Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
Example 2.4.6. There exists an operator T ∈ B(X) and a finite rank operator
F ∈ B(X) commuting with T such that Weyl’s theorem holds for T but it does not
hold for T + F .
Proof. Define on l 2 ⊕ l 2 , T := I ⊕ S and F = K ⊕ 0, where S : l 2 → l 2 is an injective
quasinilpotent operator and F : l 2 → l 2 is defined by
F (x 1 , x 2 , x 3 , · · · ) = (−x 1 , 0, 0, · · · ).
Then F is of finite rank and commutes with T . It is easy to see that
σ(T ) = ω(T ) = {0, 1} and π 00 (T ) = ∅,
which implies that Weyl’s theorem holds for T . We however have
σ(T + F ) = ω(T + F ) = {0, 1} and π 00 (T + F ) = {0},
which implies that Weyl’s theorem fails for T + F .
Theorem 2.4.5 may fail if “finite rank” is replaced by “compact”. In fact Weyl’s
theorem may fail even if K is both compact and quasinilpotent: for example, take
T = 0 and K the operator on l 2 defined by K(x 1 , x 2 , · · · ) = ( x 2
2
, x 3
3
, x 4
4
, · · · ). We will
however show that if “isoloid” condition is strengthened slightly then Weyl’s theorem
is transmitted from T to T + K if K is either a compact or a quasinilpotent operator
commuting with T . To see this we observe:
Lemma 2.4.7. If K ∈ B(X) is a compact operator commuting with T ∈ B(X) then
Proof. See [HanL2].
π 00 (T + K) ⊆ iso σ(T ) ∪ ρ(T ).
An operator T ∈ B(X) will be said to be finite-isoloid if iso σ(T ) ⊆ π 0f (T ).
Evidently finite-isoloid ⇒ isoloid. The converse is not true in general: for example,
take T = 0. In particular if σ(T ) has no isolated points then T is finite-isoloid. We
now have:
Theorem 2.4.8. Suppose T ∈ B(X) is finite-isoloid. If Weyl’s theorem holds for T
then Weyl’s theorem holds for T + K if K ∈ B(X) commutes with T and is either
compact or quasinilpotent.
Proof. First we assume that K is a compact operator commuting with T . Suppose
Weyl’s theorem holds for T . We first claim that with no restriction on T ,
σ(T + K) \ ω(T + K) ⊆ π 00 (T + K). (2.41)
For (2.41), it suffices to show that if λ ∈ σ(T + K) \ ω(T + K) then λ ∈ iso σ(T + K).
Assume to the contrary that λ ∈ acc σ(T + K). Then we have that λ ∈ σ b (T + K) =
σ b (T ), so that λ ∈ σ e (T ) or λ ∈ acc σ(T ). Remember that the essential spectrum and
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