Semigroup Uniqueness for Dirichlet Operators

Definitions and mathematical statement

Supplied question and missing setting. The record asks whether the Friedrichs extension of an operator $L_0$ is its unique closed extension generating a strongly continuous semigroup on $L^p(\mu)$, for $p\geq 1$, and describes essential self-adjointness as the equivalent question when $p=2$. The record does not specify the measure space, the operator, its initial domain, or its hypotheses. Consequently, the following expresses the intended uniqueness property conditionally; it does not assert a conjecture for arbitrary measures and operators.

Let $(X,\Sigma,\mu)$ denote the unspecified measure space. For a finite real exponent $p\geq 1$, $L^p(\mu)$ is the space of measurable functions modulo equality almost everywhere with norm \[ \|f\|_p=\left(\int_X |f|^p\,{\,\mathrm{d}}\mu\right)^{1/p}<\infty. \] The scalar field is not specified in the record. Write $L_{0,p}:D_{0,p}\subseteq L^p(\mu)\to L^p(\mu)$ for the initial realization intended by the source's notation $L_0$; neither $D_{0,p}$ nor the compatibility of these realizations across exponents is supplied.

A $C_0$-semigroup on $L^p(\mu)$ is a family of bounded linear operators $(T(t))_{t\geq0}$ satisfying $T(0)=I$, $T(s+t)=T(s)T(t)$, and $\lim_{t\downarrow0}\|T(t)f-f\|_p=0$ for every $f\in L^p(\mu)$. Its generator $A$ is defined by \[ D(A)=\left\{f\in L^p(\mu):\lim_{t\downarrow0}\frac{T(t)f-f}{t}\text{ exists in }L^p(\mu)\right\}, \qquad Af=\lim_{t\downarrow0}\frac{T(t)f-f}{t}. \] Define the class of extensions relevant to the literal wording by \[ \mathcal G_p(L_{0,p})= \left\{A: \begin{array}{l} A\text{ is a closed, densely defined linear operator on }L^p(\mu),\\ D_{0,p}\subseteq D(A),\quad A|_{D_{0,p}}=L_{0,p},\\ A\text{ generates a }C_0\text{-semigroup on }L^p(\mu) \end{array} \right\}. \] No positivity, contractivity, or Markov condition on these semigroups is stated in the supplied question.

For a densely defined symmetric operator $S$ on a Hilbert space that is bounded below, its Friedrichs extension $F(S)$ is the self-adjoint operator associated with the closure of its semibounded quadratic form. This construction is a Hilbert-space construction. If the source instead uses a nonpositive Dirichlet generator convention, the corresponding distinguished extension would be written $-F(-L_{0,2})$. The record does not specify the sign convention or the hypotheses needed for either construction. It also does not specify how the distinguished extension is realized on $L^p(\mu)$ when $p\ne2$.

Conditional formulation of the intended question. Once the missing setting and a distinguished closed extension $L_{F,p}$ have been specified, is it true, for every exponent intended by the source, that \[ \mathcal G_p(L_{0,p})=\{L_{F,p}\}? \] This equality includes both generation by $L_{F,p}$ and uniqueness among all extensions in the displayed class. The written range $p\geq1$ includes every finite real $p\geq1$; whether an additional endpoint $p=\infty$ is intended is unspecified.

For a densely defined symmetric operator $L_{0,2}$, essential self-adjointness means \[ \overline{L_{0,2}}=(\overline{L_{0,2}})^*. \] The source asks whether this holds at $p=2$, presenting it as equivalent to its semigroup uniqueness question at that exponent. That equivalence is retained as a source claim requiring the missing hypotheses and extension conventions; it is not established here. In particular, no equivalence with uniqueness at all other exponents is asserted.

Short English statement

The question asks whether the distinguished Friedrichs realization is the only extension of $L_0$ that generates a strongly continuous semigroup on each intended $L^p(\mu)$, with essential self-adjointness proposed as the formulation at $p=2$. The operator, domain, measure, sign convention, and realizations for $p\ne2$ are missing, so this is a conditional statement requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30002706$, title Semigroup Uniqueness for Dirichlet Operators, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2014 and supplies no individual proposer.
• Supplied report citation: Dirichlet Form Theory and its Applications (2015), Oberwolfach Reports, pp. 2667--2756, DOI 10.4171/owr/2014/48. https://doi.org/10.4171/owr/2014/48. The literature-source list also labels this reference Oberwolfach Report 48/2014. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Albeverio--Kondratiev--Rockner, Essential self-adjointness. https://bibos.math.uni-bielefeld.de/preprints/06-05-211.pdf. No more precise bibliographic information or result location is supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that essential self-adjointness and $L^p$-uniqueness have been proved for important concrete Gibbs and Dirichlet-operator classes, while the general uniqueness question remains a fundamental open problem. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that its cleanup is self-contained and adds no unsupported content. These are supplied verification claims; the missing mathematical setting remains unresolved in this draft. Neither linked source has been visited for this response.