Abstract:
We investigate the limiting behavior of Chernoff approximations for operator semigroups when the classical conditions of Chernoff's theorem are not fully satisfied. An example is considered in which the closure of the derivative of the Chernoff approximating operator function at zero is a symmetric operator with deficiency indices $(2,2)$, but not a semigroup generator. In this scenario, the sequence of Chernoff iterations $\{ (\mathbf F(t/n))^n \}$ does not converge, but exhibits a remarkable structure: it remains precompact in the strong operator topology, and the set of its limit points corresponds to a family of semigroups diffeomorphic to a circle.
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