24.3. OPERATOR VALUED FUNCTIONS 667

This proves the claim.Claim 2: |λ n| ≥ |λ n+1| .Proof of claim: From 24.17 and the definition of An and ek⊗ ek,

(An−1en+1,en+1) =

((A−

n−1

∑k=1

λ kek⊗ ek

)en+1,en+1

)= (Aen+1,en+1) = (Anen+1,en+1)

Thus,

λ n+1 = (Anen+1,en+1) = (An−1en+1,en+1)−λ n |(en,en+1)|2 = (An−1en+1,en+1)

By the previous claim. Therefore,

|λ n+1|= |(An−1en+1,en+1)| ≤ |(An−1en,en)|= |λ n|

by the definition of |λ n|. (en makes |(An−1x,x)| as large as possible.)Claim 3: limn→∞ λ n = 0.Proof of claim: If for some n,λ n = 0, then λ k = 0 for all k > n by claim 2. Thus, for

some n,A = ∑nk=1 λ kek⊗ek. Assume then that λ k ̸= 0 for any k. Then if limk→∞ |λ k|= ε >

0, one contradicts, ∥ek∥= 1 for all k because

∥Aen−Aem∥2 = ∥λ nen−λ mem∥2 = λ2n +λ

2m ≥ 2ε

2

which shows there is no Cauchy subsequence of {Aen}∞

n=1 , which contradicts the compact-ness of A. This proves the claim.

Claim 4: ∥An∥→ 0Proof of claim: Let x,y ∈ B

|λ n+1| ≥∣∣∣∣(An

x+ y2

,x+ y

2

)∣∣∣∣= ∣∣∣∣14 (Anx,x)+14(Any,y)+

12(Anx,y)

∣∣∣∣≥ 1

2|(Anx,y)|− 1

4|(Anx,x)+(Any,y)|

≥ 12|(Anx,y)|− 1

4(|(Anx,x)|+ |(Any,y)|)≥ 1

2|(Anx,y)|− 1

2|λ n+1|

and so 3 |λ n+1| ≥ |(Anx,y)| . It follows ∥An∥ ≤ 3 |λ n+1| . By 24.17 this proves 24.16 andcompletes the proof. ■

24.3.2 Measurable Compact OperatorsHere the operators will be of the form A(ω) where ω ∈ Ω and ω → A(ω)x is stronglymeasurable and A(ω) is a compact operator in L (H,H).

Theorem 24.3.10 Let A(ω) ∈ L (H,H) be a compact self adjoint operator andH is a separable Hilbert space such that ω → A(ω)x is strongly measurable. Then thereexist real numbers {λ k (ω)}∞

k=1 and vectors {ek (ω)}∞

k=1 such that

∥ek (ω)∥= 1