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