Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
1.7 The Calkin Algebra
We begin with:
Theorem 1.7.1. If T ∈ B(X, Y ) then
T is Fredholm ⇐⇒ ∃ S ∈ B(Y, X) such that I − ST and I − T S are finite rank.
Proof. (⇒) Suppose T Fredholm and let
X = T −1 (0) ⊕ X 0 and Y = T (X) ⊕ Y 0 .
Define T 0 := T | X0 . Since T 0 is one-one and T 0 (X 0 ) = T (X) is closed
T −1
0 : T (X) → X 0 is invertible.
Put S := T −1
0 Q, where Q : Y → T (X) is a projection. Evidently, S(Y ) = X 0 and
S −1 (0) = Y 0 . Furthermore,
I − ST is the projection of X onto T −1 (0)
I − T S is the projection of Y onto Y 0 .
In particular, I − ST and I − T S are of finite rank.
(⇐) Assume ST = I − K 1 and T S = I − K 2 , where K 1 , K 2 are finite rank. Since
T −1 (0) ⊂ (ST ) −1 (0) and (T S)X ⊂ T (X),
we have
α(T ) ≤ α(ST ) = α(I − K 1 ) < ∞
β(T ) ≤ β(T S) = β(I − K 2 ) < ∞,
which implies that T is Fredholm.
Theorem 1.7.1 remains true if the statement “I − ST and I − T S are of finite
rank” is replaced by “I − ST and I − T S are compact operators.” In other words,
T is Fredholm ⇐⇒ T is invertible modulo compact operators.
Let K(X) be the space of all compact operators on X. Note that K(X) is a closed
ideal of B(X). On the quotient space B(X)/K(X), define the product
[S][T ] = [ST ],
where [S] is the coset S + K(X).
The space B(X)/K(X) with this additional operation is an algebra, which is called
the Calkin algebra, with identity [I].
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