Abstract
An example is given which clarifies the present situation of the operator norm convergence of Trotter-Kato product formula. It shows that the rate of convergence of the formula with respect to the operator norm obtained in [NZ2] is best possible. It also yields a counter-example of the operator norm convergence of the formula in another case.
Similar content being viewed by others
References
[C] Paul. R. Chernoff: Note on product formulas for operator semigroups,J. Funct. Anal. 2 (1968), 238–242.
[DIT] Atsushi Doumeki, Takashi Ichinose and Hideo Tamura: Error bounds on exponential product formulas for Schrödinger operators,J. Math. Soc. Japan 50 (1998), 359–377.
[IT1] Takashi Ichinose and Hideo Tamura: Error estimate in operator norm for Trotter-Kato product formula,Integr. Equ. Oper. Theory 27 (1997), 195–207.
[IT2] Takashi Ichinose and Hideo Tamura: Error estimate in operator norm of exponential product formula for propagators of parabolic equations,Osaka J. Math. 35 (1998), 751–770
[K] Tosio Kato: Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups,Topics in Funct. Anal., Ad. Math. Suppl. Studies Vol. 3, 185–195 (I. Gohberg and M. Kac eds.), Acad. Press, New York 1978.
[NZ1] Hagen Neidhardt and Valentin A. Zagrebnov: On error estimates for the Trotter-Kato product formula,Lett. Math. Phys. 44 (1998), 169–186.
[NZ2] Hagen Neidhardt and Valentin A. Zagrebnov: Fractional powers of self-adjoint operators and Trotter-Kato product formula, to appear inIntegr. Equ. Oper. Theory.
[R] Dzh. L. Rogava: Error bounds for Trotter-type formulas for self-adjoint operators,Funct. Anal. Prilozhen 27 (1993), 84–86: English transl. inFunkt. Anal. Appl. 27(1993), 217–291.
[T] Hale F. Trotter: On the product of semi-groups of operators,Proc. Amer. Math. Soc. 10 (1959), 545–551.