Reciprocal Lie-Trotter formula

Let [GRAPHICS] and [GRAPHICS] be positive semidefinite matrices. The limit of the expression [GRAPHICS] as [GRAPHICS] tends to [GRAPHICS] is given by the well-known Lie-Trotter formula. A similar formula holds for the limit of [GRAPHICS] as [GRAPHICS] tends to [GRAPHICS] , where [GRAPHICS] is the geometric mean of [GRAPHICS] and [GRAPHICS] . In this paper we study the limit of [GRAPHICS] and [GRAPHICS] as [GRAPHICS] tends to [GRAPHICS] instead of [GRAPHICS] , with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie-Trotter formula. We show that the limit of [GRAPHICS] exists and find an explicit formula in a special case. The limit of [GRAPHICS] is shown for [GRAPHICS] matrices only.