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Commutators of certain holomorphic semigroups and applications to alternate directions. (Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées.) (French) Zbl 0853.47024
Summary: Let \(A\) and \(B\) be two non necessarily commuting operators. Two high-order alternate direction formulae are defined by \[ M_1(t)= \textstyle{{4\over 3}} e^{tA/4} e^{tB/2} e^{tA/2} e^{tB/2} e^{tA/4}- \textstyle{{1\over 3}} e^{tA/2} e^{tB} e^{tA/2}, \] \[ M_2(t)= \textstyle{{2\over 3}} (e^{tA/2} e^{tB} e^{tA/2}+ e^{tB/2} e^{tA} e^{tB/2})- \textstyle{{1\over 6}} (e^{tA} e^{tB}+ e^{tB} e^{tA}) \] and in the sense of formal series \[ \begin{cases} e^{t(A+ B)}- M_1(t/n)^n= O(n^{- 4})\\ e^{t(A+ B)}- M_2(t/n)^n= O(n^{- 3})\end{cases}\tag{\(*\)} \] If \(a\), \(a_0\), \(b\) and \(b_0\) are strictly positive and infinitely differentiable from \(\mathbb{T}^2= (\mathbb{R}/\mathbb{Z})^2\) to \(\mathbb{R}\), operators \(A\) and \(B\) are defined by \[ A= {\partial\over \partial x_1} \Biggl(a(x_1, x_2) {\partial\over \partial x_1}\Biggr)- a_0(x_1, x_2);\;B= {\partial\over \partial x_2} \Biggl(b(x_1, x_2) {\partial\over \partial x_2}\Biggr)- b_0(x_1, x_2). \] These two operators generate holomorphic semigroups in \(L^2(\mathbb{T}^2)\) and the following estimates hold in operator norm in \(L^2(\mathbb{T}^2)\), \[ |M_1(t)|= O(t)\quad \text{and} \quad |M_2(t)|= O(t). \] Here, there exists a constant \(c\) such that \[ |M_1(t/n)^n|\leq e^{ct}\quad \text{and} \quad |M_2(t/n)^n|\leq e^{ct}, \] which implies that formulae \((*)\) are stable.

MSC:
47D06 One-parameter semigroups and linear evolution equations
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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