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

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Published:2001-09-01
Printed: Sep 2001
• Wai-Shun Cheung
• Chi-Kwong Li
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## Abstract

Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$. The $c$-numerical range of an $n \times n$ matrix $A$ is defined by $$W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\} \text{ an orthonormal basis for } \IC^n \Bigr\},$$ and the $c$-numerical radius of $A$ is defined by $r_c (A) = \max \{|z| : z \in W_c (A)\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying $$W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or} \quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A.$$
 Keywords: linear operator, numerical range (radius), block triangular matrices
 MSC Classifications: 15A04 - Linear transformations, semilinear transformations 15A60 - Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 47B49 - Transformers, preservers (operators on spaces of operators)

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