Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
Theorem 1.2.5. Let X and Y be Banach spaces and T ∈ B(X, Y ). Then the followings
are equivalent:
(a) T has closed range;
(b) T ∗ has closed range;
(c) T ∗ (Y ∗ ) = T −1 (0) ⊥ ;
(d) T (X) = . ⊥ (T ∗−1 (0)).
Proof. (a) ⇔ (d): From Theorem 1.2.4 (b).
(a) ⇒ (c): Observe that the operator T ∧ : X/T −1 (0) → T X defined by
x + T −1 (0) ↦→ T x
is invertible by the Open Mapping Theorem. Thus we have
T −1 (0) ⊥ ∼ =
(
X/T −1 (0) )∗ ∼ = (T X)
∗ ∼ = T ∗ (Y ∗ ).
(c) ⇒ (b): This is clear because T −1 (0) ⊥ is closed.
(b) ⇒ (a): Observe that if T 1 : X → cl (T X) then T ∗ 1 : (cl T X) ∗ → X ∗ is one-one.
Since T ∗ (Y ∗ ) = ran T ∗ 1 , T ∗ 1 has closed range. Therefore T ∗ 1 is bounded below, so that
T 1 is open; therefore T X is closed.
Definition 1.2.6. If T ∈ B(X, Y ), write
α(T ) := dim T −1 (0) and β(T ) = dim Y/cl (T X).
Theorem 1.2.7. If T ∈ B(X, Y ) has a closed range then
α(T ∗ ) = β(T ) and α(T ) = β(T ∗ ).
Proof. This follows form the following observation:
T ∗−1 (0) = (T X) ⊥ ∼ = (Y/T X)
∗ ∼ = Y/T X
and
T −1 (0) ∼ = (T −1 (0)) ∗ ∼ = X ∗ /T −1 (0) ⊥ ∼ = X ∗ /T ∗ (Y ∗ ).
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