17.8.1 Nuclear Operators
Definition 17.8.5 A self adjoint operator A ∈ℒ
for H a separable
Hilbert space is called a nuclear operator if for some complete orthonormal set,
To begin with here is an interesting lemma.
Lemma 17.8.6 Suppose
is a sequence of compact operators in ℒ
Banach spaces, X and Y and suppose A ∈ℒ
Then A is also compact.
Proof: Let B be a bounded set in X such that
for all b ∈ B.
I need to verify
is totally bounded. Suppose then it is not. Then there exists ε >
0 and a sequence,
bi ∈ B
whenever i≠j. Then let n be large enough that
a contradiction to An
being compact. This proves the lemma.
Then one can prove the following lemma. In this lemma, A ≥ 0 will mean
Lemma 17.8.7 Let A ≥ 0 be a nuclear operator defined on a separable Hilbert space,
H. Then A is compact and also, whenever
is a complete orthonormal
Proof: First consider the formula. Since A is given to be continuous,
the series converging because
Then also since A is self adjoint,
Next consider the claim that A is compact. Let CA ≡
be defined by
Then An has values in span
and so it must be a compact operator
because bounded sets in a finite dimensional space must be precompact. Then
and this shows that if n
is sufficiently large,
and so A is the limit in operator norm of finite rank bounded linear operators, each of
which is compact. Therefore, A is also compact.
Definition 17.8.8 The trace of a nuclear operator A ∈ℒ
such that A ≥
defined to equal
is an orthonormal basis for the Hilbert space, H.
Theorem 17.8.9 Definition 17.8.8 is well defined and equals ∑
j=1∞λj where the
λj are the eigenvalues of A.
is some other orthonormal basis. Then
By Lemma 17.8.7 A is compact and so
where the uk are the orthonormal eigenvectors of A which form a complete orthonormal
and this proves the theorem.
This is just like it is for a matrix. Recall the trace of a matrix is the sum of the
It is also easy to see that in any separable Hilbert space, there exist nuclear operators.
be a complete orthonormal set of vectors.
It is not too hard to verify this works.
Much more can be said about nuclear operators.