Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
5.1.3 Toeplitz Operators
Let P be the orthogonal projection of L 2 (T) onto H 2 (T). For φ ∈ L ∞ (T), the Toeplitz
operator T φ with symbol φ is defined by
T φ f = P (φf) for f ∈ H 2 .
Remember that {z n : n = 0, 1, 2, · · · } is an orthonormal basis for H 2 . Thus if
φ ∈ L ∞ has the Fourier coefficients
̂φ(n) = 1
2π
∫ 2π
0
φz n dt,
then the matrix (a ij ) for T φ with respect to the basis {z n : n = 0, 1, 2, · · · } is given
by:
a ij = (T φ z j , z i ) = 1 ∫ 2π
φz i−j dt = ̂φ(i − j).
2π
Thus the matrix for T φ is constant on diagonals:
⎡
⎤
c 0 c −1 c −2 c −3 · · ·
c 1 c 0 c −1 c −2 · · ·
(a ij ) =
c 2 c 1 c 0 c −1 · · ·
, where c
⎢
⎣
c 3 c 2 c 1 c 0 · · · ⎥
j = ̂φ(j) :
⎦
.
. .. . .. . .. . ..
Such a matrix is called a Toeplitz matrix.
Lemma 5.1.10. Let A ∈ B(H 2 ). The matrix A relative to the orthonormal basis
{z n : n = 0, 1, 2, · · · } is a Toeplitz matrix if and only if
U ∗ AU = A, where U is the unilateral shift.
Proof. The hypothesis on the matrix entries a ij = ⟨Az j , z i ⟩ of A if and only if
Noting Uz n = z n+1 for n ≥ 0, we get
0
a i+1,j+1 = a ij (i, j = 0, 1, 2, · · · ). (5.1)
(5.1) ⇐⇒ ⟨U ∗ AUz j , z i ⟩ = ⟨AUz j , Uz i ⟩ = ⟨Az j+1 , z i+1 ⟩ = ⟨Az j , z i ⟩, ∀ i, j
⇐⇒ U ∗ AU = A.
Remark. AU = UA ⇔ A is an analytic Toeplitz operator (i.e., A = T φ with φ ∈ H ∞ ).
Consider the mapping ξ : L ∞
→ B(H 2 ) defined by ξ(φ) = T φ . We have:
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