Models of learning and the polar decomposition of bounded linear operators

We study systems of differential equations in $mathcal{B}(mathcal{H})$, the space of all bounded linear operators on a separable complex Hilbert space $ mathcal{H} $ equipped with the operator norm. These systems are infinite dimensional generalizations of mathematical models of learning. We use the polar decomposition of operators to find an explicit form for solutions. We also discuss the standard questions of existence and uniqueness of local and global solutions, as well as their long-term behavior.