Evolution-Semigroup Generators for Nonautonomous Operator Families

Definitions and mathematical statement

Let $X$ be a Banach space and let $(A(t),D(A(t)))_{t\in{\mathbb{R}}}$ be a family of operators on $X$, each described in the source as a generator. We interpret this terminology as generation of a strongly continuous semigroup on $X$; no common operator domain or regularity in $t$ is specified. Set \[ X_0=C_0({\mathbb{R}},X)=\left\{f\in C({\mathbb{R}},X):\lim_{|s|\to\infty}\|f(s)\|_X=0\right\}, \qquad \|f\|_{X_0}=\sup_{s\in{\mathbb{R}}}\|f(s)\|_X. \] For a norm-differentiable function $f$ satisfying $f(s)\in D(A(s))$ for every $s\in{\mathbb{R}}$, consider the formal differential expression \[ (\mathcal Af)(s)=-f'(s)+A(s)f(s),\qquad s\in{\mathbb{R}}. \] A pointwise realization on $X_0$ must choose a linear domain $D(\mathcal A)\subseteq X_0$ on which this expression is defined and belongs to $X_0$.

Problem. Determine conditions on the family $(A(t))_{t\in{\mathbb{R}}}$, together with an appropriate domain or closed realization of the displayed expression, for which $\mathcal A$ generates a strongly continuous semigroup on $X_0$. Explicitly, the requested realization should be the generator of bounded linear operators $(T(h))_{h\geq0}$ on $X_0$ satisfying \[ T(0)=I,\qquad T(h+k)=T(h)T(k),\qquad \lim_{h\downarrow0}\|T(h)f-f\|_{X_0}=0 \] for every $h,k\geq0$ and $f\in X_0$, with generator domain and action \[ D(\mathcal A)=\left\{f\in X_0:\lim_{h\downarrow0}\frac{T(h)f-f}{h}\text{ exists in }X_0\right\}, \qquad \mathcal Af=\lim_{h\downarrow0}\frac{T(h)f-f}{h}. \] The source does not specify whether the desired realization must act pointwise on its entire domain or may be obtained by closing a restriction of the formal expression. This distinction remains unresolved here. The underlying variable ranges over all of ${\mathbb{R}}$; the ambient space requires vanishing at both infinities, and no additional boundary conditions are supplied.

Short English statement

Given a time-dependent family of semigroup generators on a Banach space, determine hypotheses and a suitable operator domain that make $-\frac{{\,\mathrm{d}}}{{\,\mathrm{d}} s}+A(s)$ generate a strongly continuous semigroup on $C_0({\mathbb{R}},X)$. The intended type of realization and the admissible hypotheses remain unspecified in the source.

Sources

• Local supplied UnsolvedMath registry record, permanent ID $30001648$, exact source problem code catalogue entry, entitled Evolution-Semigroup Generators for Nonautonomous Operator Families; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The record gives proposed year 2010.
• Supplied report citation: Mini-Workshop: Wellposedness and Controllability of Evolution Equations (2011), Oberwolfach Reports, DOI 10.4171/owr/2010/56, pp. 3269--3309. https://doi.org/10.4171/owr/2010/56. This is also the supplied literature reference labelled "original Oberwolfach problem report."
• Supplied literature reference labelled "evolution semigroups for nonautonomous problems": https://studylib.net/doc/10826927/evolution-semigroups-for-nonautonomous-cauchy-problems. No author, publication date, or further bibliographic identification is supplied for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that evolution-semigroup and nonautonomous Cauchy-problem theories provide many sufficient conditions, while the requested general characterization for arbitrary generator families remains open. This is supplied dated curation, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its verification note says that the mathematical content was checked against the cited official report and that notation, context, and question form were normalized without changing the source-level claim. This is a supplied verification claim, not a source check performed for this draft.