19.9. COMPACT OPERATORS IN BANACH SPACE 557
19.9 Compact Operators in Banach SpaceIn general for A ∈L (X ,Y ) the following definition holds.
Definition 19.9.1 Let A ∈L (X ,Y ) . Then A is compact if whenever B ⊆ X is a boundedset, AB is precompact. Equivalently, if {xn} is a bounded sequence in X , then {Axn} has asubsequence which converges in Y.
An important result is the following theorem about the adjoint of a compact operator.
Theorem 19.9.2 Let A∈L (X ,Y ) be compact. Then the adjoint operator, A∗ ∈L (Y ′,X ′)is also compact.
Proof: Let {y∗n} be a bounded sequence in Y ′. Let B be the closure of the unit ballin X . Then AB is precompact. Then it is clear that the functions {y∗n} are equicontinuousand uniformly bounded on the compact set, A(B). By the Ascoli Arzela theorem, there is asubsequence
{y∗nk
}which converges uniformly to a continuous function, f on A(B). Now
define g on AX by
g(Ax) = ||x|| f(
A(
x||x||
)),g(A0) = 0.
Thus for x1,x2 ̸= 0, and a,b scalars,
g(aAx1 +bAx2) ≡ ||ax1 +bx2|| f(
A(ax1 +bx2)
||ax1 +bx2||
)≡ lim
k→∞
||ax1 +bx2||y∗nk
(A(ax1 +bx2)
||ax1 +bx2||
)= lim
k→∞ay∗nk
(Ax1)+by∗nk(Ax2)
= a limk→∞
||x1||y∗nk
(Ax1
||x1||
)+b lim
k→∞
||x2||y∗nk
(Ax2
||x2||
)= a ||x1|| f
(Ax1
||x1||
)+b ||x2|| f
(Ax2
||x2||
)≡ ag(Ax1)+bg(Ax2)
showing that g is linear on AX . Also
|g(Ax)|= limk→∞
∣∣∣∣||x||y∗nk
(A(
x||x||
))∣∣∣∣≤C ||x||∣∣∣∣∣∣∣∣A( x
||x||
)∣∣∣∣∣∣∣∣=C ||Ax||
and so by the Hahn Banach theorem, there exists y∗ extending g to all of Y having the sameoperator norm.
y∗ (Ax) = limk→∞
||x||y∗nk
(A(
x||x||
))= lim
k→∞y∗nk
(Ax)