Ideal Membership in H∞: Toeplitz Corona Approach

Michael Hartz; Brett D. Wick

doi:10.1007/s00020-018-2488-9

Ideal Membership in H^∞: Toeplitz Corona Approach

Michael Hartz
, Brett D. Wick

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the ideal membership problem in H^∞ on the unit disc. Thus, given functions f, f₁, … , f_n in H^∞, we seek sufficient conditions on the size of f in order for f to belong to the ideal of H^∞ generated by f₁, … , f_n. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space H², which is equivalent to the ideal membership problem by the Nevanlinna–Pick property of H².

Original language	English
Article number	66
Journal	Integral Equations and Operator Theory
Volume	90
Issue number	6
DOIs	https://doi.org/10.1007/s00020-018-2488-9
State	Published - Dec 1 2018

Keywords

Carleson measure
Corona problem
Ideal membership
Nevanlinna–Pick

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10.1007/s00020-018-2488-9

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title = "Ideal Membership in H∞: Toeplitz Corona Approach",

abstract = "We study the ideal membership problem in H∞ on the unit disc. Thus, given functions f, f1, … , fn in H∞, we seek sufficient conditions on the size of f in order for f to belong to the ideal of H∞ generated by f1, … , fn. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space H2, which is equivalent to the ideal membership problem by the Nevanlinna–Pick property of H2.",

keywords = "Carleson measure, Corona problem, Ideal membership, Nevanlinna–Pick",

author = "Michael Hartz and Wick, \{Brett D.\}",

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AU - Hartz, Michael

AU - Wick, Brett D.

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N2 - We study the ideal membership problem in H∞ on the unit disc. Thus, given functions f, f1, … , fn in H∞, we seek sufficient conditions on the size of f in order for f to belong to the ideal of H∞ generated by f1, … , fn. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space H2, which is equivalent to the ideal membership problem by the Nevanlinna–Pick property of H2.

AB - We study the ideal membership problem in H∞ on the unit disc. Thus, given functions f, f1, … , fn in H∞, we seek sufficient conditions on the size of f in order for f to belong to the ideal of H∞ generated by f1, … , fn. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space H2, which is equivalent to the ideal membership problem by the Nevanlinna–Pick property of H2.

KW - Carleson measure

KW - Corona problem

KW - Ideal membership

KW - Nevanlinna–Pick

UR - https://www.scopus.com/pages/publications/85053906931

U2 - 10.1007/s00020-018-2488-9

DO - 10.1007/s00020-018-2488-9

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VL - 90

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

IS - 6

M1 - 66

ER -

Ideal Membership in H^∞: Toeplitz Corona Approach

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Keywords

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