Multiplicative preservers and induced operators Original Research Article

Let V be an n-dimensional Hilbert space. Suppose H is a subgroup of the symmetric group of degree m, and image is a character of degree 1 on H. Consider the symmetrizer on the tensor space circle times operatormVimagedefined by H and χ. The subspace image of circle times operatormV spanned by S(circle times operatormV) is called the symmetry class of tensors over V associated with H and χ. The elements in image of the form S(v1 circle times operator cdots, three dots, centered circle times operator vm) are called decomposable tensors and are denoted by v1 * cdots, three dots, centered * 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 image satisfyingK(T)v1*cdots, three dots, centered*vm=Tv1*cdots, three dots, centered*Tvm.We characterize multiplicative maps phi such that F(phi(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).