Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
1.3 Definitions and Examples
In the sequel X and Y denote complex Banach spaces.
Definition 1.3.1. An operator T ∈ B(X, Y ) is called a Fredholm operator if T X
is closed, α(T ) < ∞ and β(T ) < ∞. In this case we define the index of T by the
equality
index (T ) := α(T ) − β(T ).
In the below we shall see that the condition “ T X is closed ” is automatically fulfilled
if β(T ) < ∞.
Example 1.3.2. If X and Y are both finite dimensional then any operator T ∈
B(X, Y ) is Fredholm and
indeed recall the “rank theorem”
which implies
index(T ) = dim X − dim Y :
dim X = dim T −1 (0) + dim T X,
index(T ) = dim T −1 (0) − dim Y/T X
= dim X − dim T X − (dim Y − dim T X)
= dim X − dim Y.
Thus in particular, if T ∈ B(X) with dim X < ∞ then T is Fredholm of index zero.
Example 1.3.3. If K ∈ B(X) is a compact operator then T = I − K is Fredholm of
index 0. This follows from the Fredholm theory for compact operators.
Example 1.3.4. If U is the unilateral shift operator on l 2 , then
index U = −1 and index U ∗ = −1.
With U and U ∗ , we can build a Fredholm operator whose index is equal to an arbitrary
prescribed integer. Indeed if
[ ] U
p
0
T =
0 U ∗q : l 2 ⊕ l 2 → l 2 ⊕ l 2 ,
then T is Fredholm, α(T ) = q, β(T ) = p, and hence index T = q − p.
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