Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
it follows
Let e 0 ∈ ker T φ and ||e 0 || = 1. Write
ker T φ = {0} and dim ker T φ = 1.
e n+1 :=
T φe n
||T φ e n || .
We claim that Ke n = α 2n+2 e n : indeed, Ke 0 = α 2 (1 − T φ T φ )e 0 = α 2 e 0 and if we
assume Ke j = α 2j+2 e j then
Ke j+1 = ||T φ e j || −1 (KT φ e j ) = ||T φ e j || −1 (α 2 T φ Ke j ) = ||T φ e j || −1 (α 2j+4 T φ e j ) = α 2j+4 e j+1 .
Thus we can see that
{
α 2 , α 4 , α 6 , · · · are eigenvalues of K ;
{e n } ∞ n=0 is an orthonormal set since K is self-adjoint.
We will then prove that {e n } forms an orthonormal basis for H 2 . Observe
Thus
tr(H ∗ φH φ ) = the sum of its eigenvalues.
∞∑
α 2n+2 ≤ tr(HφH ∗ φ ) = ||H φ || 2 2
n=0
Since ψ ∈ H ∞ , we have
(|| · || 2 denotes the Hilbert-Schmidt norm).
(5.31)
||H φ || 2 2 = ||H ψ + αH ψ
|| 2 2 = α 2 ||H ψ
|| 2 2 = α 2 tr(H ∗ ψ H ψ ) = α2 tr [T ψ
, T ψ ]
which together with (5.31) implies that
≤ α2
π µ(σ(T ψ)) = α2
π µ(ψ(D)) =
α2
1 − α 2 ,
∑
α 2n+2 ≤ ||H φ || 2 2 ≤
α2
∞ 1 − α 2 = ∑
α 2n+2 ,
so tr(H ∗ φH φ ) = ∑ ∞
n=0 α2n+2 , which say that {α 2n+2 } ∞ n=0 is a complete set of nonzero
eigenvalues for K ≡ H ∗ φH φ and each has multiplicity one. Now, by Beurling’s
theorem,
n=0
kerK = kerH ∗ φH φ = kerH φ = bH 2 , where b is inner or b = 0.
Since KT φ = α 2 T φ K, we see that
f ∈ kerK ⇒ T φ f ∈ kerK
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