Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
Proof. If λ /∈ σ e (T ) then by Lemma 2.2.12, T − λI has a finite ascent. Since if
S ∈ B(X) is Fredholm of finite ascent then index (S) ≤ 0: indeed, either if S has
finite descent then S is Browder and hence index (S) = 0, or if S does not have finite
descent then
n index (S) = dim N(S n ) − dim R(S n ) ⊥ → −∞ as n → ∞,
which implies that index (S) < 0. Thus we have that index (T − λI) ≤ 0. Thus T
satisfies the condition (2.28), which gives the result.
Theorem 2.3.14. If T ∈ B(X) satisfies
X T ({λ}) = N(T − λI) for every λ ∈ C,
then Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )).
Proof. By Corollary 2.2.18, Weyl’s theorem holds for T , T is isoloid, and T has the
SVEP. In particular by Corollary 2.3.13, T satisfies the spectral mapping theorem for
the Weyl spectrum. Thus the result follows from Corollary 2.3.12.
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