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A lifting theorem for symmetric commutants

Author: Gelu Popescu
Journal: Proc. Amer. Math. Soc. 129 (2001), 1705-1711
MSC (2000): Primary 47F25, 47A57, 47A20; Secondary 30E05
Published electronically: October 31, 2000
MathSciNet review: 1814100
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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-Foias 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

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
Article copyright: © Copyright 2000 American Mathematical Society

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