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Numerical Range of the Derivation of an Induced Operator
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 82 / Issue 3 / June 2007
- Published online by Cambridge University Press:
- 09 April 2009, pp. 325-344
- Print publication:
- June 2007
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- Article
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Let V be an n–dimensional inner product space over
, let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by 
(H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset 
of complex numbers such that⊆ W*(DK(T)) ⊆ W(DK(T)) = con where conv
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.
, let 
(
of complex numbers such that