, Consider the mapping given by for all w ∈ Ξ 0 . Notice that, by assumption, ξ 0 (e iθ ) 2 E = 1 for almost every e iθ ∈ T. Therefore, for any w ∈ Ξ 0 , we have since w is pointwise orthogonal to ξ 0 almost everywhere on T. Thus the mapping is an isometry. Furthermore, C ξ 0 : Ξ 0 → ξ 0∧ Ξ 0 is a surjective mapping, thus Ξ 0 and ξ 0∧ Ξ 0 are isometrically isomorphic. Therefore, since Ξ 0 is a closed subspace of to be the pointwise orthogonal complement of ξ 0 , . . . , ξ j in H 2 (D, E). Let ψ ∈ H 2 (D, E). We may write ψ as Then, for all ψ ∈ H 2 (D, E) and for almost all z ∈ T, due to the pointwise linear dependence of ξ k and ψ, ξ k L 2 (T,E) ξ k on T.
Proposition 4.11. Let E be a separable Hilbert space, let F be a subspace of L 2 (T, E) and let X be a subset of L 2 (T, E). The space is a closed subspace of F.
Proof. Let λ, µ ∈ C and let f, g ∈ POC(X, F ). Then for all Hence λf + µg ∈ POC(X, F ), and so POC(X, F ) is closed under addition and scalar multiplication. Now let (g n ) ∞ n=1 be a convergent sequence in POC(X, F ) with respect to · L 2 (T,E) and let g ∈ F be its limit. We need to show that g ∈ POC(X, F ). Since, for all n ∈ N, g n ∈ POC(X, F ), we get that for every x ∈ X g n (z), x(z) E = 0 for almost all z ∈ T. (4.4) For each fixed x ∈ X, consider the function ϕ : L 2 (T, E) → C given by Then, by equation (4.4), since g n (e iθ ), x(e iθ ) E = 0 for almost all e iθ ∈ T, we have Moreover, by Lemma 4.7, ϕ is a continuous function on (L 2 (T, E), · L 2 ), and so 1 2π Hence g(z), x(z) E = 0 for almost all z ∈ T. We can show the same way that for every x ∈ X, | g(e iθ ), x(e iθ ) E | = 0 almost everywhere on T. Hence, g ∈ POC(X, F ). We have proved that POC(X, F ) is a closed subspace of F.  By the generalized Fatou's Theorem (see [8], p. 186), the radial limit exists almost everywhere on T and defines a functionf ∈ L p (T, E). The set of points on T at which the above limit does not exist, will be called the singular set of the function f and will be denoted by N f .
Note that the singular sets of functions in H p (D, E) for 1 ≤ p ≤ ∞ are null sets with respect to Lebesgue measure. (f∧g)(z) = f (z)∧g(z) for all z ∈ D belongs to H 2 (D, ∧ 2 E), and moreover the operator for all e iθ ∈ T\(N f ∪N ξ ∪N g ), and P + : L 2 (T, E) → H 2 (D, E) is the orthogonal projection. Here N f , N g , N ξ are the singular sets of the functions f, g and ξ respectively.
Lemma 4.19. Let E and F be separable Hilbert spaces, and let G ∈ L ∞ (T, B(F, E)). For every x ∈ L 2 (T, E), the function Gx, defined almost everywhere on T by belongs to L 2 (T, E).
Proof. For almost all z ∈ T, Definition 4.20. Let E and F be separable Hilbert spaces. Let P + : L 2 (T, E) → H 2 (D, E) be the orthogonal projection operator. Corresponding to any G ∈ L ∞ (T, B(F, E)) we define the Toeplitz operator with symbol G to be the operator given by T G x = P + (Gx) for any x ∈ H 2 (D, F ).
By Theorem 4.21, C * ξ C ξ h = P + α, where, for all z ∈ T, Calculations yield, for all z ∈ T, Thus (P + α)(z) = 1 2 The latter expression is not in the pointwise orthogonal complement of ξ in C 2 , since for all z ∈ T, 1 2 x(e iθ ) 2 E dθ .
Since ξ is inner, ξ(e iθ ) E = 1 almost everywhere on T, hence the last set is equal to x ∈ H 2 (D, E) : 1 2π The following example illustrates the fact that, although C ξ is an isometry on POC{ξ, H 2 (D, C 2 )}, it is not a partial isometry on H 2 (D, C 2 ). Example 4.24. C * ξ C ξ fails to be a projection for some inner function ξ ∈ H ∞ (D, C 2 ). Let us calculate C * ξ C ξ for ξ(z) = 1 √ 2 1 z , z ∈ D. By Theorem 4.21, for h ∈ H 2 (D, E) where, for all z ∈ T, α(z) is given by Thus where S, S * denote the shift and the backward shift operators on H 2 (D, C) respectively. Hence By equation (4.5), since SS * = 1 − P 0 , where P 0 ( ∞ n=−∞ a n z n ) = a 0 . Consequently C * ξ C ξ is not a projection and hence C ξ is not a partial isometry.