Ideal Membership in \(H^\infty \): Toeplitz Corona Approach

  • Michael HartzEmail author
  • Brett D. Wick


We study the ideal membership problem in \(H^\infty \) on the unit disc. Thus, given functions \(f,f_1,\ldots ,f_n\) in \(H^\infty \), we seek sufficient conditions on the size of f in order for f to belong to the ideal of \(H^\infty \) generated by \(f_1,\ldots ,f_n\). 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 \(H^2\), which is equivalent to the ideal membership problem by the Nevanlinna–Pick property of \(H^2\).


Corona problem Ideal membership Carleson measure Nevanlinna–Pick 

Mathematics Subject Classification

Primary 30H05 Secondary 46J20 30H80 


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsWashington University in St. LouisSt. LouisUSA

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