Tensor product of linear maps

The tensor product of linear maps is determined by $(A\otimes B)(x\otimes y)=Ax\otimes By$ and linear extension.

A tensor-product operator factors as $A\otimes B$ on a tensor product of two spaces. If both factors are positive semidefinite operators, their tensor-product operator is positive: a tensor-product basis of eigenvectors has eigenvalues $a_i{b}_j\ge 0$. This positivity underlies the necessary direction of the positive partial transpose criterion.