When Chernoff Theorem Almost Works: Compactness of Non-Convergent Approximations

This paper investigates the limiting behavior of Chernoff approximations for operator semigroups when the classical conditions of the Chernoff Theorem are not fully satisfied. We construct a fundamental example where the closure of the right derivative of the approximating operator-function at zero yields a symmetric operator S with finite, equal defect indices $$(2,2)$$ , rather than a semigroup generator. In this scenario, the sequence of Chernoff iterations \(F(t/n))ⁿ\ fails to converge, yet exhibits remarkable structure: it remains precompact in the strong operator topology, and its set of limit points precisely corresponds to the family of semigroups generated by self-adjoint extensions of S .