Trotter-Kato product formula and fractional powers of self-adjoint generators

摘要

Let A and B be non-negative self-adjoint operators in a Hilbert space such that their densely defined form sum H = A + B obeys dom(H-alpha) subset of or equal to dom(A(alpha)) boolean AND dom(B-alpha) for some alpha is an element of (1/2, 1). It is proved that if, in addition, A and B satisfy dom(A(1/2)) subset of or equal to dom(B-1/2), then the symmetric and non-symmetric Trotter-Kato product formula converges in the operator norm: parallel to(e(-tB/2n)e(-tA)e(-tB/2n))(n) - e(-tH)parallel to = O(n(-(2alpha-1))) parallel to(e(-tA)e(-tB))(n) - e(-tH)parallel to = O(n(-(2alpha-1))) uniformly in t is an element of [0, T], 0 < T < infinity, as n --> infinity, 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 (.) defined on the non-negative real axis obeying 0 less than or equal to f (x) less than or equal to 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-alpha with respect to A to the milder assumption dom(A(1/2)) subset of or equal to dom(B-1/2) and extending essentially the admissible class of Kato functions. (C) 2003 Elsevier Inc. All rights reserved. [References: 18]