Inner functions and spherical isometries
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- by Michael Didas and Jörg Eschmeier PDF
- Proc. Amer. Math. Soc. 139 (2011), 2877-2889 Request permission
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.References
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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
- Received by editor(s): July 28, 2010
- Published electronically: March 29, 2011
- Communicated by: Richard Rochberg
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2877-2889
- MSC (2010): Primary 47A13, 47B20, 47L45; Secondary 47B35, 47L80
- DOI: https://doi.org/10.1090/S0002-9939-2011-11034-7
- MathSciNet review: 2801629