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Relations Among Various Versions of the Segal-Bargmann Transform

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Recent Progress in Operator Theory and Its Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 220))

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Abstract

We present various relations among Versions A, B and C of the Segal-Bargmann transform. We get results for the Segal-Bargmann transform associated to a Coxeter group acting on a finite-dimensional Euclidean space. Then analogous results are shown for the Segal-Bargmann transform of a connected, compact Lie group for all except one of the identities established in the Coxeter case. A counterexample is given to show that the remaining identity from the Coxeter case does not have an analogous identity for the Lie group case. A major result is that in both contexts the Segal-Bargmann transform for Version C is determined by that for Version A.

Mathematics Subject Classification (2000). Primary 45H05, 44A15; Secondary 46E15.

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Authors and Affiliations

  1. Centro de Investigación en Matemáticas, A.C. (CIMAT), Guanajuato, Gto., Mexico

    Stephen Bruce Sontz

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  1. Stephen Bruce Sontz
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Correspondence to Stephen Bruce Sontz .

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Editors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute, McBryde Hall 524, Blacksburg, 24061, Virginia, USA

    Joseph A. Ball

  2. Dept. Mathematics, University of Iowa, MacLean Hall 14, Iowa City, 52242-1419, Iowa, USA

    Raúl E. Curto

  3. , CINVESTAV, IPN - Instituto Politécnico Nacional, Av. IPN 2508, Mexico, D.F., 07360, Distrito Federal, Mexico

    Sergei M. Grudsky

  4. Dept. Mathematics, University of California, San Diego, Gilman Dr. 9500, La Jolla, 92093-0112, California, USA

    J. William Helton

  5. , Departamento di Matemática, Centro de Investigación en Matemáticas, Guanajuato, Gto., 36000, Guanajuato, Mexico

    Raúl Quiroga-Barranco

  6. Departamento de Matemática, CINVESTAV-IPN, Apdo. Postal, México, 07300, Distrito Federal, Mexico

    Nikolai L. Vasilevski

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Sontz, S.B. (2012). Relations Among Various Versions of the Segal-Bargmann Transform. In: Ball, J., Curto, R., Grudsky, S., Helton, J., Quiroga-Barranco, R., Vasilevski, N. (eds) Recent Progress in Operator Theory and Its Applications. Operator Theory: Advances and Applications(), vol 220. Springer, Basel. https://doi.org/10.1007/978-3-0348-0346-5_19

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