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

TL;DR

This paper establishes necessary and sufficient conditions for weak solutions of an abstract evolution equation with a scalar type spectral operator to be infinitely differentiable, revealing an inherent smoothness improvement effect in such equations.

Contribution

It provides a comprehensive characterization of differentiability properties of weak solutions for evolution equations with spectral operators, including the case of normal operators in Hilbert spaces.

Findings

Conditions for strong infinite differentiability of weak solutions are identified.

Strong differentiability at a single point implies infinite differentiability on the entire real line.

The smoothness improvement effect explains why finite differentiability assumptions are unnecessary.

Abstract

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\in \mathbb{R}, \] with scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on $\mathbb{R}$. The important case of the equation with a normal operator $A$ in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at $0$, then all of them are strongly infinite differentiable on $\mathbb{R}$.