Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 5.
TOEPLITZ THEORY
5.2 Hyponormality of Toeplitz operators
An elegant and useful theorem of C. Cowen [Cow3] characterizes the hyponormality
of a Toeplitz operator T φ on the Hardy space H 2 (T) of the unit circle T ⊂ C by
properties of the symbol φ ∈ L ∞ (T). This result makes it possible to answer an
algebraic question coming from operator theory – namely, is T φ hyponormal - by
studying the function φ itself. Normal Toeplitz operators were characterized by a
property of their symbol in the early 1960’s by A. Brown and P.R. Halmos [BH], and
so it is somewhat of a surprise that 25 years passed before the exact nature of the
relationship between the symbol φ ∈ L ∞ and the positivity of the selfcommutator
[Tφ, ∗ T φ ] was understood (via Cowen’s theorem). As Cowen notes in his survey paper
[Cow2], the intensive study of subnormal Toeplitz operators in the 1970’s and early
80’s is one explanation for the relatively late appearance of the sequel to the Brown-
Halmos work. The characterization of hyponormality via Cowen’s theorem requires
one to solve a certain functional equation in the unit ball of H ∞ . However the
case of arbitrary trigonometric polynomials φ, though solved in principle by Cowen’s
theorem, is in practice very complicated. Indeed it may not even be possible to find
tractable necessary and sufficient conditions for the hyponormality of T φ in terms of
the Fourier coefficients of φ unless certain assumptions are made about φ. In this
chapter we present some recent development in this research.
5.2.1 Cowen’s Theorem
In this section we present Cowen’s theorem. Cowen’s method is to recast the operatortheoretic
problem of hyponormality of Toeplitz operators into the problem of finding
a solution of a certain functional equation involving its symbol. This approach has
been put to use in the works [CLL, CuL1, CuL2, CuL3, FL1, FL2, Gu1, HKL1, HKL2,
HwL3, KL, NaT, Zh] to study Toeplitz operators.
We begin with:
Lemma 5.2.1. A necessary and sufficient condition that two Toeplitz operators commute
is that either both be analytic or both be co-analytic or one be a linear function
of the other.
Proof. Let φ = ∑ i α iz i and ψ = ∑ j β jz j . Then a straightforward calculation shows
that
T φ T ψ = T ψ T φ ⇐⇒ α i+1 β −j−1 = β i+1 α −j−1 (i, j ≥ 0).
Thus either α −j−1 = β −j−1 = 0 for j ≥ 0, i.e., φ and ψ are both analytic, or
α i+1 = β i+1 = 0 for i ≥ 0, i.e., φ and ψ are both co-analytic, or there exist i 0 , j 0
such that α i0 +1 ≠ 0 and α −j0 −1 ≠ 0. So for the last case, if the common value of
β −j0−1/α −j0−1 and β i0+1/α i0+1 is denoted by λ, then
β i+1 = λ α i+1 (i ≥ 0) and β −j−1 = λ α −j−1 (j ≥ 0).
Therefore, β k = λ α k (k ≠ 0).
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