Operator properties of T and K(T)

Let V be an n-dimensional inner product space over C, and let H be a subgroup of the symmetric group on {1,…,m}. Suppose χ:H→C is an irreducible character (not necessarily linear). Let Vχm(H) denote the symmetry class of tensors over V associated with H and χ and let K(T)∈End(Vχm(H)) be the induced operator of T∈End(V).It is known that if T is normal, unitary, positive (semi-)definite, Hermitian, then K(T) has the corresponding property. Furthermore, if T1=ξT2 for some ξ∈C with ξm=1, then K(T1)=K(T2). The converse of these statements are not valid in general. Necessary and sufficient conditions on χ and the operators T,T1,T2 ensuring the validity of the converses of the above statements are given. These extend the results of those on linear characters by Li and Zaharia.