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

Abstract

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 α )⊆dom( A α )∩dom( B α ) for some α∈(1/2,1). It is proved that if, in addition, A and B satisfy dom( A 1/2)⊆dom( B 1/2), then the symmetric and non-symmetric Trotter–Kato product formula converges in the operator norm: ||(e −tB/2ne −tA/ne −tB/2n) n−e −tH||=O(n −(2α−1)) ||(e −tA/ne −tB/n) n−e −tH||=O(n −(2α−1)) uniformly in t∈[0, T], 0< T<∞, as n→∞, 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⩽ f( x)⩽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 α with respect to A α to the milder assumption dom( A 1/2)⊆dom( B 1/2) and extending essentially the admissible class of Kato functions.