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The 3-Point Spectral Pick Interpolation Problem and an Application to Holomorphic Correspondences

  • Vikramjeet Singh ChandelEmail author
Article
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Abstract

We provide a necessary condition for the existence of a 3-point holomorphic interpolant \(F:{\mathbb {D}}\longrightarrow \Omega _n\), \(n\ge 2\). Our condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma. We combine some of the ideas and techniques used in our result on the \({\mathcal {O}}({\mathbb {D}},\,\Omega _n)\)-interpolation problem to establish a Schwarz lemma—which may be of independent interest—for holomorphic correspondences from \({\mathbb {D}}\) to a general planar domain \(\Omega \Subset {\mathbb {C}}\).

Keywords

Spectral unit ball Minimal Blaschke product Holomorphic functional calculus Holomorphic correspondences 

Mathematics Subject Classification

Primary 30E05 32H35 47A56 Secondary 32F45 

Notes

Acknowledgements

I wish to thank my thesis adviser Gautam Bharali for the many helpful discussions during the course of this work. I am especially grateful to him for supporting me as a Research Fellow under his Swarnajayanti Fellowship (Grant No. DST/SJF/MSA-02/2013-14).

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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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