Definition 17.7.1 Let A ∈ℒ
|
It is clear y → A∗y is linear and continuous. A∗ is called the adjoint of A. A is a self adjoint operator if A = A∗. Thus for a self adjoint operator,
The big result is called the Hilbert Schmidt theorem. It is a generalization to arbitrary Hilbert spaces of standard finite dimensional results having to do with diagonalizing a symmetric matrix. There is another statement and proof of this theorem around Page 2056.
Theorem 17.7.2 Let A be a compact self adjoint operator defined on a Hilbert space, H. Then there exists a countable set of eigenvalues,
| (17.7.27) |
and either
| (17.7.28) |
or for some n,
| (17.7.29) |
In any case,
| (17.7.30) |
and for all x ∈ H,
| (17.7.31) |
This sequence of eigenvectors and eigenvalues also satisfies
| (17.7.32) |
and
| (17.7.33) |
where H ≡ H1 and Hn ≡
Proof: If A = 0 then pick u ∈ H with
Assume from now on A≠0. Let λ1 be real and λ12 ≡
|
Since A is compact, there exists a subsequence of
|
in addition to being self adjoint, it follows x,y →
|
Since A2xn converges, it follows since λ1≠0 that
|
Now
|
If
Suppose
|
this yields the conclusion of the theorem in the situation of 17.7.29. Therefore, assume the span of these vectors is always a proper subspace of H. It is shown next that An+1 : Hn+1 → Hn+1. Let
|
Then for k ≤ n
|
showing An+1 : Hn+1 → Hn+1 as claimed. There are two cases. Either λn = 0 or it is not. In the case where λn = 0 it follows An = 0. Every element of H is the sum of one in span
Now consider the case where λn≠0. In this case repeat the above argument used to find un+1 and λn+1 for the operator, An+1. This yields un+1 ∈ Hn+1 ≡
|
and if it is ever the case that λn = 0, it follows from the above argument that the conclusion of the theorem is obtained.
I claim limn→∞λn = 0. If this were not so, then for some ε > 0, 0 < ε = limn→∞
It remains to verify that span
|
Therefore, Aw = 0. Now every vector from H can be written as a sum of one from
|
and one from span
|
Therefore, from Corollary 17.5.3,
|
Therefore,
Define v ⊗ u ∈ℒ
|
then 17.7.31 is of the form
|
This is the content of the following corollary.
Corollary 17.7.3 The main conclusion of the above theorem can be written as
|
where the convergence of the partial sums takes place in the operator norm.
Proof: Using 17.7.31
|
and this proves the corollary.
Corollary 17.7.4 Let A be a compact self adjoint operator defined on a separable Hilbert space, H. Then there exists a countable set of eigenvalues,
| (17.7.34) |
| (17.7.35) |
Furthermore, if λi≠0, the space, V λi ≡
Proof: In the proof of the above theorem, let W ≡span
It remains to verify the space, V λi, is finite dimensional. First observe that A : V λi → V λi. Since A is continuous, it follows that A : V λi →V λi. Thus A is a compact self adjoint operator on V λi and by Theorem 17.7.2, 17.7.29 holds because the only eigenvalue is λi. This proves the corollary.
Note the last claim of this corollary holds independent of the separability of H. This proves the corollary.
Suppose λ
Corollary 17.7.5 Let A be a compact self adjoint operator and let λ
| (17.7.36) |
Proof: Let m < n. Then since the
|
and so for m large enough, the first term in 17.7.37 is smaller than ε. This shows the infinite series in 17.7.36 converges. It is now routine to verify that the formula in 17.7.36 is the inverse.
+ class=”left” align=”middle”(U)17.8. STURM LIOUVILLE PROBLEMS