doi:10.1017/S0024609305018060
FREDHOLM THEORY OF TOEPLITZ OPERATORS ON THE HARDY SPACE H
1J. A. VIRTANEN
Abstract
The Fredholm properties of Toeplitz operators Ta on Hardy spaces Hp (1 < p < ∞) with continuous symbols a are well understood. We consider Ta acting on H1, where the operator is bounded provided that a belongs to the class of symbols given by Janson and Stegenga’s result on the pointwise multipliers on H1. A necessary and sufficient condition for Ta to be a Fredholm operator is given when a is continuous and satisfies a mild additional condition (much weaker than H¨older continuity). A formula for the index of Ta is also derived. In addition, we study the case of matrix-valued symbols and Toeplitz operators on BMOA.
1. Introduction
Let C denote the complex plane and T the unit circle {|z| = 1}. If f is analytic inD, 0 < p < ∞, and
sup
0r<1
1 2π
π
−π
f (reiθ)pdθ <∞,
then we say that f belongs to the Hardy space Hp(D). The space of all bounded analytic functions inD is denoted by H∞(D). We also set
Hp(T) = {f∗:T −→ C | f ∈ Hp(D)} ,
where f∗ ∈ Lp(T) denotes the nontangential boundary function of f ∈ Hp(D) (see [2, Section II.3]).
If f ∈ L1(T), we define
(Sf ) (t) = 1 πi
T
f (τ )
τ− tdτ, t∈ T, which exists in the Cauchy principal value sense; that is, the limit
εlim→0(Sf ) (t) = 1 πi
T\D (t,ε)
f (τ ) τ− tdτ
exists, where D(t, ε) denotes the disk centered at t with radius ε. The operator S is referred to as the Cauchy singular integral operator; it is bounded on Lp(T) if 1 < p <∞ according to a theorem of M. Riesz, and is unbounded if p = 1 or p = ∞.
Let HC1 ={f ∈ L1(T) | Sf ∈ L1(T)} with the norm f = f1+Sf1. It is not hard to see that S is bounded on HC1 and that S2 = I, where I is the identity operator. Therefore we can define a bounded projection P on HC1 and on Lp(T) (1 < p <∞) by P = 12(I + S). Note that P (Lp(T)) = Hp(T) (1 < p < ∞) and
Received 1 June 2004, revised 13 January 2005.
2000 Mathematics Subject Classification 47B35, 47A53 (primary), 45E10, 30D50 (secondary).
Supported by EPSRC grant GR/R81749/02.
P (HC1) = H1(T). The Toeplitz operator Ta with symbol a∈ L∞(T) is defined by Ta: Hp(T) −→ Hp(T), f → P (af). (1.1) Let a∈ L∞(T). Then the Toeplitz operator Ta is bounded on Hp(T), provided that 1 < p <∞. In order for T : H1(T) −→ H1(T) to be bounded, we require that the symbol a be in the class of pointwise multipliers on HC1, described by Theorem 1 below, due to Janson and Stegenga. We are concerned with Toeplitz operators on H1(T) and their Fredholm properties when the symbol a is continuous and satisfies a mild additional condition (much weaker than H¨older continuity).
When 1 < p <∞ and a is continuous, the Fredholm properties of Ta on Hp(T) are well understood [1, Section 2.42]: a necessary and sufficient condition for Ta to be Fredholm is that a is nowhere zero, in which case Ind Ta=− ind a. In the present paper, an analogous result is given for p = 1, in the case of both scalar-valued and matrix-valued symbols. In Section 3.7, it is pointed out that similar results also hold for Toeplitz operators acting on BMOA.
The precise class of symbols that we consider is the algebra C ∩ VMO|log|−1, defined in (2.4); note that VMOϕ occupies a similar natural position as a subspace of BMOϕ as does VMO in BMO; the latter relation is studied in [10]. In Section 2, we develop some necessary theory of this algebra; in particular, its characterization as the closure of the set of all smooth functions will play an important role in proving the Fredholm properties of Ta.
2. Preliminary results
2.1. Multipliers on HC1
Let X ⊂ L1(T). Then the set of pointwise multipliers on X ⊂ L1(T) is defined by M (X) ={f | fg ∈ X for all g ∈ X} with the norm
fM X = sup
gX1fgX.
In order to describe M (HC1), we need the following notation: for an arc I ⊂ T, let fI be the average of f over I,
O(f, I) = 1
|I|
I
|f − fI| , (2.1)
and
Oδ(f, ϕ) = sup
|I |<δ
O(f, I) ϕ(|I|) ,
where ϕ : (0, 1/2)−→ R is nondecreasing. Let f ∈ L1(T). If supI⊂TO(f, I) < ∞, we say that f is of bounded mean oscillation, and write f ∈ BMO. If, in addition, Oδ(f, 1)→ 0 as δ → 0, then f is said to have vanishing mean oscillation. We also define a family of spaces that includes BMO:
BMOϕ =
f ∈ L1(T) | O1/2(f, ϕ) <∞ with the norm
fBMOϕ =f1+O1/2(f, ϕ).
Note that if ϕ is a constant function, then BMO = BMOϕ. Let B = L∞ ∩ BMO|log|−1. The following result gives us a class of symbols a for which the Toeplitz operator Ta acting on H1(T) is bounded.
Theorem 1 (see [5, 11]). Both M (BMO) and M (HC1) coincide with B.
2.2. Distributions
Let Ω⊂ R be a nonempty open set. Let D(Ω) = C0∞(Ω) be the usual space of test functions (smooth functions with compact supports), andD(Ω) its dual space of distributions (see, for example, [9, Definition 6.7]).
If ϕ∈ D(R), then we set ϕ−(x) = ϕ(−x) and define the shift τyϕ, y∈ R, by
τyϕ
(x) = ϕ(x− y), x∈ R.
If we also have f∈ D(R), then we set
τyf
(ϕ) = f (τ−yϕ), f−(ϕ) = f (ϕ−). (2.2) The convolution f∗ ϕ is defined by
f∗ ϕ
(y) = f (τyϕ−), y∈ R.
Proposition 1(see [9, Theorem 6.30]). Let f ∈ D(R) and ϕ, ψ ∈ D(R); then (f∗ ϕ) ∗ ψ = f ∗ (ϕ ∗ ψ) = f ∗ (ψ ∗ ϕ) (2.3) and f∗ ϕ ∈ C∞.
2.3. Functions of vanishing mean oscillation
We are concerned with the properties of VMOϕ, a closed subspace of BMOϕ, defined by
VMOϕ={f ∈ BMOϕ| Oδ(f, ϕ)→ 0 as δ → 0} , (2.4) where ϕ : (0, 1/2)−→ R is nondecreasing. Note that when ϕ is a constant function, we have VMOϕ= VMO. Note also that it is not important that we subtract exactly fI in (2.1) when defining the set VMOϕ. Suppose that for each bounded arc I, there is a constant αI ∈ C such that
sup
|I |<δ
1 ϕ(|I|) |I|
I
|f(x) − αI| dx → 0 (2.5) as δ→ 0. Then it is easy to see that f ∈ VMOϕ.
When dealing with Toeplitz operators Ta in the next section, we will assume that the symbol a is in the spaceV = C ∩ VMO|log|−1. The properties of this space will now be studied.
Lemma 1. If a∈ C ∩ VMOϕ and a is nowhere zero, then 1/a∈ C ∩ VMOϕ. Proof. Let us first show that aI = 0 when |I| is sufficiently small. Since a is nowhere zero, there is ε > 0 such that |a| ε. By the continuity of a, there is δ0> 0 such that
|a(t) − a(s)| < ε/2 whenever|t − s| < δ0.
Suppose that|I| < δ0, and let τ ∈ I. Then
|aI− a(τ)| = 1
|I|
I
a(t)− a(τ) dt
1
|I|
I
|a(t) − a(τ)| dt ε/2, which implies that|aI| ε/2.
Since a∈ VMOϕ, sup
|I |<δ
1 ϕ(|I|) |I|
I
1 a(t)− 1
aI
dt = sup|I |<δ 1 ϕ(|I|) |I|
I
|a(t) − aI|
|a(t)aI| dt
4 ε2 sup
|I |<δ
1 ϕ(|I|) |I|
I
|a(t) − aI| dt → 0
as δ→ 0. This completes the proof; see (2.5).
Theorem 2. The set C∩ VMOϕ is a Banach algebra.
Proof. It is enough to show that for f, g∈ C ∩VMOϕ, we have f g∈ C ∩VMOϕ. This, however, follows easily from the following estimate
|fg − fIgI| = |(f − fI)g + (g− gI)fI| |g| |f − fI| + |fI| |g − gI| and (2.5).
We finish this section by showing that each function in the space V can be approximated by smooth functions in theB norm. To do this, we need to introduce some theory of Bochner integrals.
Let X be a Banach space, and let (Ω, M, µ) be a space with a measure. We call a function f : Ω −→ X a step function if there is a finite number of measurable sets Ej ∈ M such that f is constant on each Ej, say f = cj ∈ X, and f = 0 on Ω\ ∪Ej. Let f : Ω−→ X be a step function. If supp f has a finite measure, we say that f is a Bochner integrable step function, and we set
Ω
f dµ =
j
cjµ(Ej).
If there exists a sequence of Bochner integrable step functions (fk) that converges to a function f : Ω−→ X almost everywhere and satisfies
nlim→∞
Ω
f − fnX dµ = 0, then we say f is Bochner integrable, and define
Ω
f dµ = lim
n→∞
Ω
fndµ.
Theorem 3 (Hille [4, Theorem 3.7.12]). Let X and Y be Banach spaces, and let A : X −→ Y be a closed linear operator. If the functions f : R −→ X and Af :R −→ Y are both Bochner integrable, then
A
Ω
f dµ
=
Ω
Af dµ for all Ω⊂ R.
Lemma 2. Let a Banach space Y be continuously embedded into D(R), and let ϕ∈ D(R). Suppose that R y → τyf ∈ Y is continuous everywhere. If f ∈ Y , then the convolution f∗ ϕ can be represented as the following Bochner integral
Rϕ(y)τyf dy.
Proof. Note first that ϕ(y)τyf is Bochner integrable, due to the continuity assumption. Using (2.2), (2.3) and Hille’s Theorem 3, we obtain
Rϕ(y)τyf dy, ψ
=
Rϕ(y) τyf, ψ dy =
Rϕ(y) f, τ−yψ dy
=
Rϕ−(y) f, τyψ dy =
Rϕ−(y)(f∗ ψ−)(y) dy
=
(f∗ ψ−)∗ ϕ (0) =
f ∗ (ϕ ∗ ψ−) (0)
=
(f∗ ϕ) ∗ ψ−
(0) = f ∗ ϕ, ψ for all ψ∈ D(R).
Theorem 4. Let Y be continuously embedded intoD(R). Suppose that f ∈ Y and τyf ∈ Y for all y ∈ R. If
(1) τyfY const fY for all y∈ R, and (2) τyf − fY → 0 as |y| → 0,
then there are fε ∈ C∞(R) ∩ Y such that
f − fεY → 0 as ε→ 0 (2.6)
withfεY const fY.
Proof. Let ϕ ∈ C0∞(R) be such that 0 ϕ 1, supp ϕ ⊂ (−1, 1) and
Rϕ(x) dx = 1. Define
ϕε(x) = 1 εϕ
x ε
,
and let fε = f∗ϕε. Then fε∈ C∞(R). According to Lemma 2, fεcan be represented as the following Bochner integral:
fε =
Rϕε(y)τyf dy.
Now
fεY =
R
ϕε(y)τyf dy Y
R
ϕε(y)τyfY dy
const fY
Rϕε(y) dy const fY, and
f − fεY = f
Rϕε(y) dy−
Rϕε(y)τyf dy Y
Rϕε(y)f − τyfY dy
sup
|y|<εf − τyfY
Rϕε(y) dy→ 0 as ε→ 0.
We can now prove the desired result on the approximation of continuous VMO|log|−1 functions.
Lemma 3. The set V equals the closure of C∞(T) in B, where V = C ∩ VMO|log|−1 andB = L∞∩ BMO|log|−1.
Proof. Let f be in the closure of C∞(T). Then there is a sequence (fn) in C∞(T) such that
f − fn∞+f − fnBMO
|log|−1 → 0
as n→ ∞. Obviously, f is continuous. Let us show that f ∈ VMO|log|−1. Let ε > 0.
Choose n∈ N such that
f − fnBMO
|log|−1 < ε/2.
Note also that for sufficiently small δ0, Oδ(fn,|log|−1) = sup
|I |<δ
|log |I||
|I|
I
|fn(x)− fn I| dx sup
|I |<δ
|log |I||
|I|
I
fn∞|I| dx
= sup
|I |<δ|log |I|| |I| fn∞ δ0|log δ0| fn∞< ε/2 whenever δ < δ0. Therefore, there exists δ0 such that
Oδ(f,|log|−1) =Oδ(f− fn+ fn,|log|−1)
Oδ(f− fn,|log|−1) +Oδ(fn,|log|−1) < ε whenever δ < δ0. So f ∈ V.
Let f ∈ V. Define
τhf
(x) = f (x− h).
In order to prove that f is in the closure of C∞(T), we will show that
τhf− fBMO
|log|−1 → 0
as |h| → 0. Then, by Theorem 4, it follows immediately that there is fε ∈ C∞(T) such that
f − fεBMO
|log|−1 → 0
as ε→ 0, so that f is in the closure of C∞(T). Let ε > 0. Since f ∈ VMO|log|−1, there is δ > 0 such thatOδ(f,|log|−1) < ε and Oδ(τhf,|log|−1) < ε. Also, since f is uniformly continuous,
|I |δsup
|log |I||
|I|
I
τhf− f (x)−
τhf− f
Idx < ε for sufficiently small |h|. Therefore, τhf − fBMO
|log|−1 < ε if |h| is sufficiently small; that is,τhf− fBMO
|log|−1 → 0 as |h| → 0.
Corollary 1. Let a∈ V. Assume that a(t0) = 0 for some t0∈ T. Then for all ε > 0, there is aε∈ C∞(T) such that
a − aε = a − aε∞+a − aεBMO
|log|−1 < ε and aε(t0) = 0.
Proof. By the preceding theorem, there exists ˜aε∈ C∞(T) such that a − ˜aε <
ε/2. Then|˜aε(t0)| < ε/2, and we can set aε= ˜aε− ˜aε(t0).
3. Fredholm theory
Let X and Y be Banach spaces, and let A : X −→ Y be a bounded operator.
Then we say that A is Fredholm and write A ∈ Φ(X, Y ) if both dim(ker A) and dim(Y / im A) are finite. Note that we write Φ(X) for Φ(X, X). The index of a Fredholm operator is defined by
Ind A = dim(ker A)− dim(Y/ im A).
We also define an index of a nonvanishing continuous function a by ind a = [arg a]T
2π ,
where [arg a]T denotes the total increment of arg a(t) when t ranges over T. The essential spectrum of A is defined by
σess(A) ={λ ∈ C | A − λI /∈ Φ (X)} , whereI : X −→ X is the identity operator.
The following stability result on the index of a Fredholm operator will be employed several times.
Theorem 5(see [3, Section 4.6, Theorem 6.4]). Let X and Y be Banach spaces, and let A∈ Φ(X, Y ). Then there exists δ > 0 such that any bounded linear operator B : X−→ Y satisfying the condition A − B < δ is also Fredholm with Ind B = Ind A.
3.1. Main results
Our main result concerns the Fredholm properties of Ta and a formula for its index. The proofs are given below in the following subsections.
Theorem 6. Let a∈ V. Then Ta is Fredholm if and only if a is nowhere zero, in which case
Ind Ta=− ind a.
The following characterization of the essential spectrum of Tafollows immediately from the previous theorem.
Corollary 2. If a∈ V, then σess(Ta) = a(T).
3.2. Sufficient condition of Theorem 6 We start with a preliminary lemma.
Lemma 4. Let a∈ C∞(T), and denote the commutator of two operators A, B by [A, B] = AB− BA. Then
[S, aI]f(t) = 1 πi
T
a(τ )− a(t) τ− t f (τ ) dτ is compact in HC1.
Proof. Since
a(t)− a(s) =
1
0
a
s + θ(t− s) dθ
(t− s), we can write the kernel of the above integral operator as follows:
K(s, t) =
1 0
a
s + θ(t− s) dθ
. (3.1)
Since
C1⊂ C ⊂ Lp(T) ⊂ HC1 (1 < p <∞)
and the first embedding is compact, it is sufficient to show that K∈ C1([−π, π]2), but this follows immediately from (3.1), and so we are done.
Corollary 3. Let a∈ V. Then
[aI, P ]f
(t) = 1 2πi
T
a(t)− a(τ) τ− t f (τ ) dτ is compact.
Proof. The preceding lemma implies that [aI, P ] is compact if a ∈ C∞(T). In the case where a∈ V, Lemmas 1 and 3 show the compactness of [aI, P ].
Now we can easily obtain a sufficient condition for Ta to be Fredholm.
Theorem 7. If a∈ V is nowhere zero, then the operator Ta : H1(T) −→ H1(T) is Fredholm.
Proof. According to Lemma 1, T1/a∈ L(H1(T)). Since
TaT1/a =I − P [aI, P ]a−1I, T1/aTa=I − P [a−1I, P ]aI,
and [aI, P ] is compact, the operator Ta has a regularizer and is hence Fredholm.
3.3. A proof of the index formula
Theorem 8. Let a∈ V. If a is nowhere zero and κ = ind a, then Ind (Ta) = Ind(Ttκ).
Proof. By Theorem 7, Ta is Fredholm. According to Theorem 5 and Lemma 3, there is a1∈ C∞(T) such that Ta1 is Fredholm, ind a1= ind a = κ and
Ind(Ta) = Ind(Ta1).
Let b(t) = a1(t)t−κ, and define Fτ(t) = tκeτ log b(t). Since a1∈ C∞(T) and ind b = 0, the function log b is continuously differentiable, and so Fτ(t) = 0 for any t, τ and Fτ is a homotopy in C1 and hence in V. Therefore, the operators TFτ (0 τ 1) belong to the same component of Φ
H1(T)
. Since in every component of the set Φ(H1(T)) the index of operators takes a constant value (see [3, Section 4.10, Theo- rem 10.1]), we have Ind(Ta) = Ind(Ta1) = Ind(TF1) = Ind(TF0) = Ind(Ttκ).
We will use the following remark to compute the index of Ttκ.
Remark 1. As is known in the case of 1 < p <∞ (see [3, Section 3.15]), the Riemann–Hilbert problem of finding ϕ, ψ∈ H1(D) such that
aϕ∗= ψ∗+ g, ψ(0) = 0, (3.2)
and the problem of finding f∈ H1(T) such that Taf = g are equivalent in the sense that a solution of one equation can be used to obtain a solution of the other in a one-to-one manner.
Also, note that the problem of finding an analytic function Φ :C \ T −→ C such that
aΦ+= Φ−+ g, Φ(∞) = 0, (3.3)
where Φ±(t) (t∈ T) denote the nontangential boundary values of Φ(z) when z ∈ D or z∈ C \ D, can be reduced to the Riemann–Hilbert problem (3.2) by setting
ψ(z) = Φ(1/z) z∈ D.
Lemma 5. If κ 0, then
dim ker(Ttκ) = 0, dim
H1(T)/ im Ttκ
= κ.
If κ < 0, then
dim ker(Ttκ) =−κ, dim
H1(T)/ im Ttκ
= 0.
Proof. We first consider ker Ttκ. According to Remark 1, it is sufficient to study the problem in (3.3) with g≡ 0. Suppose that Φ is a solution. Let
X(z) =
1 if z∈ D, zκ if z∈ C \ D.
Then tκX+= X− and
Φ+ X+ = Φ−
X−.
Assume that κ < 0. By Carleman’s theorem [6, Section III.E.2], the function Φ/X has an analytic continuation to the whole plane. Since |Φ(z)/X(z)| O(|z|−κ), the entire function Φ/X is a polynomial P of degree at most−κ. Now Φ(∞) = 0 implies that we have−κ linearly independent solutions of the form Φ = XP . For κ 0, since we require Φ(∞) = 0, we have no solutions aside from the trivial solution Φ = 0.
Let us next consider the codimension of im Ttκ. If Φ is a solution of (3.3), then Φ+
X+ − Φ− X− = g
tκX+
and Φ/X has a finite degree at infinity. Using [3, Section 2.4], we can show that Φ(z) =X(z)
2πi
T
g(w)
wκX+(w)(w− z)dw + X(z)P (z) (3.4) for some polynomial P . Some of these solutions Φ have to be disregarded due to the requirement that Φ(∞) = 0. Indeed, if κ 0, then the solution in (3.4) vanishes if and only if the degree of P is not greater than −κ − 1; for κ = 0, P ≡ 0. In particular, dim(H1(T)/ im Ttκ) = 0. Suppose now that κ > 0. Then P ≡ 0, and the
coefficients of z−1, z−2, . . . , z−κ in the expansion 1
2πi
T
g(w)
wκX+(w)(w− z)dw =
n1
−z−n 2πi
T
wn−1g(w) wκX+(w)dw have to be zero. Consequently, (3.3) has a solution if and only if
T
wng(w)
wκX+(w)dw = 0 (3.5)
for all n = 0, 1, . . . , κ− 1. Therefore, the equation Ttκf = g has a solution if and only if the conditions in (3.5) hold, which implies that the codimension of im Ttκ
is κ, by [9, Theorem 4.9].
Corollary 4. Let a∈ V be nowhere zero. Then Ind Ta=− ind a.
3.4. Necessary condition of Theorem 6
Theorem 9. Let a ∈ V, and assume that a(t0) = 0 for some t0 ∈ T. Then Ta: H1(T) −→ H1(T) is not Fredholm.
Proof. Suppose that Ta is Fredholm. By Theorems 1 and 5 and Corollary 1, there is a1∈ C∞(T) such that a1(t0) = 0 and Ta1 is Fredholm with the same index as the operator Ta. Using Theorem 5 again, we can show that there is a2∈ C∞(T) such that a2(t0) = 0 is a simple point of the curve t→ a(t) and Ta2is still Fredholm with the same index.
Next we will deform a2 in two directions in such a way that the deformed coefficients are nowhere zero. Indeed, use Theorem 5 to construct two coefficients b1, b2 ∈ C∞(T) such that Tbk is Fredholm with the same index as Ta and so that the coefficients have the following properties. On a sufficiently small arc, say I⊂ T, in the neighborhood of t0, the curve b1 will remain on the left of a2(I) without intersecting a2(T) \ a2(I). Similarly, the curve b2will remain on the right of the arc a2(I). Then bk(t) = 0 for all t ∈ T, k = 1, 2, and ind b1 = ind b2. This, however, contradicts the fact that we should have Ind Tb1 = Ind Tb2 and Ind Tbk = ind bk, k = 1, 2.
3.5. Matrix-valued symbols
Our objective here is to generalize Theorem 6 to the case where the given coefficient is an n× n matrix instead of a scalar-valued function a ∈ V. More specifically, let a = (aij)ni,j =1 with aij ∈ V. Define Ta: Hn1(T) −→ Hn1(T) by
Ta(f1, . . . , fn) =
P aa11f1 + . . . + P aa1nfn
... . .. ...
P aan 1f1 + . . . + P aan nfn
;
that is, Ta = (P aijI). The space Hn1(T) is equipped with the norm
f = max
1infi1.
We start with a useful formula for the determinant of Ta.
Lemma 6. There is a compact operator K such that det Ta = Tdet a+ K.
Proof. It is well known that for an n× n matrix (Cj k), det(Cj k) =
σ∈Sn
sgn(σ)
n i=1
(Ciσ (i)),
where Sn is the group of n-permutations and sgn(σ) denotes the sign of each permutation. In particular, we can write
det Ta=
σ∈Sn
sgn(σ)P a1σ (1)I . . . P an σ (n )I
=
σ∈Sn
sgn(σ)P (a1σ (1). . . an σ (n ))I + K
for some compact operator K, since Tb1Tb2= Tb1b2+ P [b1I, P ]b2I (b1, b2∈ V) and [b1I, P ] is compact.
Theorem 10. The operator Ta : Hn1(T) −→ Hn1(T) is Fredholm if and only if det a(t) = 0 for any t ∈ T, in which case Ind Ta =− ind det a.
Proof. By Atkinson’s theorem, the operator Ta is Fredholm if and only if Ta+K(Hn1(T)) is invertible in the Calkin algebra.
It follows from [7, Section I.1, Theorem 1.1] and Corollary 3 that Ta is Fredholm if and only if det Ta+K(H1(T)) is invertible, which can happen if and only if det Ta
is Fredholm; note that, due to the preceding lemma, det Ta = Tdet a+ K for some compact operator K. Consequently, by Theorem 6, Ta is Fredholm if and only if det a(t) = 0 for any t ∈ T.
It remains to prove the index formula. Let A =
P bI + K | b ∈ V, K ∈ K(H1(T)) ,
and let b∈ V. By Theorem 5 and Lemma 3, there is ˜b ∈ C∞(T) such that ˜b(t) = 0 for any t ∈ T and P˜bI is Fredholm approximating the operator Tb. Therefore, Φ(HC1)∩ A is dense in A; the other conditions of [7, Section I.3, Theorem 3.1] are easily verified. Thus
Ind Ta= Ind det Ta= Ind(Tdet a + K) = Ind Tdet a =− ind det a.
3.6. Alternative proof of the index formula
Next we will give an alternative proof of the index formula. The proof is based on the following result.
Theorem 11(see [8, Appendix VI, Lemma 6]). Let A1∈ C1(T, Cn×n). Assume that det A1(t) = 0 for any t ∈ T, and let κ = ind det A1. If A2 = diag[bj]nj =1 with b1(t) = tκ and bj = 1 (j = 2, . . . , n), then A1 and A2 are homotopic in GC1(T, Cn×n).
Proof of the index formula. Let ε > 0. By Lemma 3, for each i, j = 1, . . . , n, there is aij ∈ C1 such that
aij− aij
B< ε nQ.
Therefore, setting a= (aij)ni,j =1, we have
Ta− Ta = P (a − a)I
and
P (a − a)(f1, . . . , fn) const ε (f1, . . . , fn) .
Thus Ind Ta = Ind Ta for sufficiently small ε > 0, by Theorem 5. Moreover, since
det a − det a∞=
σ∈Sn
sgn(σ)
a1σ (1). . . an σ (n )− a1σ (1). . . an σ (n )
∞
σ∈Sn
a1σ (1). . . an σ (n )− a1σ (1). . . an σ (n )
∞
and
a1σ (1)a2σ (2)− a1σ (1)a2σ (2)
a1σ (1)− a1σ (1)∞
∞ a2σ (2)
∞+ a1σ (1)
∞ a2σ (2)− a2σ (2)
∞, we can show, by induction, thatdet a − det a∞< ε ifaij− aij∞is sufficiently small, which we can achieve according to the Weierstrass theorem. Therefore, by the continuity of the index of a continuous function, ind det a = ind det a.
Let κ = ind det a. Then Theorem 11 implies that a and the matrix diag(bi)ni=1 with b1 = tκ and bi = 1 (i > 1) are homotopic in GC1(T, Cn×n) and hence in V(T, Cn×n). Therefore the operators Ta and Tdiag(bi)have the same index. Since
Tdiag(bi)= diag(Bj)nj =1, where Bj = Ttκ and Bj =I (j > 1), the index Ind(Tdiag(bi)) = Ind(Ttκ). Consequently,
Ind(Ta) = Ind(Ta) = Ind(Ttκ) =−κ = − ind det a.
3.7. Toeplitz operators on BMOA
It follows from Theorem 1, as for Toeplitz operators on H1(T), that Tais bounded on BMOA = P (BMO) when a∈ B.
Let a ∈ V. Then it can be shown that Ta : BMOA −→ BMOA is Fredholm if and only if a is nowhere zero, in which case Ind Ta = − ind a. In addition, if a = (ai,j)ni,j =1 with aij ∈ V, then the sufficient and necessary condition for Ta to be Fredholm is that det a(t) = 0 for any t ∈ T, in which case Ind Ta=− ind det a.
References
1. A. B¨ottcherand B. Silbermann, Analysis of Toeplitz operators (Springer, Berlin, 1990).
2. J. B. Garnett, Bounded analytic functions (Academic Press, New York, 1981).
3. I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations (Birkh¨auser, Basel, 1992).
4. E. Hille and R. S. Phillips, Functional analysis and semi-groups (Amer. Math. Soc., Providence, RI, 1957).
5. S. Janson, ‘On functions with conditions on the mean oscillation’, Ark. Mat. (2) 14 (1976) 189–196.
6. P. Koosis, Introduction to Hp spaces, 2nd edn (Cambridge University Press, Cambridge, 1998).
7. N. Krupnik, Banach algebras with symbol and singular integral operators (Birkh¨auser, Boston, 1987).
8. N. I. Muskhelishvili, Singular integral equations, 3rd edn (Nauka, Moscow, 1968) (in Russian).
9. W. Rudin, Functional analysis (McGraw-Hill, New York, 1973).
10. D. Sarason, ‘Functions of vanishing mean oscillation’, Trans. Amer. Math. Soc 207 (1975) 391–405.
11. D. A. Stegenga, ‘Bounded Toeplitz operators on H1and applications of the duality between H1and the functions of bounded mean oscillation’, Amer. J. Math. (3) 98 (1976) 573–589.
J. A. Virtanen
Department of Mathematics
King’s College, University of London The Strand
London WC2R 2LS United Kingdom [email protected]
