Interpolation problems on the spectrum of H ∞

Mortini, Raymond

doi:10.1007/s00605-008-0040-8

Cited by 4 publications

(2 citation statements)

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“…Finally, let us mention that in [13] thin sequences (a n ) and the associated Gleason parts, P (m), of their cluster points m in the spectrum of H ∞ were used together with Theorem 3.3 to solve interpolation problems of the form B(x) = f n (x) for every x ∈ P (x n ) and n ∈ N, where (f n ) is an arbitrary sequence of functions in the unit ball of H ∞ , B is a Blaschke product and (x n ) is a discrete sequence of cluster points of (a n ). As a corollary, we could deduce a result of Gallardo and Gorkin that a Blaschke product B admits (within the Sarason algebra H ∞ +C(T)) a universal Blaschke factor if and only if B is not a finite product of interpolating Blaschke products.…”

Section: Bhmentioning

confidence: 99%

Thin interpolating sequences in the disk

Mortini¹

2009

Arch. Math.

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Abstract. Using a kind of summability procedure we give, within this survey, a new proof of a result by Sundberg and Wolff on large perturbations of thin interpolating sequences and present a detailed proof of the statement of Dyakonov and Nicolau that asymptotic interpolation problems in H ∞ can be solved by thin Blaschke products. Mathematics Subject Classification (2000). Primary 30D50; Secondary 46J15.

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Section: Bhmentioning

confidence: 99%

Thin interpolating sequences in the disk

Mortini¹

2009

Arch. Math.

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“…The first part of item (5) appears in Hedenmalm [15] and goes back to a general result of Sundberg and Wolff [25]. The rest of item (5) and item (6) are due to Izuchi [18, p. 1014] (see also [21,Lemma 2.6]). Item (7) is in [12].…”

Section: Is a Thin Blaschke Product Then Each Of Its Frostman Transformsmentioning

confidence: 99%

Quasi-Universal Functions for Sequences of Composition Operators on H 2 Via Hoffman’s Theory

Mortini

2011

Integr. Equ. Oper. Theory

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Let φ be a non-elliptic automorphism of the unit disk D. Gallardo and Gorkin (respectively Gallardo, Gorkin and Suárez) showed that there exists an interpolating Blaschke product b (respectively a thin Blaschke product) such that the linear span of its orbit under the composition operator C φ is dense in H 2 (in other words that b is a cyclic vector for C φ ). Using Hoffman's theory, we extend their result from the automorphic case to the selfmap case and show that for any sequence (φn) of holomorphic selfmaps of D with |φn(0)| → 1 and lim sup n→∞ |φ n (0)| 1 − |φn(0)| 2 = 1, there exists a thin Blaschke product B with zero distribution as sparse as one wishes such that the linear span of the orbit {B • φn : n ∈ N} of B is dense in H 2 . We will dub such a function a quasi-universal function. We also describe the limit points in the topological space M (H ∞ ) D of these admissible sequences (φn). This approach will enable us to give shorter proofs of the Gallardo-Gorkin result as well as the result of Bayart, Gorkin, Grivaux and the author on the existence of B-universal functions for admissible sequences. Finally we discuss the problem whether interpolating Blaschke products can ever be B-universal for admissible sequences (φn).Mathematics Subject Classification (2010). Primary 47B33; Secondary 46J15, 47A16, 30J10.

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