Multiplicative preservers and induced operators

Let V be an n-dimensional Hilbert space. Suppose H is a subgroup of the symmetric group of degree m, and χ:H→C is a character of degree 1 on H. Consider the symmetrizer on the tensor space ⊗mVS(v1⊗⋯⊗vm)=1|H|∑σ∈Hχ(σ)vσ-1(1)⊗⋯⊗vσ-1(m)defined by H and χ. The subspace Vχm(H) of ⊗mV spanned by S(⊗mV) is called the symmetry class of tensors over V associated with H and χ. The elements in Vχm(H) of the form S(v1 ⊗ ⋯ ⊗ vm) are called decomposable tensors and are denoted by v1 ∗ ⋯ ∗ vm. For any linear operator T acting on V, there is an (unique) induced operator Kχ(T) (or just K(T) for notational simplicity) acting on Vχm(H) satisfyingK(T)v1∗⋯∗vm=Tv1∗⋯∗Tvm.We characterize multiplicative maps ϕ such that F(ϕ(T)) = F(T) for all operators T acting on V, where F are various scalar or set valued functions including the spectral radius, (decomposable) numerical radius, spectral norm, spectrum, (decomposable) numerical range of T or K(T).