Tensor product of linear maps

Question

Let $\phi:U\rightarrow X$ and $\psi:V\rightarrow Y$ be linear maps. Then we have $\phi\otimes \psi:U\otimes V\rightarrow X\otimes Y$. The following (1 to 4) are all clear:

1. $\ker\left(\phi\right)\otimes\ker\left(\psi\right)\leq\ker\left(\phi\otimes\psi\right)$
2. $\textrm{im}\left(\phi\right)\otimes\textrm{im}\left(\psi\right)\leq\textrm{im}\left(\phi\otimes\psi\right)$

and if $U=X$ and $V=Y$

3. $E_{\lambda}\left(\phi\right)\otimes E_{\mu}\left(\psi\right)\leq E_{\lambda\mu}\left(\phi\otimes\psi\right)$

and if $\dim\left(U\right)=n$ and $\dim\left(V\right)=m$

4. $\chi_{\phi\otimes\psi}=\chi_{\phi}^{m}\chi_{\psi}^{n} $

Where $E_\lambda\left(\phi\right)=\ker\left(\phi-\lambda I\right)$ is the eigenspace, and $\chi_\phi\left(x\right)=\det\left(\phi-x I\right)$ is the characteristic polynomial.

The question then is which if any of these ought to hold in some reverse form. For instance, do we have $$\ker\left(\phi\right)\otimes\ker\left(\psi\right)=\ker\left(\phi\otimes\psi\right)$$ or $$\textrm{im}\left(\phi\right)\otimes\textrm{im}\left(\psi\right)=\textrm{im}\left(\phi\otimes\psi\right)$$

Futhermore, for the minimal polynomials $m_{\phi\otimes\psi}$, $m_{\phi}$ and $m_{\psi}$, is there a relation between $m_{\phi\otimes\psi}$ and $m_{\phi}^mm_{\psi}^n$ like we have for the characteristic polynomials? Essentially, do the properties of the constituent maps entirely determine the properties for the tensor-product map?

Answer 1

In general, you won't have the first of these, since $$\ker\phi\otimes V+U\otimes\ker\psi\subseteq\ker(\phi\otimes\psi).$$