Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
1.8 The Punctured Neighborhood Theorem
If T ∈ B(X, Y ) then
(a) T is said to be upper semi-Fredholm if T (X) is closed and α(T ) < ∞;
(b) T is said to be lower semi-Fredholm if T (X) is closed and β(T ) < ∞.
(c) T is said to be semi-Fredholm if it is upper or lower semi-Fredholm.
Theorem 1.6.1 remains true for semi-Fredholm operators. Thus we have:
Lemma 1.8.1. Suppose T ∈ B(X, Y ) is semi-Fredholm. If ||S|| < γ(T ) then
(i) T + S has a closed range;
(ii) α(T + S) ≤ α(T ), β(T + S) ≤ β(T );
(iii) index (T + S) = index T .
Proof. This follows from a slight change of the argument for Theorem 1.6.1.
We are ready for the punctured neighborhood theorem; this proof is due to Harte
and <strong>Lee</strong> [HaL1].
Theorem 1.8.2. (Punctured Neighborhood Theorem) If T ∈ B(X) is semi-Fredholm
then there exists ρ > 0 such that α(T −λI) and β(T −λI) are constant in the annulus
0 < |λ| < ρ.
Proof. Assume that T is upper semi-Fredholm and α(T ) < ∞. First we argue
Indeed,
Next we claim that
(T − λI) −1 (0) ⊂
∞∩
T n (X) =: T ∞ (X). (1.11)
n=1
x ∈ (T − λI) −1 (0) =⇒ T x = λx, and hence x ∈ T (X)
=⇒ Note that λx = T x ∈ T (T X) = T 2 (X)
=⇒ By induction, x ∈ T n (X) for all n.
T ∞ (X) is closed:
indeed, since T n is upper semi-Fredholm for all n, T n (X) is closed and hence T ∞ (X)
is closed.
If S commutes with T , so that also S(T ∞ (X)) ⊂ T ∞ (X), we shall write ˜S :
T ∞ (X) → T ∞ (X). We claim that
To see this, let y ∈ T ∞ (X) and thus
˜T : T ∞ (X) → T ∞ (X) is onto. (1.12)
∃ x n ∈ T n (X) such that T x n = y (n = 1, 2, . . .).
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