Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
Theorem 2.2.7. [Co] Weyl’s theorem holds for every Toeplitz operator T φ .
Proof. It was known [Wi2] that σ(T φ ) is always connected. Since there are no
quasinilpotent Toeplitz operators except 0, σ(T φ ) can have no isolated eigenvalues
of finite multiplicity. Thus Weyl’s theorem is equivalent to the fact that
σ(T φ ) = ω(T φ ). (2.13)
Since T φ − λI = T φ−λ , it suffices to show that if T φ is Weyl then T φ is invertible.
If T φ is not invertible, but is Weyl then it is easy to see that both T φ and Tφ ∗ = T φ
must have nontrivial kernels. Thus we want to show that this can not happen, unless
φ = 0 and hence T φ is the non-Weyl operator.
Suppose that there exist nonzero functions φ, f, and g (φ ∈ L ∞ and f, g ∈ H 2 )
such that T φ f = 0 and T φ g = 0. Then P (φf) = 0 and P (φg) = 0, so that there exist
functions h, k ∈ H 2 such that
∫ ∫
h dθ = k dθ = 0 and φf = h, φg = k.
Thus by the F. and M. Riesz’s theorem, φ, f, g, h, k are all nonzero except on a set
of measure zero. We thus have that f/g = h/k pointwise a.e., so that fk = gh a.e.,
which implies gh = 0 a.e. Again by the F. and M. Riesz’s theorem, we can conclude
that either g = 0 a.e. or h = 0 a.e. This contradiction completes the proof.
We review here a few essential facts concerning Toeplitz operators with continuous
symbols, using [Do1] as a general reference. The sets C(T) of all continuous complexvalued
functions on the unit circle T and H ∞ (T) = L ∞ ∩H 2 are Banach algebras, and
it is well-known that every Toeplitz operator with symbol φ ∈ H ∞ is subnormal. The
C ∗ -algebra A generated by all Toeplitz operators T φ with φ ∈ C(T) has an important
property which is very useful for spectral theory: the commutator ideal of A is the
ideal K(H 2 ) of compact operators on H 2 . As C(T) and A/K(H 2 ) are ∗-isomorphic
C ∗ -algebras, then for every φ ∈ C(T),
T φ is a Fredholm operator if and only if φ is invertible (2.14)
index T φ = −wn(φ) , (2.15)
σ e (T φ ) = φ(T) , (2.16)
where wn(φ) denotes the winding number of φ with respect to the origin. Finally,
we make note that if φ ∈ C(T) and if f is an analytic function defined on an open
set containing σ(T φ ), then f ◦ φ ∈ C(T) and f(T φ ) is well-defined by the analytic
functional calculus.
We require the use of certain closed subspaces and subalgebras of L ∞ (T), which
are described in further detail in [Do2] and Appendix 4 of [Ni]. Recall that the
subspace H ∞ (T) + C(T) is a closed subalgebra of L ∞ . The elements of the closed
selfadjoint subalgebra QC, which is defined to be
QC = ( H ∞ (T) + C(T) ) ∩ ( H ∞ (T) + C(T) ) ,
45