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Multipliers of Embedded Discs

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An Erratum to this article was published on 25 July 2014

Abstract

We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on complete Nevanlinna–Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of \(\ell ^2\) which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of \(\ell ^2\) which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov by providing a proper rational biholomorphism of the disc onto a variety \(V\) in \({\mathbb {B}}_2\) such that the multiplier algebra is not all of \(H^\infty (V)\). We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada

    Kenneth R. Davidson & Michael Hartz

  2. Department of Mathematics, Ben-Gurion University of the Negev, Beersheva, Israel

    Orr Moshe Shalit

Authors
  1. Kenneth R. Davidson
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  2. Michael Hartz
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  3. Orr Moshe Shalit
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Correspondence to Orr Moshe Shalit.

Additional information

Communicated by Mihai Putinar.

K. R. Davidson is partially supported by an NSERC grant. M. Hartz is partially supported by an Ontario Trillium Scholarship. O. M. Shalit is partially supported by ISF Grant no. 474/12, by EU FP7/2007-2013 Grant no. 321749, and by GIF Grant no. 2297-2282.6/20.1. The authors thank the Faculty of Natural Science’s Distinguished Scientist Visitors Program as well as the Center for Advanced Mathematical Studies at Ben-Gurion University of the Negev, for supporting the first author’s visit to Ben-Gurion University during April 2013.

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Davidson, K.R., Hartz, M. & Shalit, O.M. Multipliers of Embedded Discs. Complex Anal. Oper. Theory 9, 287–321 (2015). https://doi.org/10.1007/s11785-014-0360-8

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  • DOI: https://doi.org/10.1007/s11785-014-0360-8

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