# Read e-book online Algebraic Theory of D-modules [Lecture notes] PDF By J. Bernstein

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23) Ft = [cxpf:fePK(t)l for every t > 0. We now define an isomorphism W\ E —• F. ,vn belong to %fe. Let fo,go: (0,t) —• % be the corresponding step functions Define f,gePK(t) Ms) = uj, Sj-i

Now oo n=0 and in particular expft - exp(0) - f, is orthogonal to exp(0). 15) is not changed if we replace u{t) with u(t) - exp(0) = u(t) - e(t). 16) | (u(t) - e{t), exp ft - exp(0) - /,) | < ||M(0 - e(/)|| • || exp>J - exp(0) - yj|| /oo 2 <\\u{t)-e{t)\\y£-,\\ft\\ N'/2 Using \\f,\\2 = H? 16) becomes „„<„_,<„„ g « ! 17) is seen to be of order o(t), as required. D CONTINOUS ANALOGUES OF FOCK SPACE 41 S. Dimension of a Product System. In this section we show that, starting with an arbitrary product system E, one can construct a natural (separable) Hilbert space H.

15], p. 282). The solution {U(t): t > 0} can be shown to have the required properties. 17) with a bounded operator B\ in fact, the domains of the respective generators need not be the same. 18. Let a, ft be two unital semigroups of endomorphisms of 3S[H\ Then atP if and only if their associated product systems Ea,Ep are isomorphic. PROOF. Assume first that a and /? 16). First, we claim that for every t > 0, one has UtEa(t) = Ep(t). Indeed, an opertaor T belongs to Ep{t) iff fit(A)T = TA for every A e 3§{H).