Trotter-Kato product formula and symmetrically normed ideals
Given selfadjoint nonnegative operators \(A\) and \(B\) on a Hilbert space \({\mathcal H}\), \(H:= A+B\) is well-defined on \({\mathcal H}_0= \overline{{\mathcal D}(A^{1/2})\cap{\mathcal D}(B^{1/2})}\). On \({\mathcal H}_0\), the semigroup generated by \(H\) is approximated by the ones for \(A\) and \(B\) in the sense that \[ s\text{-} \lim_n(e^{-tA/n} e^{-tB/n})^n= e^{-tH} P_0,\quad t\geq 0, \] where \(P_0:{\mathcal H}\to{\mathcal H}_0\) is the orthogonal projection. The authors strengthen and extend this ``Trotter-Kato product formula'' to convergence results for symmetrically normed ideals (like the trace class ideal of operators) and replace the exponential function \(e^{-x}\) to more general functions \(f\) and \(g\). Taking \(0\leq f\), \(g\leq 1\), \(f(0)= g(0)= 1\), \(f'(0)= g'(0)= -1\) and letting \(F(t)= g(tB)^{1/2} f(tA) g(tB)^{1/2}\) or \(\widetilde F(t):= f(tA) g(tB)\), they give conditions when \[ \lim_{r\to\infty} \sup_{t\in [\tau_0,\tau]}\|F(t/r)^r- e^{-tH} P_0\|_\phi= 0 \] converges uniformly in the ideal norm \(\|\cdot\|_\phi\) away from \(0\) in the sense that \([\tau_0, \tau]\) is a compact interval in \((0,\infty)\). Convergence for \(F\) is equivalent to one for \(\widetilde F\) or similar constructs.
scientific article; zbMATH DE number 1346904
scientific article