1. VI.3 Spectrum of compact operators

2. Spectrum of T
Let
is called the resolvent set of T
: spectrum of T

3. Eigenvalue and Eigenspace
: eigenvalue of T
the eigenspace associated
then If

4. Remark 5 In general the inclusion is strict (except when ):
there may exists
such that
and
( such
but is not
belongs to
an eigenvalue)

5. Example
Let
then
but

6. Proposition VI.7
is compact and

9. Lemma 1.2 Suppose that is a sequence consisting of totally different
numbers such that
then
i.e.
consists only
isolated elements.

13. Theorem VI. 8 Let T is compact and Then (a) (b) (c) is finite or
is a sequence tending 0.

18. Remark
Given
Then there is a compact operator T such that

21. VI.4 Spectral decomposition of self-adjoint operators

22. Sesquilinear p.1 Let X be a complex Hilbert space.
is called sesquilinear if

23. Sesquilinear p.2 B is called bounded if there is r>0 such that
B is called positive definite if there is ρ>0 s.t.

24. Theorem 5.1 The Lax-Milgram Theorem p.1
Let X be a complex Hilbert space and B a
a bounded, positive definite sesquilinear
functional on X x X , then there is a unique
bounded linear operator S:X →X such that
and

25. Theorem 5.1 The Lax-Milgram Theorem p.2
Furthermore
exists and is bounded with

26. Self-Adjoint E=H is a Hilbert space is called self-adjoint
Definition : if
i.e.

27. Proposition VI.9
T : self-adjoint,
then

28. Remark of Proposition
This Proposition is better than Thm VI. 7

32. Corollary VI.10
Let
and
then T=0

34. Propositions p.1 be an orthogonal system in a Let
Hilbert space X, and let U be the closed vector
subspace generated by
Let
be the orthogonal projection onto U
where
and

35. Proposition (1)

36. Proposition (2)

37. Proposition (3)
For each k and x,y in X

38. Proposition (4)
For any x,y in X

39. Proposition (5)
Bessel inequality

40. Proposition (6) ( Parseval relation) An orthonormal system
is called complete and a Hilbert basis if U=X

41. Separable A Hilbert space is called separable
if it contains a countable dense subset

42. Theorem VI.11 H: a separable Hilbert space
T: self-adjoint compact operator. Then it
admits a Hilbert basis formed by eigenvectors
of T.

49. VI.1 Definition. Elementary Properties Adjoint

50. Lemma VI.1 (Riesz-Lemma)
Let
For any
fixed , apply Green’s second
identity to u and
in the domain
we have
and then let