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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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##### References:
 [1] N. BOURBAKI, 1953, Espaces vectoriels topologiques, Livre V. Herman, Paris. MR54161 · Zbl 0050.10703 [2] N. BOURBAKI, 1969, Intégration, Livre VI. Herman, Paris. MR276436 · Zbl 0189.14201 [3] D. O. DIA, Analyse numérique de certains schémas de directions alternées d’ordre élevé, en préparation. [4] T. KATO, 1966, Perturbation Theory for Linear Operators, Springer, Berlin. Zbl0148.12601 · Zbl 0148.12601 [5] F. RIESZ, 1952, Leçons d’Analyse Fonctionnelle, Akadémiai Kiadó, Budapest. Zbl0122.11205 MR55417 · Zbl 0122.11205 [6] M. SCHATZMAN, Higher order formulae for time integration, International Conference on spectral and higher order methods, Montpellier, juin 1992, à paraître dans les ’Proceedings ICOSAHOM 92’. [7] Q. SHENG, 1989, Solving Linear Partial Differential Equations by Exponential Splitting, IMA J. Numer. Anal., 9, pp. 199-212. Zbl0676.65116 MR1000457 · Zbl 0676.65116 · doi:10.1093/imanum/9.2.199
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