On Hermitian operator in tensor product space
Let V be an n-dimensional Hilbert space, \(A_ i\) \((i=1,...,n)\) linear operators on V and \(L=A_ 1\otimes...\otimes A_ n\) their tensor product acting in the \(n^{th}\) tensor power of V. The numerical range of L is defined as the set of (Lx,x), where \(x=x_ 1\otimes...\otimes x_ n\) and \((x_ 1,...,x_ n)\) spans the orthonormal bases of V. It is proved that, for \(n\geq 3\), the numerical range of L is real if and only if L is hermitian.
scientific article; zbMATH DE number 4040620