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 $\mathbb{C}$, let H be a subgroup of the symmetric group on $\{1,\dots,m\}$, and let $\chi: H \rightarrow \mathbb{C}$ be an irreducible character. Denote by $V_\chi^m(H)$ the symmetry class of tensors over V associated with H and $\chi$. Let $K(T) \in \operatorname{End} (V_\chi^m(H))$ be the operator induced by $T\in \operatorname{End}(V)$, and let $D_K(T)$ be the derivation operator of T. The decomposable numerical range $W^*(D_K(T))$ of $D_K(T)$ is a subset of the classical numerical range $W(D_K(T))$ of $D_K(T)$. It is shown that there is a closed star-shaped subset $\mathcal{S}$ of complex numbers such that
 
\begin{align*}\mathcal{S} \subseteq W^*(D_K(T)) \subseteq W(D_K(T)) = \operatorname{conv} \mathcal{S}, \end{align*}
 
where $\operatorname{conv} \mathcal{S}$ denotes the convex hull of $\mathcal{S}$ . In many cases, the set $\mathcal{S}$ is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.
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