Factorization theory for a class of Toeplitz + Hankel operators

Abstract

In this paper we study operators of the form $M(\phi)=T(\phi)+H(\phi)$ where $T(\phi)$ and $H(\phi)$ are the Toeplitz and Hankel operators acting on $H^p(\T)$ with generating function $\phi\in L^\iy(\T)$. It turns out that $M(\phi)$ is invertible if and only if the function $\phi$ admits a certain kind of generalized factorization.

Full text

Page 1

arXiv:math/0204038v1 [math.FA] 2 Apr 2002
Factorization theory for a class of Toeplitz + Hankel
operators
Estelle L. Basor∗
Department of Mathematics
California Polytechnic State University
San Luis Obispo, CA 93407, USA
Torsten Ehrhardt†
Fakult¨ at f¨ ur Mathematik
Technische Universit¨ at Chemnitz
09107 Chemnitz, Germany
Abstract
In this paper we study operators of the form M(φ) = T(φ)+H(φ) where T(φ) and
H(φ) are the Toeplitz and Hankel operators acting on Hp(T) with generating function
φ ∈ L∞(T). It turns out that M(φ) is invertible if and only if the function φ admits a
certain kind of generalized factorization.
1Introduction
This paper is devoted to the study of operators of the form
M(φ) = T(φ) + H(φ)(1)
acting on the Hardy space Hp(T) where 1 < p < ∞. Here φ ∈ L∞(T) is a Lebesgue
measurable and essentially bounded function on the unit circle T. The Toeplitz and Hankel
operators are defined by
T(φ) : f ?→ P(φf),H(φ) : f ?→ P(φ(Jf)),f ∈ Hp(T),(2)
∗ebasor@calpoly.edu. Supported in part by NSF Grant DMS-9970879.
†tehrhard@mathematik.tu-chemnitz.de.
1

Page 2

where is J the following flip operator,
J : f(t) ?→ t−1f(t−1),t ∈ T,(3)
acting on the Lebesgue space Lp(T). The operator P stands for the Riesz projection,
P :
∞
?
n=−∞
fntn?→
∞
?
n=0
fntn,t ∈ T,(4)
which is bounded on Lp(T), 1 < p < ∞, and whose image is Hp(T). The complex conjugate
Hardy space Hp(T) is the set of all functions f whose complex conjugate belongs to Hp(T).
For a Banach algebra B (such as L∞(T) or H∞(T)) we will denote by GB the group of
all invertible elements.
It is well known that a necessary and sufficient condition for the Fredholmness and
the invertibility of the Toeplitz operator T(φ) with φ ∈ L∞(T) can be given in terms of
the Wiener-Hopf factorability of the generating function φ. Let us recall the underlying
definitions and results [3, 5, 6] (see also [8] for the case p = 2).
A function φ ∈ L∞(T) is said to admit a factorization in Lp(T) if one can write
φ(t) = φ−(t)tκφ+(t),t ∈ T,(5)
where κ is an integer and the factors φ−and φ+satisfy the following conditions:
(i) φ−∈ Hp(T), φ−1
− ∈ Hq(T),
(ii) φ+∈ Hq(T), φ−1
+ ∈ Hp(T),
(iii) The linear operator f ?→ φ−1
polynomials and takes values in Lp(T), can be extended by continuity to a linear
bounded operator acting from Lp(T) into Lp(T).
+P(φ−1
−f), which is defined on the set of all trigonometric
Here p−1+ q−1= 1. A factorization where merely (i) and (ii) but not necessarily condition
(iii) is fulfilled is called a weak factorization in Lp(T). The number κ is called the index of
the (weak) factorization and is uniquely determined.
The crucial result is that for given φ ∈ L∞(T) the operator T(φ) is a Fredholm operator
on the space Hp(T) if and only if the function φ admits a factorization in Lp(T). In this case
the defect numbers are given by
dimkerT(φ) = max{0,−κ},dimker(T(φ))∗= max{0,κ}. (6)
Hence T(φ) is invertible if and only if φ admits a factorization in Lp(T) with index κ = 0.
The goal of the present paper is to obtain a corresponding results for operators M(φ)
with φ ∈ L∞(T). We will encounter another type of factorization of the function φ which is
related to the Fredholm theory of these operators. The investigations taken up in this paper
are motivated by the results obtained in [1, 4].
2

Page 3

2Basic properties of M(φ)
It is well known that for Toeplitz and Hankel operators the following relations hold:
T(φψ) = T(φ)T(ψ) + H(φ)H(?ψ),
H(φψ) = T(φ)H(ψ) + H(φ)T(?ψ).
(7)
(8)
Here?ψ(t) = ψ(t−1). Adding both equations, it follows that
M(φψ) = T(φ)M(ψ) + H(φ)M(?ψ),
and hence
M(φψ) = M(φ)M(ψ) + H(φ)M(?ψ − ψ).
In some particular cases, this identity simplifies to a multiplicative relation:
(9)
M(φψ) = M(φ)M(ψ)(10)
if φ ∈ H∞(T) or ψ =?ψ. Based on this identity one can establish a sufficient invertibility
in this paper. Assume that φ ∈ L∞(T) admits a factorization φ = φ−φ0, where φ− ∈
GH∞(T) and φ0∈ GL∞(T) such that?φ0= φ0. Then M(φ) is invertible and its inverse its
It is occasionally convenient to introduce the multiplication operator on Lp(T),
criteria for M(φ), which anticipates to some extent the factorization result that we establish
given by M(φ−1
0)M(φ−1
−).
L(φ) : Lp(T) → Lp(T),f ?→ φf,(11)
and then consider Toeplitz and Hankel operators as restrictions of the following operators
onto Hp(T):
T(φ) = PL(φ)P|Hp(T),H(φ) = PL(φ)JP|Hp(T).(12)
The Toeplitz + Hankel operators M(φ) are of the form
M(φ) = PL(φ)(I + J)P|Hp(T).(13)
The following results gives estimates for the norm of M(φ), where the constant Cpdepends
only on the parameter p.
Proposition 2.1 Let φ ∈ L∞(T). Then ?φ?L∞(T)≤ ?M(φ)?L(Hp(T))≤ Cp?φ?L∞(T).
3

Page 4

Proof.
bounded on Lp(T). In order to prove the lower estimate, put Un= M(tn). Since UnU−n= I
and JUn= U−nJ, it is easy to see that
The upper estimate follows from (13) Note that the operators P and J are both
U−nM(φ)Un = (U−nPUn)L(φ)(U−nPUn) + (U−nPU−n)L(φ)J(U−nPUn).
Using the fact that U−nPUn→ I and U−nPU−n→ 0 strongly on Lp(T) as n → ∞, it follows
that
U−nM(φ)Un → L(φ) (14)
strongly on Lp(T) as n → ∞. Since U±n are isometries on Lp(T), this implies the lower
estimate.
2
Now we obtain a necessary condition for the Fredholmness of M(φ).
Proposition 2.2 Suppose that M(φ) is Fredholm on Hp(T). Then φ ∈ GL∞(T).
Proof. The proof is based on standard arguments. If M(φ) is a Fredholm operator, then
there exist a δ > 0 and a finite rank projection K on the kernel of M(φ) such that
?M(φ)f?Hp(T)+ ?Kf?Hp(T)
≥ δ?f?Hp(T)
for all f ∈ Hp(T). Putting Pf instead of f, this implies that
?M(φ)f?Lp(T)+ ?KPf?Lp(T)+ δ?(I − P)f?Lp(T) ≥ δ?f?Lp(T)
for all f ∈ Lp(T). Replacing f by Unf and observing again that U±nare isometries on Lp(T),
it follows that
?U−nM(φ)Unf?Lp(T)+ ?KPUnf?Lp(T)+ δ?U−n(I − P)Unf?Lp(T)
≥ δ?f?Lp(T).
Now we take the limit n → ∞. Because Un→ 0 weakly, we have KPUn→ 0 strongly. It
remains to apply (14) and again the fact that U−nPUn→ I strongly. We obtain
?L(φ)f?Lp(T) ≥ δ?f?Lp(T).
This implies that φ ∈ GL∞(T).
The following lemma, which appears in slightly less general form as Lemma 2.9 in [2],
turns out to be useful.
2
Lemma 2.3 Let X1 and X2 be linear spaces, A : X1 → X2 be a linear and invertible
operator, P1: X1→ X1and P2: X2→ X2be linear projections, and Q1= I − P1and Q2=
I − P2. Then P2AP1: imP1→ imP2is invertible if and only if Q1A−1Q2: imQ2→ imQ1
is invertible.
4

Page 5

Proof.
by a direct calculation:
The following formulas, which relate both inverses with each other, can be verified
(P2AP1)−1
(Q1A−1Q2)−1
= P1A−1P2− P1A−1Q2(Q1A−1Q2)−1Q1A−1P2,
= Q2AQ1− Q2AP1(P2AP1)−1P2AQ1.
The validity of these formulas implies the desired assertion.
2
In what follows, we are going to relate M(φ) with two other operators Φ(φ) and Ψ(φ).
First of all note that J2= I. Hence
PJ=I + J
2
,QJ=I − J
2
(15)
are complementary projections acting on the space Lp(T) and decompose this space into the
direct sum Lp(T) = imPJ|Lp(T)∔ imQJ|Lp(T). Let Lp
define
Φ(φ) = PL(φ)PJ,
J(T) = imPJ|Lp(T). Given φ ∈ L∞(T),
Ψ(φ) = PJL(φ)P, (16)
where we consider these operators as acting between the following spaces:
Φ(φ) : Lp
J(T) → Hp(T),Ψ(φ) : Hp(T) → Lp
J(T),(17)
Proposition 2.4 Let φ ∈ GL∞(T). Then the following assertions are equivalent:
(i) M(φ) is invertible in L(Hp(T)),
(ii) Φ(φ) is invertible in L(Lp
J(T),Hp(T)),
(iii) Ψ(ψ) is invertible in L(Hp(T),Lp
J(T)), where ψ(t) = φ−1(−t−1).
Proof.
and (16) of these operators. Moreover, note that A = (I + J)P : Hp(T) → Lp
B =1
In order to prove the equivalence of (i) and (ii) we refer to the representation (13)
J(T) and
2P(I + J) : Lp
J(T) → Hp(T) are inverse to each other:
AB =
1
2(I + J)P(I + J) =
1
2P(I + J)2P = P(I + J)P = P
P + JP + Q + JQ
2
=
I + J
2
BA =
Here we have used the basic relations, J2= I and JPJ = Q, where Q = I − P.
In order to prove the equivalence of (ii) and (iii) we apply Lemma 2.3 with P1 = PJ,
P2= P, Q1= QJand Q2= Q. We obtain that Φ(φ) is invertible if and only if the operator
QJL(φ−1)Q : imQ|Lp(T)→ imQJ|Lp(T)
5