Inner functions and zero sets for ℓ^{𝑝}_{𝐴}

Abstract

In this paper we characterize the zero sets of functions from ℓ p A (the analytic functions on the open unit disk D whose Taylor coefficients form an ℓ p sequence) by developing a concept of an "inner function" modeled by Beurling's discussion of the Hilbert space ℓ 2 A , the classical Hardy space. The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, we give an alternative proof of a result of Vinogradov [34] which says that when p > 2, there are zero sets for ℓ p A which are not Blaschke sequences.This work was supported by NSERC (Canada). 1 (4.1), provides many examples. Definition 5.3. For f ∈ ℓ p A , let f be the metric projection of f onto [Sf ] from Definition 3.6. Proposition 5.4. For f ∈ ℓ p A , let J = f − f be the co-projection of f . Then J is p-inner and any p-inner function arises in this way.Proof. First observe that f is the unique vector in [Sf ] which satisfieswhere P denotes the set of analytic polynomials. Then for any t ∈ C and n ∈ N we see that sinceFrom the definition of Birkhoff-James orthogonality from (4.1), J ⊥ p S n J for all n ∈ N and thus J is p-inner.Conversely, suppose that J is p-inner. Then J ⊥ p S n J for all n ∈ N. But since the criterion for Birkhoff-James orthogonality in the ℓ p A setting (4.2) is linear in the second argument, we see that J ⊥ p QJ for all Q ∈ P, Q(0) = 0. This means that J − J = inf J + QJ p : Q ∈ P, Q(0) = 0} J p .By the definition of J, we see that J = 0 and so the p-inner function J = J − J is of the desired form.

Inner Functions in Reproducing Kernel Spaces

“…We have used Birkhoff-James orthogonality in several recent papers to discuss problems involving the ℓ p A spaces of analytic functions whose power series coefficients belong to the sequence space ℓ p . In [9] we use this orthogonality to give some new bounds on the zeros of an analytic function while in [6] we use this orthogonality, and the concept of an ℓ p A -inner function, to describe the zeros sets of ℓ p A . Still further, we use orthogonality in [8] to give a factorization theorem for ℓ p A functions.…”

“…The unilateral shift (T f )(z) = zf is an isometry on ℓ p A and the notion of T -inner was studied in [6]. The condition for f ∈ ℓ p A to be T -inner is k 0 |a k | p−2 a k a N +k = 0, N 1, but this condition can be difficult to work with.…”

“…Inspired by Beurling's analysis of the structure of the shift invariant subspaces of the classical Hardy space H 2 [4,11], and by similar analysis in other settings [1,3,22,23,24,27], we explored a notion of "inner function" in the sequence space ℓ p A and used it to characterize its zero sets [6,9]. As this "Beurling approach" seems to be ubiquitous, we will survey a method from [27] to the setting of reproducing kernel Hilbert spaces of analytic functions, as we head towards an analogous result for Banach spaces of analytic functions.…”

“…When Ω is the open unit disk D and X is the classical Hardy space H 2 , basic Fourier analysis will show that an S H 2 -inner function is a bounded analytic function on D for which the radial boundary function has constant modulus almost everywhere, in agreement with the classical and well-known notion of inner. Similarly defined inner functions were explored in other spaces [1,3,6,26]. As a topic to be explored in future work, a more general notion of T -inner vector will be presented in this paper, in which T is a bounded linear transformation on a Banach space X , and a vector x ∈ X is said to be T -inner if x ⊥ X T n x for all n 1.…”

Polynomial approach to cyclicity for weighted $$\ell ^p_A$$

“…Thus we will need a generalized concept of orthogonality due to Birkhoff and James, as well as properties of the metric projections on uniformly convex Banach spaces. The study of invariant subspaces of the shift in ℓ p A (1) has proved fruitful in [5,6] and we will make use of many of the ideas in those articles. All these preliminaries, arithmetic properties related to the notation (1.3), general properties of the spaces in study, and the lack of uniqueness of optimal polynomial approximants for p = 1 and p = ∞ are introduced in Section 2 below.…”