 By Gross L.

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Extra info for Harmonic analysis on Hilbert space

Example text

7) is almost automorphic. Proof. 7). 1 Lineax Equations 55 Clearly X{s) e D(A) D D{B) = D(A) x D{B) (observe that the algebraic sum A + B exists since D(A + B) = H x H). Now decompose X{s) as follows X{s) = PsX{s) + {1x1where PsX{s) e R{Ps) = NiQs), Ps)Xis), and QsX{s) € N{Ps) = R(Qs). We have ±iPsXis)) = Pslxis) = PsAXis) + PsBX{s) = APsX{s) + PsBX{s) = APsX{s) Prom —(PsX(s)) as (by (by (it)) {w)) = APsX(5), it follows that PsX{s) = r{t)PsXio). Now ax^cording to (i), the vector-valued function s H-> PSX{S) T{t)PsX{0) = is ahnost automorphic.

We have proved that {T{t))t^^+ is a Co-semigroup. Conversely, suppose we have a Co-semigroup (T(i))tGR+ and define uiR-^ xX^Xhy u{t, x) = T{t)x, t € R+, xeX. 42 are obviously true. The mapping u is then a dynamical system. 43 tells us that the notions of abstract dynamical systems and Co-semigroups are equivalent. This fact provides a solid groimd to study Co-semigroups of linear operators as an independent topic. In the rest of the section, we will consider a Co-semigroup of linear operators T — {T{t))teR+ such that the motion T{t)xo :R-^ is in AAA(X) ^X with principal term f{t).

Let H = Z/^(E^) and let A, B be the operators given by Au = -Au, D{A) = H''(W) and Bu = Q% D{B) = {ue L 2 ( R ^ ) : Qu e ^^(R^)}, where Q : R^ H-> C is a measurable function satisfying (H) Re Q{x) > 0, Q € L\W), and Q ^ ^LCR"") 0^-. / ^ l / P ^ = oo for every compact Q C R^). Then: 1. If n < 3, the previous proposition is still valid. 5 . 2. 5 cannot be used here anymore). This can be proved using the boimdedness of the fractional operator I^u := {—A)^^'^Uy when a. la : 1/2 (R^) h-> BMO(R^) and b.