MAT4340 - H?st 2006

Undervisningsplan

DatoUndervises avStedTemaKommentarer / 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 p? L_2[a,b]. Konvekse mengder.

Oppgaver.

Kap.1: 18, 20

Kap.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 line?rt rom. Fors?k ? omdefinere Rp[a,b] slik at det blir line?rt (og fremdeles inneholder C[a,b]).?

22.08.2006Terje Sund? B 70? 1.1 - 1.4. ? Normerte rom. Kvotientrom.?
Publisert 14. aug. 2006 16:26 - Sist endret 14. nov. 2006 14:40
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