Woo Young Lee Lecture Notes on Operator Theory

Chapter 1
Fredholm Theory
1.1 Introduction
If k(x, y) is a continuous complex-valued function on [a, b] × [a, b] then K : C[a, b] →
C[a, b] defined by
(Kf)(x) =
∫ b
a
k(x, y)f(y)dy
is a compact operator. The classical Fredholm integral equations is
λf(x) −
∫ b
a
k(x, y)f(y)dy = g(x), a ≤ x ≤ b,
where g ∈ C[a, b], λ is a parameter and f is the unknown. Using I to be the identity
operator on C[a, b], we can recast this equation into the form (λI − K)f = g. Thus
we are naturally led to study of operators of the form T = λI − K on any Banach
space X. Riesz-Schauder theory concentrates attention on these operators of the
form T = λI − K, λ ≠ 0, K compact. The Fredholm theory concentrates attention
on operators called Fredholm operators, whose special cases are the operators λI −
K. After we develop the “Fredholm Theory”, we see the following result. Suppose
k(x, y) ∈ C[a, b] × C[a, b] (or L 2 [a, b] × L 2 [a, b]). The equation
λf(x) −
∫ b
a
k(x, y)f(y)dy = g(x), λ ≠ 0 (1.1)
has a unique solution in C[a, b] for each g ∈ C[a, b] if and only if the homogeneous
equation
λf(x) −
∫ b
a
k(x, y)f(y)dy = 0, λ ≠ 0 (1.2)
has only the trivial solution in C[a, b]. Except for a countable set of λ, which has
zero as the only possible limit point, equation (1.1) has a unique solution for every
g ∈ C[a, b]. For λ ≠ 0, the equation (1.2) has at most a finite number of linear
independent solutions.
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