Download Analysis of Hamiltonian PDEs by Kuksin, Sergej B PDF

By Kuksin, Sergej B

For the final 20-30 years, curiosity between mathematicians and physicists in infinite-dimensional Hamiltonian structures and Hamiltonian partial differential equations has been turning out to be strongly, and plenty of papers and a couple of books were written on integrable Hamiltonian PDEs. over the past decade even though, the curiosity has shifted gradually in the direction of non-integrable Hamiltonian PDEs. the following, no longer algebra yet research and symplectic geometry are definitely the right analysing instruments. the current booklet is the 1st one to exploit this method of Hamiltonian PDEs and current an entire facts of the "KAM for PDEs" theorem. it will likely be a useful resource of data for postgraduate arithmetic and physics scholars and researchers.

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Example text

PROOF It is easy to see that (T—A)* = T*-A for all scalars A. 16 implies the rest. 14 If A, B are linear operators in a Banach space such that A is an extension of B and p(A) P\ p(B) is not empty, then A = B. PROOF If x G T)(A) and A G p(A) n p(B) then (A - X)x = (B - X)y for some y G Tf(B). Hence (A — X)(x — y) = 0 and, therefore, x = y, implying A = B. 15 Let T be a linear operator in a Banach space X. A set of scalars f(Tx) - where x ranges over all x G T)(T) with ||a;|| = 1, and / is a normalized tangent functional corresponding to this x - is called a numerical range of T.

Hence if g = T * - 1 / , then ||z|| = f(x) = (T*g)(x) = g(Tx) = (T*~lf)(Tx) < H^" 1 1|||^o;||. 19) Thus T is one-to-one. 19) implies that {xn} converges to some x E X. Closedness of T implies that x E X>(T), Tx = y and therefore ft(T) is closed. 7 would imply existence of g E Y* such that g(y) > 0 and g(Tx) = 0 for all x E D(T). However, this implies g E T>(T*), T*^ = 0, and, since T* is one-to-one we would have g = 0 which contradicts #(y) > 0. 19). 17 Suppose T : T>(T) c £ p (0,l) -> LP(0,1), p e [l,oo), is given by 77 = /' for / E D(T) = {/ E AC[0,1] | / ' E Lp(0,1), /(0) = 0}.

Un} be a basis for N. Each x G N has a unique repre­ sentation of the form x = Xi(x)ui H + \n(x)un, Xi(x) G K. 1, Aj G N*. The Hahn-Banach Theorem gives extensions Aj G X*of Xim Let M = nf=1'N(Ai). If x G X and y = Ai(o;)in + • ■ • + K(x)un G N, then x — y G M because Ai(x) = Xi(y) — A^(y) for 1 < i < n. 10 If X is a Banach space and T G 95(X) zs compact, then %{T - X) is closed for every scalar A ^ O . 7. 5, N = N(T — A) is a finite dimensional subspace of X. 9. Define S G 53(M, X) by So; = Tx - Xx.

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