Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
Proof. Newburgh’s theorem is stated as follows: if in a Banach algebra A, {a i } i is a
sequence of elements commuting with a ∈ A and such that a i → a, then lim σ(a i ) =
σ(a). If π denotes the canonical homomorphism of B(X) onto the Calkin algebra
B(X)/K(X), then the assumptions give that π(T n ) → π(T ) and [π(T n ), π(T )] = 0
for each n. Hence, lim σ(π(T n )) = σ(π(T )); that is, lim σ e (T n ) = σ e (T ).
Theorem 2.3.3. Suppose that T, T n ∈ B(X), for n ∈ Z + , are such that T n converges
to T . If [T n , T ] ∈ K(X) for each n, then
lim ω(f(T n )) = ω(f(T )) for every f ∈ H(σ(T )). (2.24)
Remark. Because T n → T , by the upper-semicontinuity of the spectrum, there is
a positive integer N such that σ(T n ) ⊆ V whenever n > N. Thus, in the left-hand
side of (2.24) it is to be understood that n > N.
Proof. If T n and T commute modulo the compact operators then, whenever Tn
−1 and
T −1 exist, T n , T, Tn
−1 and T −1 all commute modulo the compact operators. Therefore
r(T n ) and r(T ) also commute modulo K(X) whenever r is a rational function with no
poles in σ(T ) and n is sufficiently large. Using Runge’s theorem we can approximate
f on compact subsets of V by rational functions r who poles lie off of V . So there
exists a sequence of rational functions r i whose poles lie outside of V and r i → f
uniformly on compact subsets of V . If n > N, then by the functional calculus,
f(T n )f(T ) − f(T )f(T n ) = lim
i
(
ri (T n )r i (T ) − r i (T )r i (T n ) ) ,
which is compact for each n. Furthermore,
∫
1
||f(T n ) − f(T )|| = || f(λ) ( (λ − T n ) −1 − (λ − T ) −1) dλ||
2πi Γ
≤ 1 l(Γ) max |f(λ)| · max
2πi ||(λ − T n) −1 − (λ − T ) −1 || ,
λ∈Γ λ∈Γ
where Γ is the boundary of V and l(Γ) denotes the arc length of Γ. The righthand
side of the above inequality converges to 0, and so f(T n ) → f(T ). By Lemma
2.25, lim σ e (f(T n )) = σ e (f(T )). The arguments used by J.B. Conway and B.B.
Morrel in Proposition 3.11 of [CoM] can now be used here to obtain the conclusion
lim ω(f(T n )) = ω(f(T )).
In general there is only inclusion for the Weyl spectrum:
Theorem 2.3.4. If T ∈ B(X) then
ω(p(T )) ⊆ p(ω(T )) for every polynomial p.
Proof. We can suppose p is nonconstant. Suppose λ /∈ pω(T ). Writing p(µ) − λ =
a(µ − µ 1 )(µ − µ 2 ) · · · (µ − µ n ), we have
p(T ) − λI = a(T − µ 1 I) · · · (T − µ n I). (2.25)
For each i, p(µ i ) = λ /∈ pω(T ), so that µ i /∈ ω(T ), i.e., T − µ i I is weyl. It thus follows
from (2.25) that p(T ) − λI is Weyl since the product of Weyl operators is Weyl.
54