Approximation by invertible functions of $H^{\infty}$

Nicolau, Artur; Suárez, Fernando Daniel

doi:10.7146/math.scand.a-15013

Cited by 7 publications

(9 citation statements)

References 11 publications

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“…To prove the second identity, consider the functions Given a Blaschke product u, we will construct an interpolating Blaschke product b and a Carleson contour Γ = ∂Ω verifying Lemma 3.4. The system of rectifiable Jordan curves Γ j appearing in Lemma 3.4 is presented in the following result which is part of the proof of [22,Lemma 3.2]. An explicit proof can be found in [11,Lemma 2].…”

Section: )mentioning

confidence: 99%

Paths of inner-related functions

Nicolau

Suárez

2012

Journal of Functional Analysis

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We characterize the connected components of the subset CN * of H ∞ formed by the products bh, where b is Carleson-Newman Blaschke product and h ∈ H ∞ is an invertible function. We use this result to show that, except for finite Blaschke products, no inner function in the little Bloch space is in the closure of one of these components. Our main result says that every inner function can be connected with an element of CN * within the set of products uh, where u is inner and h is invertible. We also study some of these issues in the context of Douglas algebras. Problem 1.1. For any inner function u and ε > 0, does there exist an interpolating Blaschke product b such that u − b ∞ < ε? This question was posed in [13] and [5, pp. 420], and it is one of the most important open problems in the area. The following weaker version of Problem 1.1 is also open. Problem 1.2. For any inner function u and ε > 0, does there exist an interpolating Blaschke product b and an invertible function h ∈ H ∞ such that u − bh ∞ < ε? This is really a question of approximation in BMO. Recall that a function f ∈ L 1 (∂D) is in the space BMO if f * = sup 1 |I| I |f − f I | < ∞, where the supremum is taken over all arcs I ⊂ ∂D of the unit circle and f I = |I| −1 I f is the mean of f over the arc I. A classical result by Fefferman and Stein says that a function f ∈ L 1 (∂D) is in BMO if and only if f can be written as f = r +s, where r, s ∈ L ∞ (∂D). Heres means the harmonic conjugate of s. Moreover, f * is comparable to f BM O = inf{ r ∞ + s ∞ }, where the infimum is taken over all possible decompositions f = r +s + c, where c is a constant. It is easy to see that Problem 1.2 has a positive answer if and only if for any inner function u and any ε > 0, there exists an interpolating Blaschke product b such that a suitable branch Arg(u/b) of the argument of the function u(ξ)/b(ξ), ξ ∈ ∂D, satisfies Arg(u/b) BM O ≤ ε.

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Section: )mentioning

confidence: 99%

Paths of inner-related functions

Nicolau

Suárez

2012

Journal of Functional Analysis

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“…As noted in Section 1, this is mentioned for complex uniform algebras in [16]. A proof based on the original definitions of the notions of bsr(A) and dsr(A) in the category of surjective, respectively dense algebra morphisms, was communicated to me by Daniel Suárez.…”

Section: Appendixmentioning

confidence: 99%

“…In [16] it is mentioned (without proof) that appsr(A) coincides in case of a uniform algebra A over C with the dense stable rank, dsr(A), of A that was introduced by Corach and Suárez [7, p. 542]. Moreover, they noticed that bsr(A) ≤ dsr(A) ≤ tsr(A).…”

Section: Introduction and Notationmentioning

confidence: 99%

Approximation by invertible elements and the generalized $E$-stable rank for $A({\boldsymbol D})_{\mathsf R}$ and $C({\boldsymbol D})_{\mathrm{sym}}$

Mortini¹,

Rupp²

2011

MATH. SCAND.

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We determine the generalized E-stable ranks for the real algebra, C(D) sym , of all complex valued continuous functions on the closed unit disk, symmetric to the real axis, and its subalgebra A(D) R of holomorphic functions. A characterization of those invertible functions in C(E) is given that can be uniformly approximated on E by invertibles in A(D) R . Finally, we compute the Bass and topological stable rank of C(K) sym for real symmetric compact planar sets K.

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“…Let E be an H ∞ + C-convex subset of M (H ∞ + C). By [13] it is sufficient to prove that any function ϕ ∈ H ∞ + C that does not vanish on E can be uniformly approximated on E by an invertible function in H ∞ + C.…”

Section: The Stable Ranks Of H ∞ + Cmentioning

confidence: 99%