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.