On Numerical Ranges of Nilpotent Elements of C*-Algebra | Science Publications

Journal of Mathematics and Statistics

A. Abdollahi¹  M. Moosavi¹  M. T. Heydari¹ 

关键词: Numerical range;    numerical radius;    C*-algebra;    nilpotent;    C*-algebra;    nilpotent;   

DOI  :  10.3844/jmssp.2009.348.351

学科分类：社会科学、人文和艺术（综合）

来源: Science Publications

【 摘 要 】

Problem statement: Let A be a C*-algebra with unit 1. For each aεA, let V(a), v(a) v₀(a) and denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. Approach: If a is a nilpotent element of A with the power of nilpotency n, i.e., aⁿ = 0 and v(a) = (n-1) v₀(a). Results: We proved that V(a) = bW(A_n), where b is a scalar and An is the strictly upper triangular n-by-n matrix with all entries above the main diagonal equal to one. Conclusion/Recommendations: We also completely determined the numerical range of such elements, by determining the numerical range of W(A_n) and showed that the boundary of it does not contain any arc of circle.

【 授权许可】

Unknown