Characteristic equation in the problem of asymptotic stability in periodic systems with aftereffect

In linear periodic systems with aftereffect, a motion is asymptotically stable, if all eigenvalues of the monodromy operator are less than one in absolute value. Procedures of constructing the characteristic equation for the monodromy operator are connected with finite-dimensional approximations of this operator. The characteristic equation on the complex plane is given by an entire function. For nuclear operators in a separable Hilbert space, this function is uniformly approximable by polynomials in any bounded closed region of the complex plane. Conditions for the nuclearity of the monodromy operator, its conjugate operator, and the regularized monodromy operator are obtained in this work.