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

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.

Minor readability improvement. arXiv admin note: substantial text overlap with arXiv:1707.09359, arXiv:1706.08014, arXiv:1803.10038, arXiv:1708.05067