On Numerical Ranges of Nilpotent Elements of C*-Algebra
A. Abdollahi, M. T. Heydari and M. Moosavi
DOI : 10.3844/jmssp.2009.348.351
Journal of Mathematics and Statistics
Volume 5, 2009
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.
© 2009 A. Abdollahi, M. T. Heydari and M. Moosavi. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.