On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one

TL;DR

This paper proves that if all weak solutions of a certain abstract evolution equation with a scalar type spectral operator are Gevrey ultradifferentiable of orders less than one, then the operator must be bounded.

Contribution

It establishes a necessary condition linking the Gevrey ultradifferentiability of solutions to the boundedness of the spectral operator.

Findings

All weak solutions are Gevrey ultradifferentiable of order less than one only if the operator is bounded.

Unbounded scalar type spectral operators cannot produce solutions with such ultradifferentiability.

The result characterizes the spectral properties of operators based on solution regularity.

Abstract

It is shown that, if all weak solutions of the evolution equation \begin{equation*} y'(t)=Ay(t),\ t\ge0, \end{equation*} with a scalar type spectral operator $A$ in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator $A$ is necessarily bounded.