Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
Theorem 1.7.8. If T ∈ B(X, Y ) is Fredholm with generalized inverse T ′ ∈ B(Y, X)
in the strong sense then
Proof. Observe that
which gives that β(T ) = α(T ′ ).
index (T ) = dim T −1 (0) − dim (T ′ ) −1 (0).
(T ′ ) −1 (0) = (T T ′ ) −1 (0) ∼ = X/T T ′ (X) ∼ = X/T (X),
Theorem 1.7.9. If T ∈ B(X, Y ) is Fredholm with generalized inverse T ′ ∈ B(Y, X),
then
index (T ) = trace (T T ′ − T ′ T ).
Proof. If T = T T ′ T is Fredholm then
Observe that
I − T ′ T and I − T T ′ are both finite rank.
dim (I − T ′ T )(X) = dim (T ′ T ) −1 (0) = dim T −1 (0) = α(T );
dim (I − T T ′ )(Y ) = dim (T T ′ ) −1 (0) = dim X/T T ′ (Y ) = dim X/T (X) = β(T ).
Thus we have
trace (T T ′ − T ′ T ) = trace ( (I − T ′ T ) − (I − T T ′ ) )
= trace (I − T ′ T ) − trace (I − T T ′ )
= rank (I − T ′ T )(X) − dim (I − T T ′ )(X)
= α(T ) − β(T )
= index (T ).
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