J. Aust. Math. Soc.  82 (2007), 325-343
Numerical range of the derivation of an induced operator

 Randall R. Holmes   Department of Mathematics and Statistics   Auburn University   Auburn   Alabama 36849-5310   USA  holmerr@auburn.edu
 Chi-Kwon Li   Department of Mathematics   College of William and Mary   PO Box 8795, Williamsburg   Virginia 23187-8795   USA  ckli@math.wm.edu
 and
 Tin-Yau Tam   Department of Mathematics and Statistics   Auburn University   Auburn   Alabama 36849-5310   tamtiny@auburn.edu

Abstract
Let V be an n-dimensional inner product space over , let H be a subgroup of the symmetric group on , and let be an irreducible character. Denote by the symmetry class of tensors over V associated with H and . Let be the operator induced by , and let be the derivation operator of T. The decomposable numerical range of is a subset of the classical numerical range of . It is shown that there is a closed star-shaped subset of complex numbers such that

where denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.