00-114 Cachia V., Neidhardt H., Zagrebnov V.A.

Accretive perturbations and error estimates for the Trotter product formula (54K, LATeX 2e) Mar 15, 00

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Abstract. We study the error bound estimate in the operator-norm topology for the exponential Trotter product formula in the case of accretive perturbations. Let $A$ be a semibounded from below self-adjoint operator on a separable Hilbert space. Let $B$ be a closed maximal accretive operator which is, together with $B^*$, Kato-small with respect to $A$ with relative bounds less than one. We show that in this case the operator-norm error bound estimate for the exponential Trotter product formula is the same as for the self-adjoint $B$ \cite{NZ1}: $$\left\|\left(e^{-tA/n}e^{-tB/n}\right)^n - e^{-t(A+B)}\right\| \leq L {\ln n\over n},\ n = 2,3,\ldots\,.$$ We verify that the operator $-(A+B)$ generates a holomorphic contraction semigroup. One gets a similar result when $B$ is substituted by $B^*$.

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