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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Free outer functions in complete Pick spaces
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by Alexandru Aleman, Michael Hartz, John E. McCarthy and Stefan Richter HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 1929-1978 Request permission


Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function $f$ in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type $f=\varphi g$, where $g$ is cyclic, $\varphi$ is a contractive multiplier, and $\|f\|=\|g\|$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.
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Additional Information
  • Alexandru Aleman
  • Affiliation: Department of Mathematics, Lund University, Faculty of Science, P.O. Box 118, S-221 00 Lund, Sweden
  • MR Author ID: 230039
  • Email:
  • Michael Hartz
  • Affiliation: Fachrichtung Mathematik, Universität des Saarlandes, 66123 Saarbrücken, Germany
  • MR Author ID: 997298
  • ORCID: 0000-0001-6509-9062
  • Email:
  • John E. McCarthy
  • Affiliation: Department of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130
  • MR Author ID: 271733
  • ORCID: 0000-0003-0036-7606
  • Email:
  • Stefan Richter
  • Affiliation: Department of Mathematics, University of Tennessee, 1403 Circle Drive, Knoxville, Tennessee 37996-1320
  • MR Author ID: 215743
  • ORCID: 0000-0003-1188-8589
  • Email:
  • Received by editor(s): March 18, 2022
  • Received by editor(s) in revised form: August 22, 2022
  • Published electronically: December 16, 2022
  • Additional Notes: The second author was partially supported by a GIF grant and by the Emmy Noether Program of the German Research Foundation (DFG Grant 466012782). The third author was partially supported by National Science Foundation Grant DMS 2054199.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 1929-1978
  • MSC (2020): Primary 46E22; Secondary 47A16, 47B32
  • DOI:
  • MathSciNet review: 4549696