  In this paper we revisit the problem of decoherence applying the block
operator matrices analysis. Riccati algebraic equation associated with the
Hamiltonian describing the process of decoherence is studied. We prove that if
the environment responsible for decoherence is invariant with respect to the
antylinear transformation then the antylinear operator solves Riccati equation
in question. We also argue that this solution leads to neither linear nor
antilinear operator similarity matrix. This fact deprives us the standard
procedure for solving linear differential equation (e.g, Schrodinger equation).
Furthermore, the explicit solution of the Riccati equation is found for the
case where the environment operators commute with each other. We discuss the
connection between our results and the standard description of decoherence (one
that uses the Kraus representation). We show that reduced dynamics we obtain
does not have the Kraus representation if the initial correlations between the
system and its environment are present. However, for any initial state of the
system (even when the correlations occur) reduced dynamics can be written in a
manageable way.
