Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
Theorem 1.4.5. Let K ∈ B(X). If K is compact then T = I − K has closed range.
Proof. Let V be a closed bounded set in X and let
y = lim
n→∞ (I − K)x n, where x n ∈ V. (1.5)
We have to prove that y = (I − K)x 0 for some x 0 ∈ V . Since V is bounded and K
is compact the sequence {Kx n } has a convergent subsequence {Kx ni }. By (1.5), we
see that
( )
x 0 := lim x ni = lim (I − K)xni + Kx ni exists.
i→∞ i→∞
But then y = (I − K)x 0 ∈ (I − K)V ; thus (I − K)V is closed. Therefore by Theorem
1.4.4, I − K has closed range.
Corollary 1.4.6. If K ∈ B(X) is compact then I − K is Fredholm.
Proof. From Theorem 1.4.5 we see that (I − K)(X) is closed. Since x ∈ (I − K) −1 (0)
implies x = Kx, the identity operator acts as a compact operator on (I − K) −1 (0);
thus α(I − K) < ∞. To prove that β(I − K) < ∞, recall that K ∗ : X ∗ → X ∗ is also
compact. Since (I − K)(X) is closed it follows from Theorem 1.2.7 that
β(I − K) = α(I − K ∗ ) < ∞.
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