Stability of the Peaceman–Rachford Approximation

LetAanBbe two self-adjoint operators in a Hilbert space; it is assumed thatAandBare bounded from below. The following operator expressionp(t)=(1+tA)−1(1−tB)(1+tB)−1(1−tA) called the Peaceman–Rachford formula, satisfies formally fortsmall the relationp(t)−exp(−2t(A+B))=O(t2). IfAandBdo not commute, it is not true in general thatp(t) is a stable approximation: we give an example ofAandBsuch that |p(t)|B(H), the operator norm ofp(t), is not bounded by 1+O(t); sufficient conditions of stability are given; they involve conditions on the commutators ofAandB. The proof relies on estimates on sandwiches, i.e., products of a finite number of copies ofA,B, (1+tA)−1, (1+tB)−1 and elements of an algebra M of bounded operators inH.