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# Undervisningsplan

 Dato Undervises av Sted Tema Kommentarer / ressurser 16.11.2006 7.3 Spectral family and spectrum of self-adjoint operators. (7.4 Simple spectrum, will not be included in the pensum) Exercises: Chap. 7: 1, 2 (, 3) 14.11.2006 7.2, 7.3 Hilbert´s theorem. Spectral decomposition of selfadjoint (bounded) operators. 09.11.2006 7.1, 7.2 Spectral decomposition. Spectral integral. Hilbert´s theorem.Exercises: 7.04, page 107 07.11.2006 7.0, 6.4 Functional calculus. Orthoprojections. 02.11.2006 6.3, 6.4a) Ordering. Projections in linear spaces.Exercises: Chap.6: 7, 11, 12 31.10.2006 6.3 Ordering in the space of self-adjoint operators. 26.10.2006 6.2c), 6.3 The second Hilbert-Schmidt Theorem. (Ordering in the space of self-adjoint operators.)Exercises: Chap.6: 5, and in addition: Exercise. Let K be an integral operator on H=L_2[a,b] with a continuous kernel function k. Assume that K has a Riemann square-integrable eigenfunction f in H with a nonzero eigenvalue. Prove that(a) f is continuous(b) If k has continuous partial derivatives of order 1 (respectively n) then f is continuously differentiable (resp. n times continuously differentiable). 24.10.2006 6.2 b, c The Minimax principle. The second Hilbert-Schmidt theorem. 19.10.2006 6.2a The first Hilbert-Schmidt theorem.Exercises: Chap. 6: 2, 3 17.10.2006 6.1, 6.2a Spectral theory for compact self-adjoint operators. 12.10.2006 5.2, 4.4, 6.1 Fredholm´s theory of compact operators. Adjoint operators.Exercises: Chap. 4: 23; Chap.5: 9, 11 10.10.2006 5.2 Compact operators. Fredholm´s theory. 05.10.2006 5.2 Continuous spectrum. Compact operators.Exercises: Chap. 5: 3, 4, 5, and in addition:Let X and Y be normed spaces, Y complete, D a dense subspace of X. Assume A : D -->Y is a bounded linear operator. Prove that A has an extension to an operator in L(X,Y) with the same norm as A. Also show that this extension of A is unique. 03.10.2006 4.7, 5.1, 5.2 Invertible operators. The spectral theory of compact operators: Point spectrum. 28.09.2006 4.6 K(X,Y) is closed in L(X,Y) : completion of proof (there is a gap at the end of the argument given in the book). Different types of operator convergences. If there is time: A in L(X,Y) compact implies the adjoint (dual operator) A* is compact.Exercises: Chap. 4: 8, 12, 20 26.09.2006 4.4, 4.5a Approximation of compact operators by operators of finite rank. Dual operators. 21.09.2006 4.5 Finite rank operators.Exercises: Chap. 4: 1, 3, 15, (19: uses exercise 8 of Chap.3) 19.09.2006 4.2, 4.3a Examples of bounded linear operators. Precompact sets. Compact operators. 14.09.2006 3.1, 4.1 Corollaries to the Hahn-Banach theorem. Bounded linear operators.Exercises: Chap. 2: 24, Chap. 3: 5 12.09.2006 3.1, 3.2 The Hahn-Banach theorem. Examples of dual spaces. 07.09.2006 2.2 c, 2.3 a-c, 3.2 Linear functionals. Dual spaces. Examples.Exercises: Chap. 2: 10, 11, 12, 13, 29 05.09.2006 2.1 e, 2.2 a, b, c Application of Parseval´s identity to L_2[a,b]. Convex sets. Orthogonal projections, orthogonal complements. 31.08.2006 2.1 e), 2.2 b) Anvendelse av Parsevals identitet på L_2[a,b]. Konvekse mengder.Oppgaver. Kap.1: 18, 20Kap.2: 1, 4, 5, 8 c). 29.08.2006 2.1 b)- 2.1 e) Bessels ulikhet og Parsevals identitet. Eksistens av ortonormal basis. 24.08.2006 1.5, 2.1 a) Komplettering av normerte rom. Hilbertrom.Oppgaver. Kap.1: 1, 11, 12, 22, 23,24.Dessuten:(I) Vis at rommet Rp[a,b] (side 16) ikke er et lineært rom. Forsøk å omdefinere Rp[a,b] slik at det blir lineært (og fremdeles inneholder C[a,b]). 22.08.2006 Terje Sund B 70 1.1 - 1.4. Normerte rom. Kvotientrom.
Publisert 14. aug. 2006 16:26 - Sist endret 14. nov. 2006 14:40