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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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