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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A lifting theorem for symmetric commutants
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by Gelu Popescu PDF
Proc. Amer. Math. Soc. 129 (2001), 1705-1711 Request permission

Abstract:

Let $T_{1},\dots , T_{n}\in B(\mathcal {H})$ be bounded operators on a Hilbert space $\mathcal {H}$ such that $T_{1} T_{1}^{*}+\cdots + T_{n} T_{n}^{*}\leq I_{\mathcal {H}}$. Given a symmetry $j$ on $\mathcal {H}$, i.e., $j^{2}=j^{*} j=I_{\mathcal {H}}$, we define the $j$-symmetric commutant of $\{T_{1},\dots , T_{n}\}$ to be the operator space \begin{equation*}\{A\in B(\mathcal {H}): \ T_{i} A=jA T_{i}, \ i=1,\dots , n\}. \end{equation*} In this paper we obtain lifting theorems for symmetric commutants. The result extends the Sz.-Nagy–Foiaş commutant lifting theorem ($n=1, j=I_{\mathcal {H}}$), the anticommutant lifting theorem of Sebestyén ($n=1, j=-I_{\mathcal {H}}$), and the noncommutative commutant lifting theorem ($j=I_{\mathcal {H}}$). Sarason’s interpolation theorem for $H^{\infty }$ is extended to symmetric commutants on Fock spaces.
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Additional Information
  • Gelu Popescu
  • Affiliation: Division of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, Texas 78249
  • MR Author ID: 234950
  • Email: gpopescu@math.utsa.edu
  • Received by editor(s): March 1, 1999
  • Received by editor(s) in revised form: September 16, 1999
  • Published electronically: October 31, 2000
  • Additional Notes: The author was partially supported by NSF Grant DMS-9531954.
  • Communicated by: David R. Larson
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 1705-1711
  • MSC (2000): Primary 47F25, 47A57, 47A20; Secondary 30E05
  • DOI: https://doi.org/10.1090/S0002-9939-00-05750-6
  • MathSciNet review: 1814100