Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
Proof. The spectral mapping theorem for the Weyl spectrum may be rewritten as
implication, for arbitrary n ∈ N and λ ∈ C n ,
(T − λ 1 I)(T − λ 2 I) · · · (T − λ n I) Weyl =⇒ T − λ j I Weyl for each j = 1, 2, · · · , n.
(2.30)
Now if index(T − zI) ≥ 0 on C \ σ e (T ) then we have
n∑
n∏
index(T −λ j I) = index (T −λ j I) = 0 =⇒ index(T −λ j I) = 0 (j = 1, 2, · · · , n),
j=1
j=1
and similarly if index (T − zI) ≤ 0 off σ e (T ). If conversely there exist λ, µ for which
then
index(T − λI) = −m < 0 < k = index(T − µI) (2.31)
(T − λI) k (T − µI) m (2.32)
is a Weyl operator whose factors are not Weyl. This together with Theorem 2.3.9
proves the equivalence of the conditions (2.28) and (2.29).
Corollary 2.3.11. If X is a Hilbert space and T ∈ B(X) is hyponormal then
f(ω(T )) = ω(f(T )) for every f ∈ H(σ(T )). (2.33)
Proof. Immediate from Theorem 2.3.10 together with the fact that if T is hyponormal
then index (T − λI) ≤ 0 for every λ ∈ C \ σ e (T ).
Corollary 2.3.12. Let T ∈ B(X). If
(i) Weyl’s theorem holds for T ;
(ii) T is isoloid;
(iii) T satisfies the spectral mapping theorem for the Weyl spectrum,
then Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )).
Proof. A slight modification of the proof of Lemma 2.3.6 shows that if T is isoloid
then
f ( σ(T ) \ π 00 (T ) ) = σ(f(T )) \ π 00 (f(T )) for every f ∈ H(σ(T )).
It thus follows from Theorem 1.7.8 and Corollary 2.3.11 that
σ(f(T )) \ π 00 (f(T )) = f ( σ(T ) \ π 00 (T ) ) = f(ω(T )) = ω(f(T )),
which implies that Weyl’s theorem holds for f(T ).
Corollary 2.3.13. If T ∈ B(X) has the SVEP then
ω(f(T )) = f(ω(T )) for every f ∈ H(σ(T )).
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