Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
1.11 Spectral Mapping Theorems
Recall the Calkin algebra B(X)/K(X). The Calkin homomorphism π is defined by
π : B(X) −→ B(X)/K(X)
π(T ) = T + K(X).
Evidently, by the Atkinson’s Theorem,
T is Fredholm ⇐⇒ π(T ) is invertible.
Theorem 1.11.1. If T ∈ B(X) and f is analytic in a neighborhood of σ(T ), then
f(σ e (T )) = σ e (f(T ))
Proof. Since f(π(T )) = f(T + K(X)) = f(T ) + K(X) = π(f(T )) it follows that
f(σ e (T )) = f(σ(π(T ))) = σ(f(π(T ))) = σ(π(f(T ))) = σ e (f(T )).
Theorem 1.11.2. If T ∈ B(X) and f is analytic in a neighborhood of σ(T ), then
f(σ b (T )) = σ b (f(T ))
Proof. Since by the analyticity of f, f(acc K) = acc f(K), it follows that
f(σ b (T )) = f(σ e (T ) ∪ acc σ(T ))
= f(σ e (T )) ∪ f(acc σ(T ))
= σ e (f(T )) ∪ acc σ(f(T ))
= σ b (f(T )).
Theorem 1.11.3. If T ∈ B(X) and p is a polynomial then
ω(p(T )) ⊆ p(ω(T )).
Proof. Let p(z) = a 0 + a 1 z + · · · + a n z n ; thus p(z) = c 0 (z − α 1 ) · · · (z − α n ). Then
p(T ) = c 0 (T − α 1 I) · · · (T − α n I).
We now claim that
0 /∈ p(ω(T )) =⇒ c 0 (z − α 1 ) · · · (z − α n ) ≠ 0 for each λ ∈ ω(T )
=⇒ λ ≠ α i for each λ ∈ ω(T )
=⇒ T − α i I is Weyl for each i = 1, 2, . . . n
=⇒ c 0 (T − α 1 I) · · · (T − α n I) is Weyl
=⇒ 0 /∈ ω(p(T ))
33