Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
Proof. Let X = T −1 (0) ⊕ X 0 and Y = T (X) ⊕ Y 0 . Define ˜T , ˜K : X0 × Y 0 → Y by
˜T (x 0 , y 0 ) = T x 0 + y 0 , ˜K(x0 , y 0 ) = Kx 0 + y 0 .
Therefore ˜K is compact since K is compact and dim Y 0 < ∞. From ( ˜T + ˜K)(x 0 , 0) =
(T + K)x 0 and Lemma 1.5.1 it follows that
T + K is Fredholm ⇐⇒
˜T + ˜K is Fredholm.
But ˜T is invertible. So
˜T + ˜K = ˜T ( I + ˜T −1 ˜K) .
Observe that ˜T −1 ˜K is compact. Thus by Corollary 1.4.6, I + ˜T
−1 ˜K is Fredholm.
Hence T + K is Fredholm.
To prove the statement about the index consider the integer valued function
F (λ) := index (T + λK). Applying Theorem 1.6.1 to T + λK in place of T shows that
f is continuous on [0, 1]. Consequently, f is constant. In particular,
index T = f(0) = f(1) = index (T + K).
Corollary 1.6.3. If K ∈ B(X) then
K is compact =⇒ I − K is Fredholm with index (I − K) = 0.
Proof. Apply the preceding theorem with T = I and note that index I = 0.
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