TY  - JOUR
AU  - Abdollahi, A. 
AU  - Heydari, M. T. 
AU  - Moosavi, M. 
PY  - 2009
TI  - On Numerical Ranges of Nilpotent Elements of C*-Algebra
JF  - Journal of Mathematics and Statistics
VL  - 5
IS  - 4
DO  - 10.3844/jmssp.2009.348.351
UR  - https://thescipub.com/abstract/jmssp.2009.348.351
AB  - Problem statement: Let A be a C*-algebra with unit 1. For each a∈A, 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(An), 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(An) and showed that the boundary of it does not contain any arc of circle.