Trotter–Kato Product Formula and Symmetrically Normed Ideals

It is proven that the Trotter product formula converges in the norm of a symmetrically normed ideal of compact operators away from t0>0 if the Kac operator (the transfer matrix) F(t)=e−tB/2e−tAe−tB/2 belongs to this ideal for t=t0. The result is generalized to the Trotter–Kato product formula. Moreover, if the perturbation B is small relative to A, then error bounds for convergence are obtained. The results apply to the Dixmier trace.