17.9 Compact Operators In Banach Space
In general for A ∈ℒ
the following definition holds.
Definition 17.9.1 Let A ∈ℒ
. Then A is compact if whenever B ⊆ X is a
bounded set, AB is precompact. Equivalently, if
is a bounded sequence in X,
has a subsequence which converges in Y.
An important result is the following theorem about the adjoint of a compact
Theorem 17.9.2 Let A ∈ℒ
be compact. Then the adjoint operator, A∗ ∈
is also compact.
be a bounded sequence in
be the closure of the unit ball
is precompact. Then it is clear that the functions
equicontinuous and uniformly bounded on the compact set,
By the Ascoli Arzela
theorem, there is a subsequence
which converges uniformly to a continuous
Now define g
Thus for x1,x2≠0, and a,b scalars,
showing that g
is linear on AX
and so by the Hahn Banach theorem, there exists y∗ extending g to all of Y having the
same operator norm.
In addition to this, for x ∈ B,
and this is uniformly small for large k
due to the uniform convergence of ynk∗
+ class=”left” align=”middle”(U)17.10. THE FREDHOLM ALTERNATIVE