Numerical Range of the Derivation of an Induced Operator
نویسندگان
چکیده
Let V be an n-dimensional inner product space over C, let H be a subgroup of the symmetric group on {1, . . . , m}, and let χ : H → C be an irreducible character. Denote by V m χ (H) the symmetry class of tensors over V associated with H and χ. Let K(T ) ∈ End (V m χ (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 S of complex numbers such that S ⊆ W ∗(DK(T )) ⊆ W (DK(T )) = convS, where convS denotes the convex hull of S. In many cases, the set S is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed. 2000 Mathematics Subject Classification. 15A69, 15A42
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