Anti Lie-Trotter formula

الأصل · EN

Let A and B be positive semidefinite matrices. The limit of the expression Zₚ:=(Aᵖ/²BᵖAᵖ/²)¹/ᵖ as p tends to 0 is given by the well known Lie-Trotter-Kato formula. A similar formula holds for the limit of Gₚ:=(Aᵖ#Bᵖ)²/ᵖ as p tends to 0, where X#Y is the geometric mean of X and Y. In this paper we study the complementary limit of Zₚ and Gₚ as p tends to ∞, with the ultimate goal of finding an explicit formula, which we call the anti Lie-Trotter formula. We show that the limit of Zₚ exists and find an explicit formula in a special case. The limit of Gₚ is shown for 2×2 matrices only.