Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
Corollary 2.5.5. A commuting n-tuple of normal operators satisfies Weyl’s theorem
(I) and hence Weyl’s theorem (II).
Proof. Immediate from (2.51) and 2.5.4.
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Corollary 2.5.6. (Riesz-Schauder theorem in several variables) Let T = (T 1 , · · · , T n )
be a doubly commuting n-tuple of hyponormal operators. If T has the quasitriangular
property then
ω(T ) = σ Tb (T ).
Proof. In view of refthm5.63, we need to show that σ Tb (T ) ⊂ ω(T ). Indeed if
λ ∈ σ T (T ) \ ω(T ) then by (2.53), λ ∈ iso σ T (T ), and hence T − λ is Taylor-Browder.
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