Semigroup Generation by Sums of Unbounded Operators

Definitions and mathematical statement

Let $X$ be a Banach space, and let $A\colon D(A)\subseteq X\to X$ and $B\colon D(B)\subseteq X\to X$ be unbounded linear operators with linear domains. The supplied source does not specify whether $X$ is real or complex, or assume that either operator is closed, densely defined, or itself a semigroup generator.

The algebraic sum has the natural definition \[ Sx=Ax+Bx,\qquad D(S)=D(A)\cap D(B). \] For any linear subspace $D\subseteq D(A)\cap D(B)$, one may also consider the realization \[ S_D\colon D\to X,\qquad S_Dx=Ax+Bx. \] A $C_0$-semigroup on $X$ is a family $(T(t))_{t\geq0}$ of bounded linear operators satisfying $T(0)=I_X$, $T(t+s)=T(t)T(s)$ for $s,t\geq0$, and continuity of $t\mapsto T(t)x$ on $[0,\infty)$ for every $x\in X$. Its generator $G$ is defined by \[ D(G)=\left\{x\in X:\lim_{h\downarrow0}\frac{T(h)x-x}{h}\text{ exists in }X\right\},\qquad Gx=\lim_{h\downarrow0}\frac{T(h)x-x}{h}. \] Problem. Determine conditions on $A$, $B$, and their domains, together with an appropriate choice of domain for their sum, under which $A+B$ generates a $C_0$-semigroup on $X$. For a realization $S_D$ as above, this means that there exists such a semigroup whose generator satisfies $D(G)=D$ and $Gx=Ax+Bx$ for every $x\in D$.

Unresolved scope. The source leaves the admissible domain choices open. It does not specify whether the intended object must be the algebraic sum on the full intersection, a restriction to a smaller domain, the closure of a closable sum, or some extension. The realization $S_D$ describes the direct sum interpretation only; closure and extension interpretations require additional specifications. No such choice or additional operator hypothesis is imposed here.

Short English statement

For two unbounded linear operators on a Banach space, find conditions and a suitable domain making their sum the generator of a strongly continuous semigroup. The source leaves the admissible domain construction and additional operator assumptions unspecified. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30001647$, title Semigroup Generation by Sums of Unbounded Operators, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2010 and names no individual proposer.
• Supplied report citation: Mini-Workshop: Wellposedness and Controllability of Evolution Equations (2011), Oberwolfach Reports, pp. 3269--3309, DOI 10.4171/owr/2010/56. https://doi.org/10.4171/owr/2010/56. This is also the supplied literature source labelled "original Oberwolfach problem report."
• Supplied literature source labelled "perturbations of semigroup generators": https://doi.org/10.1155/2014/213020. No authors or further bibliographic details are supplied for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that perturbation theorems cover important special cases, while no necessary-and-sufficient criterion for arbitrary unbounded $A$ and $B$ on Banach spaces is known. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is supplied provenance and has not been independently checked here.