2. OPERATORS L = A∗A + B 9 The scalar product of two vectors a and b in Rn or Cn will be denoted either by a, b or by a · b. The norm of a vector a in Rn or Cn will be denoted simply by |a|, and the Hilbert–Schmidt norm of an n × n matrix M (with real or complex entries) by |M|. The usual Brownian process in Rn will be denoted by (Bt)t≥0. The notation Hk will stand for the usual Sobolev space in Rn: explicitly, u 2 Hk = j≤k ∇ju 2 L2 . Sometimes I shall use subscripts to emphasize that the gradient is taken only with respect to certain variables and sometimes I shall indicate a reference measure if it is not the Lebesgue measure. For instance, u 2 Hv 1(µ) = |u|2 + |∇vu|2 dµ. 2. Operators L = A∗A + B For the moment we shall be concerned with linear operators of the form (2.1) L := A∗A + B, B∗ = −B, to be thought as the negative of the generator of a certain semigroup (St)t≥0 of interest: St = e−tL. (Of course, up to regularity issues, any linear operator L with nonnegative symmetric part can be written in the form (2.1) but this will be interesting only if A and B are “simple enough”.) In Proposition 2 below I have gathered some properties of L which can be expressed quite simply in terms of A and B. 2.1. Dirichlet form and kernel of L. Introduce K := Ker L, Π := orthogonal projection on K, Π⊥ = I Π. Proposition 2. With the above notation, (i) ∀h D(A∗A) D(B), Lh, h = Ah 2 (ii) K = Ker A Ker B. Proof. The proof of (i) follows at once from the identities A∗Ah, h = Ah, Ah = Ah 2, Bh, h = 0. It is clear that Ker A Ker B K. Conversely, if h belongs to K, then 0 = Lh, h = Ah 2, so h Ker A, and then Bh = Lh A∗Ah = 0. This concludes the proof of (ii). 2.2. Nonexpansivity of the semigroup. Now it is assumed that one can define a semigroup (e−tL)t≥0, i.e. a mapping (t, h) −→ e−tLh, continuous as a function of both t and h, satisfying the usual rules e0L = Id , e−(t+s)L = e−tLe−sL for t, s 0 (semigroup property), and ∀h D(L), d dt t=0+ e−tLh = −Lh. As an immediate consequence, for all h D(A∗A) D(B), 1 2 d dt t=0+ e−tLh 2 = Lh, h = −Ah 2 0. This, together with the semigroup property, the continuity of the semigroup and the density of the domain, implies that the semigroup is nonexpansive, i.e. its operator norm at any time is bounded by 1: ∀t 0 e−tL H→H 1.
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