SEPARATION OF ROOTS OF MATRIX AND OPERATOR POLYNOMIALS

In a complex Hilbert space X for an arbitrary operator polynomial L(la mbda) (lambda is an element of C) of degree m the following theorem is proved. If the equation (L(lambda)x, x) = 0 has m distinct roots at e very point x is an element of X, \\x\\ = 1, then there exist m pairwis e disjoint connected sets in C such that each set contains a root at e very x. The minimal distance between the roots is separated from zero under the same assumption on the discriminant and the leading coeffici ent of that equation.