Product manifolds and the curvature operator of the second kind

We investigate the curvature operator of the second kind on product Riemannian manifolds and obtain some optimal rigidity results. For instance, we prove that the universal cover of an $n$-dimensional non-flat complete locally reducible Riemannian manifold with $(n+\frac{n-2}{n})$-nonnegative (respectively, $(n+\frac{n-2}{n})$-nonpositive) curvature operator of the second kind must be isometric to ${\mathbb{S}}^{n-1}\times \mathbb{R}$ (respectively, ${\mathbb{H}}^{n-1}\times \mathbb{R}$) up to scaling. We also prove analogous optimal rigidity results for ${\mathbb{S}}^{n_1}\times {\mathbb{S}}^{n_2}$ and ${\mathbb{H}}^{n_1}\times {\mathbb{H}}^{n_2}$, $n_1,n_2\ge 2$, among product Riemannian manifolds, as well as for ${\mathbb{C}\mathbb{P}}^{m_1}\times {\mathbb{C}\mathbb{P}}^{m_2}$ and ${\mathbb{C}\mathbb{H}}^{m_1}\times {\mathbb{C}\mathbb{H}}^{m_2}$, $m_1,m_2\ge 1$, among product Kähler manifolds. Our approach is pointwise and algebraic.