Abstract:
F. Kühnemund and M. Wacker proposed commutativity conditions that allow proving Trotter
product formulas directly, without using the powerful Chernoff theorem. For the applicability of
the Chernoff theorem (in the context of the Trotter formula for semigroups T and S with
generators A and B), a condition that is difficult to verify is required: the image $(\lambda - A -
B)D$ must be dense in the Banach space. It turns out that if we require the smallness of the
difference $T(t)S(t)f - S(t)T(t)f$, it becomes possible to prove the convergence of the
iterations $[T(t/n)S(t/n)]^n f$ to the action of some strongly continuous semigroup on $f$ in a
way different from the classical approach.
[1] Kühnemund F., Wacker M. Commutator conditions implying the convergence of the
Lie–Trotter products //Proceedings of the American Mathematical Society. – 2001. – Vol. 129.
– No. 12. – P. 3569-3582.
