On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

TL;DR

This paper characterizes when weak solutions of an abstract evolution equation with a scalar type spectral operator are Gevrey ultradifferentiable, based solely on the spectrum's location, and reveals smoothness improvement effects.

Contribution

It provides necessary and sufficient spectral conditions for Gevrey ultradifferentiability of solutions, including the case of bounded operators and normal operators in Hilbert spaces.

Findings

Weak solutions are Gevrey ultradifferentiable if and only if the spectrum satisfies certain conditions.

All solutions are analytic or entire if the spectrum is suitably located.

If solutions are of order less than one, the operator must be bounded.

Abstract

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\in \mathbb{R}, \] with a scalar type spectral operator $A$ in a complex Banach space, we find conditions on $A$, formulated exclusively in terms of the location of its spectrum in the complex plane, necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic or entire, on $\mathbb{R}$. We also reveal certain inherent smoothness improvement effects and show that, if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator $A$ is necessarily bounded. The important particular case of the equation with a normal operator $A$ in a complex Hilbert space follows immediately.