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

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(Hα)⊆dom(Aα)∩dom(Bα) for some α∈(1/2,1). It is proved that if, in addition, A and B satisfy dom(A1/2)⊆dom(B1/2), 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