Book chapter
The Extended Aluthge Transform
Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology, pp.55-76
Operator Theory: Advances and Applications, Springer International Publishing
12/12/2020
DOI: 10.1007/978-3-030-43380-2_3
Abstract
Given a bounded linear operator T with canonical polar decomposition T≡VT\documentclass[12pt]{minimal}
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$$T \equiv V\left |T\right |$$
\end{document}, the Aluthge transform of T is the operator Δ(T):=TVT\documentclass[12pt]{minimal}
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$$\Delta (T):=\sqrt {\left |T\right |} V \sqrt {\left |T\right |}$$
\end{document}. For P an arbitrary positive operator such that V P = T, we define the extended Aluthge transform of T associated with P by ΔP(T):=PVP\documentclass[12pt]{minimal}
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$$\Delta _P(T):=\sqrt {P} V \sqrt {P}$$
\end{document}. First, we establish some basic properties of ΔP; second, we study the fixed points of the extended Aluthge transform; third, we consider the case when T is an idempotent; next, we discuss whether ΔP leaves invariant the class of complex symmetric operators. We also study how ΔP transforms the numerical radius and numerical range. As a key application, we prove that the spherical Aluthge transform of a commuting pair of operators corresponds to the extended Aluthge transform of a 2 × 2 operator matrix built from the pair; thus, the theory of extended Aluthge transforms yields results for spherical Aluthge transforms.
Details
- Title: Subtitle
- The Extended Aluthge Transform
- Creators
- Chafiq Benhida - UFR de Mathématiques, Université des Sciences et Technologies de Lille, Villeneuve d’Ascq, FranceRaul E Curto - University of Iowa
- Resource Type
- Book chapter
- Publication Details
- Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology, pp.55-76
- Publisher
- Springer International Publishing; Cham
- Series
- Operator Theory: Advances and Applications
- DOI
- 10.1007/978-3-030-43380-2_3
- eISSN
- 2296-4878
- ISSN
- 0255-0156
- Language
- English
- Date published
- 12/12/2020
- Academic Unit
- Mathematics
- Record Identifier
- 9984241044802771
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