554 CHAPTER 19. HILBERT SPACES

It is also easy to see that in any separable Hilbert space, there exist nuclear operators.Let ∑

∞k=1 |λ k|< ∞. Then let {ek} be a complete orthonormal set of vectors. Let

A≡∞

∑k=1

λ kek⊗ ek.

It is not too hard to verify this works.Much more can be said about nuclear operators.

19.8.2 Hilbert Schmidt Operators

Definition 19.8.10 Let H and G be two separable Hilbert spaces and let T map H to G belinear. Then T is called a Hilbert Schmidt operator if there exists some orthonormal basisfor H,

{e j}

such that

∑j

∥∥Te j∥∥2

< ∞.

The collection of all such linear maps will be denoted by L2 (H,G) .

Theorem 19.8.11 L2 (H,G)⊆L (H,G) and L2 (H,G) is a separable Hilbert space withnorm given by

∥T∥L2≡

(∑k∥Tek∥2

)1/2

where {ek} is some orthonormal basis for H. Also L2 (H,G)⊆L (H,G) and

∥T∥ ≤ ∥T∥L2. (19.8.64)

All Hilbert Schmidt opearators are compact. Also for X ∈H and Y ∈G,X⊗Y ∈L2 (H,G)and

∥X⊗Y∥L2= ∥X∥H ∥Y∥G (19.8.65)

Proof: First I want to show L2 (H,G) ⊆ L (H,G) and ∥T∥ ≤ ∥T∥L2. Pick an or-

thonormal basis for H,{ek} and an orthonormal basis for G,{ fk}. Then letting

x =n

∑k=1

xkek,

T x = T

(n

∑k=1

xkek

)=

n

∑k=1

xkT (ek)