Characterisation of Log-Convex Decay in Non-Selfadjoint Dynamics

TL;DR

This paper investigates the conditions under which solutions to certain linear evolution equations exhibit log-convex decay, providing a comprehensive analysis of their short-time and global behavior in Hilbert spaces.

Contribution

It introduces a general necessary and sufficient condition for the norm of solutions to be log-convex and strictly decreasing, advancing understanding of non-selfadjoint dynamics.

Findings

Norm of solutions is log-convex and strictly decreasing under certain conditions.

Derivative of the norm at initial time is controlled by the generator's lower bound.

Injectivity of holomorphic semigroups is key to the analysis.

Abstract

The short-time and global behaviour are studied for an autonomous linear evolution equation, which is defined by a generator inducing a uniformly bounded holomorphic semigroup in a Hilbert space. A general necessary and sufficient condition is introduced under which the norm of the solution is shown to be a log-convex and strictly decreasing function of time, and differentiable also at the initial time with a derivative controlled by the lower bound of the generator, which moreover is shown to be positively accretive. Injectivity of holomorphic semigroups is the main technical tool.