ON HYPONORMAL TOEPLITZ OPERATORS WITH POLYNOMIAL ...

Theorem 2. Let a 0 , a 1 , . . . , a n be fixed complex numbers and let α ∈ C be such that either α = 0
or |α| = 1. Suppose that ϕ(e iθ ) = ∑ N
−m b ne inθ is a trigonometric polynomial, with m ≤ N, whose
coefficients b n satisfy the following combinatorial constraints:
⎧
⎪⎨ a n (0 ≤ n ≤ N − m)
(2.1) b n = a n + αa n+1 (N − m + 1 ≤ n ≤ N − 1)
⎪⎩
a N (n = N)
and
{ e iω (a N−n+1 + ᾱa N−n ) (1 ≤ n ≤ m − 1)
(2.2) b −n =
e iω a N−m+1 (n = m) .
If f denotes the analytic polynomial f(z) = a N−m+1 +a N−m+2 z+· · ·+a N z m−1 , then the following
statements are equivalent.
1. T ϕ is a hyponormal operator.
2. For every root ζ of f such that |ζ| > 1, the number 1/ζ is a root of f in D of multiplicity
greater than or equal to the multiplicity of ζ.
In the cases where T ϕ is a hyponormal operator, we have that
⎧
⎨
e iω a N
E(ϕ) =
⎩ a N
m−1
∏
z N−m
j=1
( z − ζj
1 − ζ j z
) ⎫ ⎬
⎭ ,
where ζ 1 , · · · , ζ m−1 denote the roots (repeated according to multiplicity) of the analytic polynomial
f. Moreover the rank of the selfcommutator of T ϕ is computed from the formula
rank [T ∗ ϕ, T ϕ ] = N − m + Z D − Z C\D
,
where Z D and Z C\D
are the number of zeros of f in D and in C\D counting multiplicity.
Remark. If α = 0 in (2.1) and (2.2) and if a 1 = · · · = a N−m = 0 when m < N, then ϕ is a
circulant polynomial with argument ω.
Proof. In view of Theorem 1, we will assume without loss of generality that
{ ᾱe iω a N (m < N)
a 0 =
αa 1 + ᾱe iω a N (m = N),
a 1 = · · · = a N−m = 0.
The connection between the given trigonometric polynomial ϕ and the analytic polynomial f is
explained by a factorisation of z m ϕ:
(
)
z m ϕ(z) = f(z) z N (z + α) + e iω (1 + ᾱz) .
So, ϕ vanishes on at least N points on T, namely on each of the (N + 1)-roots of −e iω if α = 0
or on each of the N-roots of −ᾱe iω if |α| = 1. Consider the function |ϕ| as a continuous function
θ ↦→ |ϕ(e iθ )| on the real interval [−π, π]. Then
log |ϕ(e iθ )| = log |f(e iθ )| + log |e iNθ (e iθ + α) + e iω (1 + ᾱe iθ )|