Differential equations in spaces of compact operators (Q557857)

This paper is concerned with the differential equation \[ Z^{\prime}(t) = M Z(t) - Z(t) Z(t)^{*} M Z(t), \] with the initial condition \(\;Z(0) = Z_{0},\;\) for \(Z(t) \in \mathcal{B}(H)\), the space of bounded linear operators on a separable Hilbert space \(H\). This equation arises as a generalization of mathematical models of learning. Assuming that \(M\) is compact and selfadjoint and that \(Z_{0}\) has special properties, the authors establish a representation formula for the solution as a series of rank-one operators. As an application of this representation formula, the existence of compact solutions, differentiable solutions and existence of \(\omega\) and \(\alpha\) limit sets in the uniform, strong and weak sense are established.