Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47F25, 47A57, 47A20, 30E05

Retrieve articles in all journals with MSC (2000): 47F25, 47A57, 47A20, 30E05


Additional Information

Gelu Popescu
Affiliation: Division of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, Texas 78249
Email: gpopescu@math.utsa.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05750-6
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