1. NOTATION 7 1. Notation 1.1. Basic notation. Let H be a separable (real or complex) Hilbert space, to be thought of as L2(µ), where µ is some equilibrium measure H is endowed with a norm · coming from a scalar (or Hermitian) product ·, ·. Let V be a finite-dimensional Hilbert space (say Rm or Cm, depending on whether H is a real or complex Hilbert space). Typically, V will be the space of those variables on which a certain diffusion operator acts. The assumption of finite dimension covers all cases that will be considered in applications, but it is not essential. Let A : H H V Hm be an unbounded operator with domain D(A), and let B : H H be an unbounded antisymmetric operator with domain D(B): ∀h, h D(B), Bh, h = −h, Bh . I shall assume that there is a dense topological vector space S in H such that S D(A) D(B) and A (resp. B) continuously sends S into S V (resp. S) this assumption is here only to guarantee that all the computations that will be performed (involving a finite number of operations of A, A∗ and B) are authorized. As a typical example, S would be the Schwartz space S(RN ) of C∞ functions f : RN R whose derivatives of arbitrary order decrease at infinity faster than all inverse polynomials but it might be a much larger space in case of need. If a linear operator S is given, I shall denote by S its operator norm: S = sup h=0 Sh h = sup h , h ≤1 Sh, h . If there is need to emphasize that S is considered as a linear operator between two spaces H1 and H2, the symbol S may be replaced by S H1→H2 . The norm A of an array of operators (A1,..∑Am) . , is defined as i Ai 2 the norm of a matrix-valued operator (Ajk) by Ajk 2 etc. The identity operator X X, viewed as a linear mapping, will always be denoted by I, whatever its domain. Often a multiplication operator (mapping a function f to fm, where m is a fixed function) will be identified with the multipli- cator m itself. Throughout the text, the real part will be denoted by . 1.2. Commutators. In the sequel, commutators involving A and B will play a crucial role. Since A takes its values in H V and B is only defined in H, some notational convention should first be made precise, since [A, B], for instance, does not a priori make sense. I shall resolve this issue by just tensorizing with the identity: [A, B] = AB −(B ⊗I)A is an unbounded operator H H⊗V. In a more pedestrian writing, [A, B] is the row of operators ([A1,B],..., [Am,B]). Then A2 stands for the matrix of operators (AjAk)j,k, [A, [A, B]] for ([Aj, [Ak,B]])j,k, etc. One should be careful about matrix operations made on components: For instance, [A, A∗] stands for ([Aj,Ak]j,k), which is an operator H H V V, while [A∗,A] stands for j [Aj ∗, Aj], which is an operator H H. Also note that [A, A] stands for the array ([Aj,Ak])j,k, and is therefore not necessary equal to 0. Whenever there is a risk of confusion, I shall make the notation more explicit.
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