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

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 $\mathbb{R}$.Comment: A correction in Remarks 3.1, a few minor readability improvements. arXiv admin note: substantial text overlap with arXiv:1707.09359, arXiv:1706.08014, arXiv:1708.0506

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

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.Comment: Minor readability improvement. arXiv admin note: substantial text overlap with arXiv:1707.09359, arXiv:1706.08014, arXiv:1803.10038, arXiv:1708.0506

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

For the evolution equation ()=() with a scalar type spectral operator in a Banach space, conditions on are found that are necessary and sufficient for all weak solutions of the equation on [0,∞) to be strongly infinite differentiable on [0,∞) or [0,∞). Certain effects of smoothness improvement of the weak solutions are analyzed

On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis

Given the abstract evolution equation y′(t)=Ay(t),  t∈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 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 R