A relationship between mixed discriminants and the joint spectrum of a family of commuting operators in finite-dimensional space

We study the properties of the polynomial operator pencil
$$ L(\lambda)=\sum_{i=0}^n\lambda^{n-i}M_i,\qquad M_i\colon\mathscr H\to\mathscr H, \quad i=\overline{0,n}, $$
where $\mathscr H$ is a $k$-dimensional Hilbert space, and prove that the mixed discriminants $\{d_j\}_{j=0}^{nk}$, defined as the coefficients of the polynomial
$$ \det L(\lambda)=\sum_{j=0}^{nk}d_j\lambda^{nk-j}, $$
are completely determined by the joint spectrum of the family $\{M_i\}_{i=0}^n$. A generalization of Gershgorin's well-known theorem on the position of the eigenvalues of a matrix to the case of a polynomial matrix pencil is obtained.