On the Smoothness of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

Abstract: Given the abstract evolution equation

$$y\prime (t) = Ay(t), \quad t ≥ 0, \hskip2cm (AEE)$$

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 infinite differentiable or strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic or entire, on $[0,\infty)$ or $(0, \infty)$. 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 is necessarily bounded.

In addition, we find characterizations of the generation of strongly infinite differentiable and Gevrey ultradifferentiable $C_0$-semigroups by scalar type spectral operators.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( video )

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