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We study cocycles (non-autonomous dynamical systems) satisfying a certainsqueezing condition with respect to the quadratic form of a boundedself-adjoint operator acting in a Hilbert space. We prove that (underadditional assumptions) the orthogonal projector maps the fibres of someinvariant set, containing bounded trajectories, in a one-to-one manner onto thenegative subspace of the operator. This allows to reduce interesting dynamicsonto this invariant set, which in some cases can be considered as a kind ofinertial manifold for the cocycle. We consider applications of the reductionprinciple for periodic cocycles. For such cocycles we give an extension of theMassera second theorem, obtain the conditions for the existence of a Lyapunovstable periodic trajectory and prove convergence-type results, which we applyto study nonlinear periodic in time delayed-feedback equations posed in aproper Hilbert space and parabolic problems with a nonlinear periodic in timeboundary control. The required operator is obtained as a solution to certainoperator inequalities with the use of the Yakubovich-Likhtarnikov frequencytheorem for $C_{0}$-semigroups and its properties are established from theLyapunov inequality and dichotomy of the linear part of the problem.

A non-local reduction principle for cocycles in Hilbert spaces