Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
Proof. By Lemma 2.3.6, p(σ(T ) \ π 00 (T )) = σ(p(T )) \ π 00 (p(T )). If Weyl’s theorem
holds for T then ω(T ) = σ(T ) \ π 00 (T ), so that
p(ω(T )) = p(σ(T ) \ π 00 (T )) = σ(p(T )) \ π 00 (p(T )).
The result follows at once from this relationship.
Example 2.3.8. Theorem 2.3.7 may fail if T is not isoloid. To see this define T 1 and
T 2 on l 2 by
T 1 (x 1 , x 2 , · · · ) = (x 1 , 0, x 2 /2, x 3 /2, · · · )
and
T 2 (x 1 , x 2 , · · · ) = (0, x 1 /2, x 2 /3, x 3 /4, · · · ).
Let T := T 1 ⊕ (T 2 − I) on X = l 2 ⊕ l 2 . Then
and
σ(T ) = {1} ∪ {z : |z| ≤ 1/2} ∪ {−1}, π 00 (T ) = {1},
ω(T ) = {z : |z| ≤ 1/2} ∪ {−1},
which shows that Weyl’s theorem holds for T . Let p(t) = t 2 . Then
and
σ(p(T )) = {z : |z| ≤ 1/4} ∪ {1}, π 00 (p(T )) = {1}
ω(p(T )) = {z : |z| ≤ 1/4} ∪ {1}.
Thus 1 ∈ p(σ(T ) \ π 00 (T )), but 1 /∈ σ(p(T )) \ π 00 (p(T )). Also ω(p(T )) = p(ω(T )) but
Weyl’s theorem does not hold for p(T ).
Theorem 2.3.9. If p(ω(T )) = ω(p(T )) for every polynomial p, then f(ω(T )) =
ω(f(T )) for every f ∈ H(σ(T )).
Proof. Let (p n (T )) be a sequence of polynomials converging uniformly in a neighborhood
of σ(T ) to f(t) so that p n (T ) → f(T ). Since f(T ) commutes with each p n (T ),
it follows from Theorem 2.3.3 that
ω(f(T )) = lim ω(p n (T )) = lim p n (ω(T )) = f(ω(T )).
Theorem 2.3.10. If T ∈ B(X) then the following are equivalent:
index(T − λI) index(T − µI) ≥ 0 for each pair λ, µ ∈ C \ σ e (T ); (2.28)
f(ω(T )) = ω(f(T )) for every f ∈ H(σ(T )). (2.29)
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