02-411 Ichinose T., Neidhardt H., Zagrebnov V.A.

Trotter-Kato product formula and fractional powers of self-adjoint generators (207K, Postscript) Oct 2, 02

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Abstract. Let $A$ and $B$ be non-negative self-adjoint operators in a Hilbert space such that their densely defined form sum $H = A \stackrel{\cdot}{+} B$ obeys $\dom(H^\ga) \subseteq \dom(A^\ga) \cap \dom(B^\ga)$ for some $\ga \in (1/2,1)$. It is proved that if, in addition, $A$ and $B$ satisfy $\dom(A^{1/2}) \subseteq \dom(B^{1/2})$, then the symmetric and non-symmetric Trotter-Kato product formula converges in the operator norm: % % \bed \ba{c} \left\|\left(e^{-tB/2n}e^{-tA/n}e^{-tB/2n}\right)^n - e^{-tH}\right\| = O(n^{-(2\ga-1)}), \\[2mm] \left\|\left(e^{-tA/n}e^{-tB/n}\right)^n - e^{-tH}\right\| = O(n^{-(2\ga-1)}) \ea \eed % % uniformly in $t \in [0,T]$, $0 < T < \infty$, as $n \to \infty$, both with the same optimal error bound. The same is valid if one replaces the exponential function in the product by functions of the Kato class, that is, by real-valued Borel measurable functions $f(\cdot)$ defined on the non-negative real axis obeying $0 \le f(x) \le 1$, $f(0) = 1$ and $f'(+0) = -1$, with some additional smoothness property at zero. The present result improves previous ones relaxing the smallness of $B^\ga$ with respect to $A^\ga$ to the milder assumption $\dom(A^{1/2}) \subseteq \dom(B^{1/2})$ and extending essentially the admissible class of Kato functions. % %

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