Balanced exponential growth of operator semigroups (Q1269587)

A \(C_0\)-semigroup \(S(t)\), \(t\geq 0\), on a Banach space \(X\) (weakly, strongly, uniformly) approaches balanced (or asynchronous) exponential growth if there exists some \(s\in\mathbb{R}\) such that \[ P= \lim_{t\to\infty} e^{-st}S(t) \] exists (in the weak, strong, uniform operator topology) and \(P\) is not the \(0\) operator. The author characterizes the strong and uniform approach to balanced exponential growth and derives applicable sufficient conditions.