Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
2.4 Perturbation Theorems
In this section we consider how Weyl’s theorem survives under “small” perturbations.
Weyl’s theorem is transmitted from T ∈ B(X) to T − K for commuting nilpotents
K ∈ B(X) To see this we need:
Lemma 2.4.1. If T ∈ B(X) and if N is a quasinilpotent operator commuting with
T then ω(T + N) = ω(T ).
Proof. It suffices to show that if 0 /∈ ω(T ) then 0 /∈ ω(T + N). Let 0 /∈ ω(T ) so that
0 /∈ σ(π(T )). For all λ ∈ C we have σ(π(T +λN)) = σ(π(T )). Thus 0 /∈ σ(π(T +λN))
for all λ ∈ C, which implies T + λN is a Fredholm operator forall λ ∈ C. But since
T is Weyl, it follows that T + N is also Weyl, that is, 0 /∈ ω(T + N).
Theorem 2.4.2. Let T ∈ B(X) and let N be a nilpotent operator commuting with
T . If Weyl’s theorem holds for T then it holds for T + N.
Proof. We first claim that
π 00 (T + N) = π 00 (T ). (2.34)
Let 0 ∈ π 00 (T ) so that ker (T ) is finite dimensional. Let (T +N)x = 0 for some x ≠ 0.
Then T x = −Nx. Since T commutes with N it follows that
T m x = (−1) m N m x for every m ∈ N. (2.35)
Let n be the nilpotency of N, i.e., n be the smallest positive integer such that N n = 0.
Then by (2.35) we have that for some r with 1 ≤ r ≤ n, T r x = 0 and then T r−1 x ∈
N(T ). Thus N(T + N) ⊂ N(T n−1 ). Therefore N(T + N) is finite dimensional. Also
if for some x (≠ 0) T x = 0 then (T + N) n x = 0, and hence 0 is an eigenvalue of
T + N. Again since σ(T + N) = σ(T ) it follows that 0 ∈ π 00 (T + N). By symmetry
0 ∈ π 00 (T + N) implies 0 ∈ π 00 (T ), which proves (2.34). Thus we have
ω(T + N) = ω(T ) (by Lemma 2.4.1)
= σ(T ) \ π 00 (T ) (since Weyl’s theorem holds for T )
= σ(T + N) \ π 00 (T + N),
which shows that Weyl’s theorem holds for T + N.
Theorem 2.4.2 however does not extend to quasinilpotents: let
Q : (x 1 , x 2 , x 3 , · · · ) ↦→ ( 1 2 x 2, 1 3 x 3, 1 4 x 4, · · · ) on l 2
and set on l 2 ⊕ l 2 ,
T =
[ ] 1 0
0 0
and K =
[ ] 0 0
. (2.36)
0 Q
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