The subspace of symmetric tensors and the subspace of anti-symmetric tensors are two natural reducing subspaces of tensor product $Aotimes I+Iotimes A$ and $Aotimes A$ for any bounded linear operator $A$ on a complex separable Hilbert space $H$. We show the set of operators $A$ such that these two subspaces are the only (nontrivial) reducing subspaces of $Aotimes I+Iotimes A$ is a dense $G_{delta}$ set in $B(H).$ This generalizes Halmos`s theorem that the set of irreducible operators is a dense $G_{delta}$ set in $B(H).$ The same question for $Aotimes A$ is still open.