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

具有标量型谱算子的抽象发展方程弱解的Gevrey超可微性

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

发现的条件的标量型谱算子$A$在一个复杂的巴拿赫空间的发展方程\begin{equation*} y'(t)=Ay(t),\ t\ge 0, \end{equation*}的所有弱解的必要和充分的强Gevrey超可微的顺序$β\ge 1$，特别是分析或整个，$[0,\infty)$。某些固有的平滑度改善效果进行了分析。

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

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.

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

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\ge 0, \] 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 Gevrey ultradifferentiable of order $β\ge 1$, in particular analytic or entire, on the open semi-axis $(0,\infty)$. Also, revealed is a certain interesting inherent smoothness improvement effect.

关于标量型谱算子生成Beurling型Carleman超可微$C_0$ -半群

On the generation of Beurling type Carleman ultradifferentiable $C_0$-semigroups by scalar type spectral operators

建立了Beurling型Carleman超可微$C_0$ -半群的标量型谱生成元的一个刻划，详细地考虑了Gevrey超可微的重要情形，观察了对应于某个快速增长定义序列的一般判据的实现.

On the regularity of scalar type spectral $C_0$-semigroups

We show that, for the $C_0$-semigroups of scalar type spectral operators, a well-known necessary condition for the generation of eventually norm-continuous $C_0$-semigroups, formulated exclusively in terms of the location of the spectrum of the semigroup's generator in the complex plane, is also sufficient and, in fact, characterizes the generators of immediately norm-continuous such semigroups. Combining characterizations of the immediate differentiability and the Gevrey ultradifferentiability of scalar type spectral $C_0$-semigroups with the generation theorem, found earlier by the author, w…