It is well-known that, if $V$ is (say) a real vector space of finite dimension $n$, then there can be established an isomorphism between the space of endomorphisms $\mathrm{End}(V)$ (or the space of $n\times n$ real matrices) and the space of $(1,1)$-tensors $\mathscr{T}{^1_1}(V)=V^*\otimes V := \{\mathrm{multilinear\ } T : V\times V^* \to \mathbb R \}$. This means that there also is a well-defined notion of diagonalizability for tensors in $V^*\otimes V$, namely:
A $(1,1)$-tensor $A$ on the space $V$ is diagonalizable over $\mathbb R$ when there exists a basis $\{\mathbf e_1,\dots, \mathbf e_n \}$ of $V$ such that $$A(\mathbf v, \boldsymbol \varphi) = \sum_{k=1}^n\lambda_{(k)} (\boldsymbol\varepsilon^k \otimes \mathbf e_k)(\mathbf v, \boldsymbol \varphi) \qquad \forall \mathbf v \in V, \forall\boldsymbol\varphi \in V^*$$ where $\{\boldsymbol\varepsilon^1, \dots, \boldsymbol\varepsilon^n\}$ is the basis of $V^*$ such that $\boldsymbol\varepsilon^\ell(\mathbf e_p)= \delta^\ell_p$, and $\lambda_{(k)}$ is a real number for all relevant $k$.
Can this notion be generalized to all varieties of mixed tensors? Something like $$A(\mathbf v_1, \dots, \mathbf v_p, \boldsymbol \varphi^1, \dots, \boldsymbol \varphi^q) = \sum_{k=1}^n \lambda_{(k)} \left(\bigotimes^p\boldsymbol\varepsilon^k \otimes \bigotimes^q\mathbf e_k\right)(\mathbf v_1, \dots, \mathbf v_p, \boldsymbol \varphi^1, \dots, \boldsymbol \varphi^q)$$ where $p$ is the amount of contravariant indices, $q$ the amount of covariant indices, and $\bigotimes^m$ indicates $m$-fold iteration of the tensor product. If so, could the finite-dimensional spectral theorem be generalized also? What would Jordan canonical form look like?