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Trotter-Kato product formula and operator-norm convergence. (English) Zbl 0949.47019
For two non-negative self-adjoint operator $$A$$ and $$B$$ on a separable Hilbert space $$\mathcal{H}$$ T. Kato showed the convergence of the Trotter products $\text{strong}- \lim_{n\rightarrow \infty} \bigl( f(tA/n) g(tB/n) \bigr)^n = e^{-tH} P_0$ to the semigroup generated by the form sum $H=A\dot{+} B \quad \text{on}\quad {\mathcal{H}}_0:=\overline{D(A^{1/2})\cap D(B^{1/2})}$ with $$P_0$$ the orthogonal projection from $$\mathcal{H}$$ onto $${\mathcal{H}}_0$$ and for Kato-functions $$f,g$$. In this paper the authors investigate the operator norm convergence (away from zero) of these products. Therefore, they establish necessary and sufficient conditions for this convergence including a resolvent convergence in operator norm. They obtain the operator norm convergence of the products away from zero for every regular Kato–function $$f$$ (such as $$e^{-t}$$) and every Kato–function $$g$$ if $$f(t_0 A)$$ is compact for some $$t_0>0$$. This condition holds if the resolvent of one of the underlying operators is compact. The compactness of the resolvent turns out to be also a sufficient condition for the norm convergence of the Trotter products. On the other side, for the case $$D(A^{1/2}) \subseteq D(B^{1/2})$$ they obtain the same convergence of the Trotter products if $$A^{1/2}$$ is relatively compact to $$B^{1/2}$$, i.e. $$B^{1/2}A^{-1/2}$$ is compact.

##### MSC:
 47B25 Linear symmetric and selfadjoint operators (unbounded) 47D06 One-parameter semigroups and linear evolution equations 47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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