The exact descriptive type of the set of proper linear systems (Q5942101)

The following family of systems of the form \[ x'=A(t)x,\quad x\in \mathbb{R}^n, t\in\mathbb{R}^+,\tag{1} \] is considered. Here, \(A(t)\) is a piecewise continuous bounded operator function on \(\mathbb{R}^+\). System (1) is said to be proper if \[ \lambda_1(A)+ \cdots+\lambda_n (A)-\liminf_{t\to\infty} t^{-1} \int^t_0 \text{Sp} A(\tau)d \tau=0, \] where \(\lambda_1(A)\leq \dots\leq\lambda_n(A)\) are the Lyapunov exponents of (1). Let \({\mathcal M}_n\) denote the metric space of system (1) equipped with the metric \(\rho(A,B)=\sup\{\|A(t)-B(t)\|:t \in \mathbb{R}^+\}\). In the short communication, the author proves that for \(n>1\) the set of proper systems is a set of the type \(F_{\sigma\delta}\) and is not a set of the type \(G_{\delta\sigma}\), in the space \({\mathcal M}_n\).