Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
2.3 Spectral Mapping Theorem for the Weyl spectrum
Let S denote the set, equipped with the Hausdorff metric, of all compact subsets
of C. If A is a unital Banach algebra then the spectrum can be viewed as a function
σ : A → S, mapping each T ∈ A to its spectrum σ(T ). It is well-known that
the function σ is upper semicontinuous, i.e., if T n → T then lim sup σ(T n ) ⊂ σ(T )
and that in noncommutative algebras, σ does have points of discontinuity. The work
of J. Newburgh [Ne] contains the fundamental results on spectral continuity in general
Banach algebras. J. Conway and B. Morrel [CoM] have undertaken a detailed
study of spectral continuity in the case where the Banach algebra is the C ∗ -algebra
of all operators acting on a complex separable Hilbert space. Of interest is the identification
of points of spectral continuity, and of classes C of operators for which σ
becomes continuous when restricted to C. In [BGS], the continuity of the spectrum
was considered when restricted to certain subsets of the entire manifold of Toeplitz
operators. The set of normal operators is perhaps the most immediate in the latter
direction: σ is continuous on the set of normal operators. As noted in Solution 104 of
[Ha3], Newburgh’s argument uses the fact that the inverses of normal resolvents are
normaloid. This argument can be easily extended to the set of hyponormal operators
because the inverses of hyponormal resolvents are also hyponormal and hence normaloid.
Although p-hyponormal operators are normaloid, it was shown [HwL1] that
σ is continuous on the set of all p-hyponormal operators.
We now examine the continuity of the Weyl spectrum in pace of the spectrum.
In general the Weyl spectrum is not continuous: indeed, it was in [BGS] that the
spectrum is discontinuous on the entire manifold of Toeplitz operators. Since the
spectra and the Weyl spectra coincide for Toeplitz operators, it follows at once that
the weyl spectrum is discontinuous.
However the Weyl spectrum is upper semicontinuous.
Lemma 2.3.1. The map T → ω(T ) is upper semicontinuous.
Proof. Let λ ∈ ω(T ). Since the set of Weyl operators forms an open set, there exists
δ > 0 such that if S ∈ B(X) and ||T − λI − S|| < δ then S is Weyl. So there exists
an integer N such that ||T − λI − (T n − λI)|| < δ 2
for n ≥ N. Let V be an open
(δ/2)-neighborhhod of λ. We have, for µ ∈ V and n ≥ N,
||T − λI − (T n − µI)|| < δ,
so that T n − µI is Weyl. This shows that λ /∈ lim sup ω(T n ). Thus lim sup ω(T n ) ⊂
ω(T ).
Lemma 2.3.2. [Ne, Theorem 4] If {T n } n is a sequence of operators converging to an
operator T and such that [T n , T ] is compact for each n, then lim σ e (T n ) = σ e (T ).
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