Abstract:
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.
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