Positive expansiveness of operator semigroups: orbit growth and spectral criteria

Abstract

This paper studies positive expansiveness and positive kinematic expansiveness for strongly continuous semigroups of bounded linear operators on complex Banach spaces. We prove that positive kinematic expansiveness is equivalent to the unboundedness of every nonzero orbit, while positive expansiveness additionally requires the generator to be bounded from below. We also show that continuous changes of time can be replaced by increasing homeomorphisms without changing positive expansiveness. Uniform exponential expansion implies positive expansiveness without any surjectivity assumption; we characterize it through the minimum modulus, the lower growth bound, and the approximate point spectrum of a single semigroup operator. Under a uniform positive lower bound on the semigroup, positive expansiveness is equivalent to positive kinematic expansiveness together with boundedness below of the generator. Diagonal semigroups on $\ell_p$ then show that uniform exponential expansion, positive expansiveness and its kinematic version form a strictly decreasing hierarchy. Finally, we derive necessary spectral conditions for bounded generators and show that all three properties coincide in finite dimensions.