Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
Suppose α(λ ′ ) ≠ n 1 . Since C is connected there exists an arc Γ lying in C with
endpoints λ 0 and λ ′ . It follows from Theorem 1.8.2 and the fact that C is open that
for each µ ∈ Γ, there exists an open ball S(µ) in C such that
α(λ) is constant on the set S(µ) with the point µ deleted.
Since Γ is compact and connected there exist points λ 1 , λ 2 , · · · , λ n = λ ′ on Γ such
that
S(λ 0 ), S(λ 1 ), . . . , S(λ n ) cover Γ and S(λ i ) ∩ S(λ i+1 ) ≠ ∅ (0 ≤ i ≤ n − 1) (1.14)
We claim that α(λ) = α(λ 0 ) on all of S(λ 0 ). Indeed it follows from the Lemma 1.8.1
that
α(λ) ≤ α(λ 0 ) for λ sufficiently close to λ 0 .
Therefore, since α(λ 0 ) is the minimum of α(λ) on C,
α(λ) = α(λ 0 ) for λ sufficiently close to λ 0 .
Since α(λ) is constant for all λ ≠ λ 0 in S(λ 0 ), which is α(λ 0 ). Now α(λ) is constant
on the set S(λ i ) with the point λ i deleted (1 ≤ i ≤ n). Hence it follows from (1.14)
and the observation α(λ) = α(λ 0 ) for all λ ∈ S(λ 0 ) that α(λ) = α(λ 0 ) for all λ ≠ λ ′
in S(λ ′ ) and α(λ ′ ) > n 1 . The result just obtained can be applied to the adjoint. This
completes the proof.
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