A lifting theorem for symmetric commutants
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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.References
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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