A generalization of a theorem of Datko and Pazy (Q2746296)

An operator-valued map \(S:\mathbb{R}_+^2\to B(E)\) is called a \(C_0\)-quasisemigroup on \(E\) if \(S(0,t_0)=I\) for any \(t_0\geq 0\), \(S(t,s+t_0) S(s,t_0)= S(t+s,t_0)\) and \(\lim_{t\to 0}\|S(t,t_0) x_0-x_0 \|=0\) for all \(t_0\in\mathbb{R}_+\), \(x_0\in E\). The \(C_0\)-quasisemigroup is called a uniformly exponentially dichotomy if there are \(N\geq 1\), \(\nu>0\) and a projection-valued function \(P: \mathbb{R}_+\to B(E)\) such that \(P(t+t_0) S(t,t_0)= S(t,t_0)P(t_0)\), \(e^{\nu t}\|S(t+s,t_0)x_0 \|\leq N\|S(s,t_0) x_0\|\), \(e^{\nu t}\|S(t,t_0)y_0 \|\leq N\|S(t+s,t_0) y_0\|\) for all \(t,s,t_0\in \mathbb{R}_+^3\), \(x_0\in \text{Im} P(t_0)\), \(y_0\in \text{Ker} P(t_0)\). If \(\nu=0\) then they call it uniformly dichotomic. Theorem: Let \(S\) be a uniformly dichotomic \(C_0\)-quasisemigroup. Then \(S\) is uniformly exponentially dichotomic iff there exists \(N\geq 1\) such that NEWLINE\[NEWLINE\int^\infty_t \bigl\|S(\tau,t_0) x_0\bigr\|d\tau\leq N\bigl\|S(t,t_0)x_0 \bigr\|, \quad\int^t_0 \bigl\|S(s,t_0) y_0\bigr\|ds\leq N\bigl\|S(t,t_0)y_0 \bigr\|NEWLINE\]NEWLINE for all \((t,t_0)\in \mathbb{R}^2_+\), \(x_0\in \text{Im} P(t_0)\), \(y_0\in \text{Ker} P(t_0)\).