Metric and Hilbert Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 4 November 2014

Lecture 42: Bases of eigenvectors for compact self adjoint operators

Consider the sequence e1,e2,e3, in p where ei=(0,0,,0,1,0,0,) with 1 in the ith spot. Then

(a) e1,e2,e3, has no convergent subsequence. (The closed unit ball is not compact.)
(b) e1,e2,e3 weakly converges to 0 i.e. If x2, x=(x1,x2,), then x,e1,x,e2,x,e3 converges to 0.
(c) Let I:pp be the identity operator. Then Ie1,Ie2,Ie3, is the sequence e1,e2,e3, has no convergent subsequence and so I(S) is not compact.

Let T:HH be a compact linear operator. Let λ be a nonzero eigenvalue of T and let Xλ={xH|Tx=λx}. If dim(Xλ) is infinite then there is a sequence e1,e2,e3, in Xλ of linearly independent vectors with ej=1 and Tej=λej and ei,ej=δij. Then Tem-Ten2 = λem-λen2 = λ2 em-en2 = λ2 (em2+en2) = λ2·2. So the sequence Te1,Te2, has no Cauchy subsequence and no convergent subsequence. So T is not compact.

Let T:HH be a compact linear operator and let λ1,λ2,λ3, be distinct eigenvalues of T.
Assume limkλk0.
Then there is a subsequence λk1,λk2, and c>0 with λkj>C for j>0.
Let e1,e2, be such that ei=1 and Tej=λkjej.
Since λk1,λk2, are all distinct then ei,ej=0.
So Tem-Ten2 = λmem-λnen2 = λmem-λnen, λmem-λnen = λm2em2+ λn2en2 > 2C2. So Te1,Te2,Te3, has no Cauchy subsequence and no convergent subsequence.
So T is not compact.

Remark Let H be a Hilbert space and let T:HH be a compact operator. If H is infinite dimensional then T is not a bijection (or, better, 0·I-T is not a bijection).

Let T:HH be a compact self adjoint operator. For each eigenvalue let Bλ be an orthonormal basis of Xλ and let B=eigenvaluesBλ and let X=span(B). Then H=XXand T=TXTX with TX and TX both compact operators TX:XX. Then TX has an eigenvector with eigenvalue TX. But all eigenspaces of T are in X. So H=X.

Notes and References

These are a typed copy of Lecture 42 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on October 12, 2014.

page history