19.7. COMPACT OPERATORS 539

and eitherlimn→∞

λ n = 0, (19.7.28)

or for some n,span(u1, · · · ,un) = H. (19.7.29)

In any case,span({ui}∞

i=1) is dense in A(H) . (19.7.30)

and for all x ∈ H,

Ax =∞

∑k=1

λ k (x,uk)uk (19.7.31)

where the sum might be finite. This sequence of eigenvectors and eigenvalues also satisfies

|λ n|= ∥An∥, (19.7.32)

andAn : Hn→ Hn. (19.7.33)

where H ≡ H1 and Hn ≡ {u1, · · · ,un−1}⊥ and An is the restriction of A to Hn.

Proof: If ∥A∥ = 0 then pick u ∈ H with ∥u∥ = 1 and let λ 1 = 0. Since A(H) = 0 itfollows the span of u is dense in A(H) and this proves the theorem in this uninterestingcase.

Assume from now on A ̸= 0. Let A1 = A and let λ 1 be real and λ21 ≡ ∥A∥

2. From thedefinition of ∥A∥ there exists xn,∥xn∥ = 1, and ∥Axn∥ → ∥A∥ = |λ 1|. Now it is clear thatA2 is also a compact self adjoint operator. Consider((

λ21−A2

)xn,xn

)= λ

21 (xn,xn)−

(A2xn,xn

)= λ

21−∥Axn∥2→ 0.

Since A is compact, there exists a subsequence of {xn} still denoted by {xn} such that Axnconverges to some element of H. Thus since λ

21−A2 satisfies((

λ21−A2

)y,y)≥ 0

in addition to being self adjoint, it follows x,y→((

λ21−A2

)x,y)

satisfies all the axiomsfor an inner product except for the one which says that (z,z) = 0 only if z = 0. Therefore,the Cauchy Schwarz inequality may be used to write∣∣∣((λ

21−A2

)xn,y

)∣∣∣ ≤ ((λ

21−A2

)y,y)1/2((

λ21−A2

)xn,xn

)1/2

≤ en ∥y∥ .

where en→ 0 as n→ ∞. Therefore, taking the sup over all ∥y∥ ≤ 1,

limn→∞

∥∥∥(λ21−A2

)xn

∥∥∥= 0.