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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Inner functions and spherical isometries


Authors: Michael Didas and Jörg Eschmeier
Journal: Proc. Amer. Math. Soc. 139 (2011), 2877-2889
MSC (2010): Primary 47A13, 47B20, 47L45; Secondary 47B35, 47L80
Published electronically: March 29, 2011
MathSciNet review: 2801629
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Abstract: A commuting tuple $ T =(T_1, \ldots, T_n) \in B(H)^n$ of bounded Hilbert-space operators is called a spherical isometry if $ \sum_{i=1}^n T_i^*T_i = 1_H$. B. Prunaru initiated the study of $ T$-Toeplitz operators, which he defined to be the solutions $ X \in B(H)$ of the fixed-point equation $ \sum_{i=1}^n T_i^*XT_i = X$. Using results of Aleksandrov on abstract inner functions, we show that $ X \in B(H)$ is a $ T$-Toeplitz operator precisely when $ X$ satisfies $ J^*XJ=X$ for every isometry $ J$ in the unital dual algebra $ \mathcal{A}_T \subset B(H)$ generated by $ T$. As a consequence we deduce that a spherical isometry $ T$ has empty point spectrum if and only if the only compact $ T$-Toeplitz operator is the zero operator. Moreover, we show that if $ \sigma_p(T) = \emptyset$, then an operator which commutes modulo the finite-rank operators with $ \mathcal{A}_T$ is a finite-rank perturbation of a $ T$-Toeplitz operator.


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Additional Information

Michael Didas
Affiliation: Fachrichtung Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
Email: didas@math.uni-sb.de

Jörg Eschmeier
Affiliation: Fachrichtung Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
Email: eschmei@math.uni-sb.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11034-7
PII: S 0002-9939(2011)11034-7
Received by editor(s): July 28, 2010
Published electronically: March 29, 2011
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.