Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
Since T −1 (0) is finite dimensional and T n (X) ⊃ T n+1 (X),
∃n 0 ∈ N such that T −1 (0) ∩ T n 0
(X) = T −1 (0) ∩ T n (X) for n ≥ n 0 .
From the fact that T n (X) ⊂ T n 0
(X), we have
x n − x n0 ∈ T −1 (0) ∩ T n 0
(X) = T −1 (0) ∩ T n (X) ⊂ T n (X).
Hence
x n0
∈ ∩
n≥n 0
T n (X) = T ∞ (X) and T x n0 = y,
which says that ˜T is onto. This proves (1.12). Now observe
dim (T − λI) −1 (0) = dim ˜ T − λI −1 (0) = index ˜ T − λI = index ˜T : (1.13)
the first equality comes from (1.11), the second equality follows from the fact that
β( T˜
− λI) ≤ β( ˜T ) = 0 by Lemma 1.8.1, and the third equality follows the observation
that ˜T is semi-Fredholm. Since the right-hand side of (1.13) is independent of λ,
α(T − λI) is constant and hence also is β(T − λI).
If instead β(T ) < ∞, apply the above argument with T ∗ .
Theorem 1.8.3. Define
Then
(i) U is an open set;
U :=
{
}
λ ∈ C : T − λI is semi-Fredholm .
(ii) If C is a component of U then on C, with the possible exception of isolated
points,
α(T − λI) and β(T − λI) have constant values n 1 and n 2 , respectively.
At the isolated points,
α(T − λI) > n 1 and β(T − λI) > n 2 .
Proof. (i) For λ ∈ U apply Lemma 1.8.1 to T − λI in place of T .
(ii) The component C is open since any component of an open set in C is open.
Let α(λ 0 ) = n 1 be the smallest integer which is attained by
α(λ) = α(T − λI) on C.
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