VI.3 Spectrum of compact operators

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

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

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

Example Let then but

Proposition VI.7 is compact and

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

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

Remark Given Then there is a compact operator T such that

VI.4 Spectral decomposition of self-adjoint operators

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

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.

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

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

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

Proposition VI.9 T : self-adjoint, then

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

Corollary VI.10 Let and then T=0

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

Proposition (1)

Proposition (2)

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

Proposition (4) For any x,y in X

Proposition (5) Bessel inequality

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

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

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

VI.1 Definition. Elementary Properties Adjoint

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