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Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices

Published online by Cambridge University Press:  20 November 2018

Wai-Shun Cheung  and
Chi-Kwong Li
Wai-Shun Cheung
Affiliation:
Department of Mathematics and Statistics University of Victoria Victoria, B.C. V8W 3P4, e-mail: wshun@math.uvic.ca
Chi-Kwong Li
Affiliation:
Department of Mathematics College of William and Mary P.O. Box 8795 Williamsburg, Virginia 23187-8795 USA, e-mail: ckli@math.wm.edu
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Abstract

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Let $c\,=\,\left( {{c}_{1}},\ldots ,{{c}_{n}} \right)$ be such that ${{c}_{1}}\,\ge \,\cdots \,\ge \,{{c}_{n}}$. The $c$-numerical range of an $n\,\times \,n$ matrix $A$ is defined by

$${{W}_{c}}\left( A \right)\,=\,\left\{ \sum\limits_{j=1}^{n}{{{c}_{j}}\left( A{{x}_{j}},\,{{x}_{j}} \right)\,:\,\left\{ {{x}_{1}},\ldots ,{{x}_{n}} \right\}\,\text{an}\,\text{orthonormal basis for }{{\mathbf{C}}^{n}}} \right\}\,,$$

and the $c$-numerical radius of $A$ is defined by ${{r}_{c}}\left( A \right)\,=\,\max \left\{ \left| z \right|\,:\,z\,\in \,{{W}_{c}}\left( A \right) \right\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying

$${{W}_{c}}\left( \phi \left( A \right) \right)={{W}_{c}}\left( A \right)\text{for}\,\,\text{all}\,\,A\,\text{or}\,\,{{r}_{c}}\left( \phi \left( A \right) \right)={{r}_{c}}\left( A \right)\text{for}\,\,\text{all}\,A.$$

Keywords

15A0415A6047B49linear operatornumerical range (radius)block triangular matrices
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 3 , 01 September 2001 , pp. 270 - 281
Copyright
Copyright © Canadian Mathematical Society 2001

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