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

  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 linear operator, numerical range (radius), block triangular matrices
MSC Classifications: 15A04, 15A60, 47B49 show english descriptions Linear transformations, semilinear transformations
Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
Transformers, preservers (operators on spaces of operators)
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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