New York Journal of Mathematics

New York J. Math. 8 (2002) 215–234.

Algebras of Singular Integral Operators

on Rearrangement-Invariant Spaces and Nikolski Ideals

Alexei Yu. Karlovich

Abstract. We construct a presymbol for the Banach algebra Alg (Ω, S) generated by the Cauchy singular integral operator S and the operators of multiplication by functions in a Banach subalgebra Ω of L^{∞}. This presymbol is a homomorphism Alg (Ω, S) → Ω ⊕ Ω whose kernel coincides with the commutator ideal of Alg (Ω, S). In terms of the presymbol, necessary conditions for Fredholmness of an operator in Alg (Ω, S) are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.

Contents

1. Introduction 216 2. Rearrangement-invariant spaces and their indices 217

2.1. Rearrangement-invariant spaces 217 2.2. Boyd indices 218 2.3. Singular integral operators, Toeplitz and Hankel operators 219

3. Nikolski ideals associated with Douglas algebras 219 3.1. Definition of the Nikolski ideals 219 3.2. Properties of the Nikolski ideals 221 3.3. Estimates for quotient norms 225

4. The presymbol of the algebra Alg (Ω, S) 227 4.1. The construction of a presymbol 227 4.2. Necessary conditions for Fredholmness 229 4.3. The commutator ideal of the algebra Alg (Ω, S) 231 4.4. Singular integral operators with quasicontinuous coefficients 232

References 233

Received July 25, 2002. Mathematics Subject Classification. Primary 47B35, 47B38, 47A53; Secondary 46E30. Key words and phrases. Douglas algebra, Nikolski ideal, singular integral operator, Fredholm-

ness, rearrangement-invariant space.

The author is partially supported by F.C.T. (Portugal) grants POCTI 34222/MAT/2000 and PRAXIS XXI/BPD/22006/99.

215^{ISSN 1076-9803/02}

1. Introduction

Let 𝕋 be the unit circle equipped with the normalized Lebesgue measure dm = |dτ|/(2π). For a function φ ∈ L^{1} = L^{1}(𝕋, dm), the Cauchy singular integral is defined by

(Sφ)(t) := \frac{1}{πi}v.p. ∫_{𝕋} \frac{φ(τ)dτ}{τ − t} , t ∈ 𝕋.

Let X = X(𝕋, dm) be a reflexive rearrangement-invariant space with nontrivial Boyd indices (for the definitions, see Section 2) and let Ω be an arbitrary Banach subalgebra of L^{∞}. We denote by L(X) the Banach algebra of all bounded linear operators on X and by K(X) the closed two-sided ideal of all compact operators on X. The smallest Banach subalgebra of L(X) containing the Cauchy singular integral operator S and the operators of multiplication M_{φ} by functions φ ∈ Ω is denoted by Alg (Ω, S). The commutator ideal of Alg (Ω, S), that is, the closed two-sided ideal generated by all commutators AB − BA with A, B ∈ Alg (Ω, S) is denoted by Com Alg (Ω, S).

S. G. Mikhlin suggested [22, 23] an idea of symbol calculus for investigation of Fredholm properties of singular integral operators on Lebesgue spaces. Recall that an operator acting on a Banach space is said to be Fredholm if its image is closed and the dimensions of its kernel and cokernel are finite. In particular, S. G. Mikhlin proved [24] (see also [25]) that every operator F ∈ Alg (C, S) ⊂ L(L^{p}), 1 < p < ∞, where C = C(𝕋) stands for the C^{∗}-algebra of all continuous functions on 𝕋, admits a canonical representation of the form

F = M_{φ}P_{+} + M_{ψ}P_{−} + K, (1.1)

where

P_{+} := (I + S)/2, P_{−} := (I − S)/2 (1.2)

are the Riesz projections, I is the identity operator, φ, ψ ∈ C and K ∈ K(L^{p}). Moreover, in this case K(L^{p}) = Com Alg (C, S) and F is Fredholm if and only if φ(t) ≠ 0, ψ(t) ≠ 0 for all t ∈ 𝕋. The representation (1.1) allows us to construct a canonical homomorphism (symbol) Alg (C, S) → C ⊕ C with the kernel K(L^{p}), where A ⊕ B stands for the direct sum of Banach algebras A and B equipped with the operations (a, b) + (c, d) = (a + c, b + d), (a, b) · (c, d) = (ac, bd) and the norm ∥(a, b)∥_{A⊕B} := max{∥a∥_{A}, ∥b∥_{B}}.

The situation becomes more difficult if Ω is wider than C and X is more general than a Lebesgue space L^{p}, 1 < p < ∞. In this paper some necessary conditions for Fredholmness of F ∈ Alg (Ω, S) ⊂ L(X) are obtained in terms of a presymbol of Alg (Ω, S). The presymbol is a canonical homomorphism of Alg (Ω, S) onto the quotient algebra Alg (Ω, S)/Com Alg (Ω, S) modulo the commutator ideal Com Alg (Ω, S). In general, the latter ideal is wider than K(X). Some specific algebras Ω ⊂ L^{∞} were treated earlier in the case of (weighted) Lebesgue spaces in [2, 3, 4, 5, 7, 10, 19, 25, 29] (see also the references therein). For more general rearrangement-invariant spaces, only the algebra Ω = P C of piecewise-continuous functions was considered earlier in [13] (see also [15]).

In this paper we follow the approach of [9] and construct a presymbol of the algebra Alg (Ω, S) ⊂ L(X) for any Banach subalgebra Ω of L^{∞} and any reflexive rearrangement-invariant space X with nontrivial Boyd indices. More precisely, we describe a Banach algebra homomorphism Alg (Ω, S) → Ω ⊕ Ω with the kernel Com Alg (Ω, S) and obtain the representation (1.1) for an arbitrary operator F in Alg (Ω, S) with φ, ψ ∈ Ω and K ∈ Com Alg (Ω, S). In this construction a collection of so-called Nikolski ideals 𝔍^{±}(𝔸) (see [9, Section 2] and also [26, 27, 28]) associated with a Douglas algebra 𝔸 (see, e.g., [8, Ch. 9]) plays an important role. Another important ingredients in the construction are two-sided estimates for the norms of the Toeplitz operators P_{+}M_{φ}P_{+}, P_{−}M_{φ}P_{−} and the Hankel operators P_{−}M_{φ}P_{+}, P_{+}M_{φ}P_{−} with a symbol φ ∈ L^{∞}. These estimates were recently obtained in [16] for reflexive rearrangement-invariant spaces with nontrivial Boyd indices.

The paper is organized as follows. In Section 2 we give necessary preliminaries on rearrangement-invariant spaces and their Boyd indices. We conclude this section with the estimates for the norms of Toeplitz and Hankel operators. In Section 3 we study properties of Nikolski ideals associated with Douglas algebras. This allows us to give estimates for quotient norms modulo these ideals for Hankel and singular integral operators of the form M_{φ}P_{+} + M_{ψ}P_{−}. Our main results are concentrated in Section 4. First, we construct the presymbol for the algebra Alg (Ω, S) ⊂ L(X), where Ω is an arbitrary Banach subalgebra of L^{∞}. Secondly, we prove necessary conditions for Fredholmness of an arbitrary operator F ∈ Alg (Ω, S) and describe the commutator ideal of the algebra Alg (Ω, S). Finally, we discuss commutator ideals of algebras Alg (Ω, S) for Ω between C and QC, where QC is the algebra of all quasicontinuous functions, and give a criterion for the Fredholmness of an operator A ∈ Alg (Ω, S) in this case.

The presentation is selfcontained. We complement and extend [9] giving details in the cases which were omitted in [9] and vice versa. In places we consider topics in the same sequence in which they are considered in [9]. As a reader of both papers will see, in some cases we are able to adapt the proofs there directly to our context, however in other places we have to involve more delicate arguments, for instance, such as new analogues of classical estimates for the norms of Hankel and Toeplitz operators (see [16]). We refine also some minor inaccuracies of [9].

2. Rearrangement-invariant spaces and their indices

2.1. Rearrangement-invariant spaces. For a general discussion of rearrangement-invariant spaces, see [1, 18, 20]. In this section we collect necessary facts.

Denote by M the set of all measurable complex-valued functions on 𝕋, and let M^{+} be the subset of functions in M whose values lie in [0, ∞]. The characteristic function of a measurable set E ⊂ 𝕋 will be denoted by χ_{E}. A mapping ρ : M^{+} → [0, ∞] is called a function norm if for all functions f, g, f_{n} ∈ M^{+} (n ∈ ℕ), for all constants a ≥ 0, and for all measurable subsets E of 𝕋, the following properties hold:

(a)  ρ(f) = 0 ⇔ f = 0 a.e., ρ(af) = aρ(f), ρ(f + g) ≤ ρ(f) + ρ(g),
(b)  0 ≤ g ≤ f a.e. ⇒ ρ(g) ≤ ρ(f) (the lattice property),
(c)  0 ≤ f_{n} ↑ f a.e. ⇒ ρ(f_{n}) ↑ ρ(f) (the Fatou property),
(d)  ρ(χ_{E}) < ∞, ∫_{E} f dm ≤ C_{E}ρ(f)

with C_{E} ∈ (0, ∞) depending on E and ρ but independent of f. When functions differing only on a set of measure zero are identified, the set X of all functions f ∈ M for which ρ(|f|) < ∞, is a Banach space under the norm ∥f∥_{X} := ρ(|f|). Such a space X is called a Banach function space. If ρ is a function norm, its associate norm ρ′ is defined on M^{+} by

ρ′(g) := sup {∫_{𝕋} fg dm : f ∈ M^{+}, ρ(f) ≤ 1} , g ∈ M^{+}.

The Banach function space X′ determined by the function norm ρ′ is called the associate space (Köthe dual) of X. The associate space X′ is a subspace of the dual space X^{∗}.

Let M_{0} and M_{0}^{+} be the classes of a.e. finite functions in M and M^{+}, respectively. Two functions f, g ∈ M_{0} are said to be equimeasurable if

m{τ ∈ 𝕋 : |f(τ)| > λ} = m{τ ∈ 𝕋 : |g(τ)| > λ} for all λ ≥ 0.

A function norm ρ : M^{+} → [0, ∞] is called rearrangement-invariant if for every pair of equimeasurable functions f, g ∈ M_{0}^{+} the equality ρ(f) = ρ(g) holds. In that case, the Banach function space X generated by ρ is said to be a rearrangementinvariant space. A Banach function space X is rearrangement-invariant if and only if its associate space X′ is rearrangement-invariant too [1, p. 60].

The Lebesgue space L^{p}, 1 ≤ p ≤ ∞, is the simplest example of a rearrangementinvariant space. Orlicz and Lorentz spaces are other important classical examples of rearrangement-invariant spaces. For every rearrangement-invariant space X (see, e.g., [1, p. 78]), we have L^{∞} ⊂ X ⊂ L^{1}.

2.2. Boyd indices. By the Luxemburg representation theorem [1, Ch. 2, Theorem 4.10], there is a unique rearrangement-invariant function norm \overline{ρ} over [0, 1] with the Lebesgue measure dt such that ρ(f) = \overline{ρ}(f^{∗}) for all f ∈ M_{0}^{+}, where f^{∗} is the non-increasing rearrangement of f (see, e.g., [1, p. 39]). The rearrangementinvariant space over ([0, 1], dt) generated by \overline{ρ} is called the Luxemburg representation of X. For each s ∈ ℝ_{+} := (0, ∞), let E_{s} denote the dilation operator defined on M_{0}([0, 1], dt) by

(E_{s}f)(t) := \{ \begin{matrix}f(st), & st ∈ [0, 1] \\ 0, & st ∉ [0, 1] \end{matrix} , t ∈ [0, 1].

For every s ∈ ℝ_{+}, the operator E_{1/s} is bounded on the Luxemburg representation of X [1, p. 165], its norm is denoted by h_{X}(s). The function h_{X} : ℝ_{+} → ℝ_{+} is submultiplicative and non-decreasing. From [18, Ch. 2, Theorem 1.3] it follows that the limits

α_{X} := lim_{s→0} \frac{log h_{X}(s)}{log s}, β_{X} := lim_{s→∞} \frac{log h_{X}(s)}{log s}

exist and α_{X} ≤ β_{X}. The numbers α_{X} and β_{X} are called the lower and upper Boyd indices of the rearrangement-invariant space X, respectively [6]. For the Lebesgue spaces L^{p}, 1 ≤ p ≤ ∞, the Boyd indices coincide and equal 1/p. For an arbitrary rearrangement-invariant space, its Boyd indices belong to [0, 1]. We will say that the Boyd indices are nontrivial if α_{X}, β_{X} ∈ (0, 1). In the case of Orlicz spaces the latter condition is equivalent to the reflexivity of the space (see, e.g., [21]). One can find properties of the Boyd indices in [1, 6, 20, 21].

2.3. Singular integral operators, Toeplitz and Hankel operators. Let M_{φ} be the operator of multiplication by a function φ ∈ L^{∞}. The Calderón-Mitjagin interpolation theorem (see, e.g., [20, Theorem 2.a.10]) implies that M_{φ} is bounded on arbitrary rearrangement-invariant space and

∥M_{φ}∥_{L(X)} ≤ ∥φ∥_{∞}. (2.1)

The Cauchy singular integral operator S is bounded on a rearrangement-invariant space X if and only if X has nontrivial Boyd indices (see, e.g., [18, Ch. 2, Section 8.6] and also [1, Ch. 3, Theorem 5.18]).

Lemma 2.1 (see [16, Lemma 4.2 and Proposition 4.3]). If X is a reflexive rearrangement-invariant space with nontrivial Boyd indices, then the operators P_{+} and P_{−} given by (1.2) are bounded projections on X and on X′ and their norms are equal

γ := ∥P_{+}∥_{L(X)} = ∥P_{−}∥_{L(X)} = ∥P_{+}∥_{L(X′)} = ∥P_{−}∥_{L(X′)}.

The exact value of γ for Lebesgue spaces L^{p}, 1 < p < ∞, was recently found by B. Hollenbeck and I. E. Verbitsky [11, Theorem 2.1]: γ = γ_{L^{p}} = 1/ sin(π/p). A lower estimate of γ for an arbitrary reflexive rearrangement-invariant space with nontrivial Boyd indices was obtained in [14, Theorem 4.5]. The exact value of this constant is unknown even for reflexive Orlicz spaces.

In the following we will always assume that X is a reflexive rearrangementinvariant space with nontrivial Boyd indices.

For a set F ⊂ L^{∞}, put \overline{F} := \{\overline{f} : f ∈ F\}. Let H^{∞} be the Hardy space of all bounded analytic functions in the open unit disk 𝔻 := {z ∈ ℂ : |z| < 1}. Consider the Toeplitz operators

T_{φ}^{+} := P_{+}M_{φ}P_{+}, T_{φ}^{−} := P_{−}M_{φ}P_{−}

and the Hankel operators

H_{φ}^{+} := P_{−}M_{φ}P_{+}, H_{φ}^{−} := P_{+}M_{φ}P_{−}.

Their norms admit the following estimates.

Theorem 2.2 (see [16, Corollaries 4.6 and 5.10]). If φ ∈ L^{∞}, then

(2.2)  ∥φ∥_{∞} ≤ ∥T_{φ}^{+}∥_{L(X)} ≤ γ^{2}∥φ∥_{∞},
(2.3)  ∥φ∥_{∞} ≤ ∥T_{φ}^{−}∥_{L(X)} ≤ γ^{2}∥φ∥_{∞},
(2.4)  inf_{ψ∈H^{∞}} ∥φ − ψ∥_{∞} ≤ ∥H_{φ}^{+}∥_{L(X)} ≤ γ^{2} inf_{ψ∈H^{∞}} ∥φ − ψ∥_{∞},
(2.5)  inf_{ψ∈\overline{H^{∞}}} ∥φ − ψ∥_{∞} ≤ ∥H_{φ}^{−}∥_{L(X)} ≤ γ^{2} inf_{ψ∈\overline{H^{∞}}} ∥φ − ψ∥_{∞}.

3. Nikolski ideals associated with Douglas algebras

3.1. Definition of the Nikolski ideals. Consider the set of all inner functions, that is, the set

𝔅 := {b ∈ H^{∞} : |b(t)| = 1 a.e. on 𝕋}.

A Banach subalgebra 𝔸 of L^{∞} generated by H^{∞} and \overline{B} with B ⊂ 𝔅 is called a Douglas algebra (see, e.g., [8, Ch. 9, Section 1]). For a Douglas algebra 𝔸, put

𝔅_{𝔸} := \{b ∈ 𝔅 : \overline{b} ∈ 𝔸\}, 𝔸_{+} := 𝔸, 𝔸_{−} := \overline{𝔸}, Q_{𝔸} := 𝔸_{+} ∩ 𝔸_{−}.

The following characteristic property of the Douglas algebras can be easily deduced from the definition.

Lemma 3.1. A function f ∈ L^{∞} belongs to a Douglas algebra 𝔸 if and only if for every ε > 0 there exist h ∈ H^{∞} and b ∈ 𝔅_{𝔸} such that ∥f − h\overline{b}∥_{∞} < ε.

Example 3.2 (see [8, Ch. 9, Sections 1–2]).

(a) If 𝔸 = L^{∞}, then 𝔅_{𝔸} = 𝔅 and Q_{𝔸} = L^{∞};

(b) if 𝔸 = H^{∞}, then 𝔅_{𝔸} = 𝕋 and Q_{𝔸} = ℂ;

(c) if 𝔸 = H^{∞}+C, then 𝔅_{𝔸} is the set of all finite Blaschke products and Q_{𝔸} = QC is the algebra of all quasicontinuous functions.

For a Douglas algebra 𝔸, following [9, Section 2], put

(3.1)  𝔍^{−}(𝔸) := {F ∈ L(X) : inf_{b∈𝔅_{𝔸}} ∥P_{−}M_{b}F ∥_{L(X)} = 0},
(3.2)  𝔍^{+}(𝔸) := \{F ∈ L(X) : inf_{b∈𝔅_{𝔸}} ∥P_{+}M\overline{b}F ∥_{L(X)} = 0\},
(3.3)  𝔍(𝔸) := 𝔍^{−}(𝔸) ∩ 𝔍^{+}(𝔸).

If 𝔸_{1}, 𝔸_{2} are Douglas algebras and 𝔸_{1} ⊂ 𝔸_{2}, then 𝔅_{𝔸_{1}} ⊂ 𝔅_{𝔸_{2}}. Hence, form the definitions of the sets 𝔍^{±}(𝔸_{i}) and 𝔍(𝔸_{i}), where i = 1, 2, we get

𝔍^{−}(𝔸_{1}) ⊂ 𝔍^{−}(𝔸_{2}), 𝔍^{+}(𝔸_{1}) ⊂ 𝔍^{+}(𝔸_{2}), 𝔍(𝔸_{1}) ⊂ 𝔍(𝔸_{2}).

Lemma 3.3. The sets 𝔍^{−}(𝔸), 𝔍^{+}(𝔸), and 𝔍(𝔸) are closed right ideals in L(X).

Proof. Let F_{1} ∈ 𝔍^{+}(𝔸), F_{2} ∈ L(X), and b ∈ 𝔅_{𝔸}. Then

∥(P_{+}M\overline{b})(F_{1}F_{2})∥_{L(X)} ≤ ∥P_{+}M\overline{b}F_{1}∥_{L(X)}∥F_{2}∥_{L(X)}.

Taking the infimum over all b ∈ 𝔅_{𝔸}, we get

inf_{b∈𝔅_{𝔸}} ∥(P_{+}M\overline{b})(F_{1}F_{2})∥_{L(X)} ≤ ∥F_{2}∥_{L(X)} ( inf_{b∈𝔅_{𝔸}} ∥P_{+}M\overline{b}F_{1}∥_{L(X)}) = 0.

Hence, F_{1}F_{2} ∈ 𝔍^{+}(𝔸), that is, 𝔍^{+}(𝔸) is a right ideal.

Now we prove that 𝔍^{+}(𝔸) is closed. Let F ∈ L(X) and let {F_{n}}_{n=1}^{∞} ⊂ 𝔍^{+}(𝔸) satisfy

lim_{n→∞} ∥F − F_{n}∥_{L(X)} = 0.

Given ε > 0, we choose N ∈ ℕ such that

(3.4) ∥F − F_{n}∥_{L(X)} < \frac{ε}{2∥P_{+}∥_{L(X)}} for every n > N.

Take m > N. Since F_{m} ∈ 𝔍^{+}(𝔸), by the definition of 𝔍^{+}(𝔸), there exists b ∈ 𝔅_{𝔸} such that

∥P_{+}M\overline{b}F_{m}∥_{L(X)} < \frac{ε}{2}. (3.5)

Then, taking into account (3.4), (3.5), and (2.1), we get

∥P_{+}M\overline{b}F ∥_{L(X)} ≤ ∥P_{+}M\overline{b}(F − F_{m})∥_{L(X)} + ∥P_{+}M\overline{b}F_{m}∥_{L(X)}

≤ ∥P_{+}∥_{L(X)}∥M\overline{b}∥_{L(X)}∥F − F_{m}∥_{L(X)} + ∥P_{+}M\overline{b}F_{m}∥_{L(X)}

< \frac{ε}{2} + \frac{ε}{2} = ε.

Since ε is arbitrary, this means that

inf_{b∈𝔅_{𝔸}} ∥P_{+}M\overline{b}F ∥_{L(X)} = 0.

Thus, F ∈ 𝔍^{+}(𝔸), which shows that 𝔍^{+}(𝔸) is closed.

In the case of 𝔍^{−}(𝔸) the proof is similar. Since 𝔍^{−}(𝔸) and 𝔍^{+}(𝔸) are closed right ideals, 𝔍(𝔸) = 𝔍^{−}(𝔸) ∩ 𝔍^{+}(𝔸) is also a closed right ideal of L(X). □

We use here the terminology of [9, Section 2] and call the ideals 𝔍^{±}(𝔸) and 𝔍(𝔸) as the Nikolski ideals associated with the Douglas algebra 𝔸. Analogous ideals were used by N. K. Nikolski [26, 27, 28] for studying of Toeplitz and Hankel operators on the Hardy space H^{2}.

3.2. Properties of the Nikolski ideals. In this subsection we study properties of Nikolski ideals.

Lemma 3.4. Let 𝔸 be a Douglas algebra and let F ∈ L(X). Then

(a) P_{∓}F ∈ 𝔍^{±}(𝔸);

(b) F ∈ 𝔍^{±}(𝔸) if and only if P_{±}F ∈ 𝔍^{±}(𝔸);

(c) F ∈ 𝔍(𝔸) if and only if P_{+}F ∈ 𝔍(𝔸) and P_{−}F ∈ 𝔍(A);

(d) if F ∈ 𝔍(𝔸), then SF ∈ 𝔍(𝔸).

Proof. (a) If b ∈ 𝔅_{𝔸}, then b ∈ H^{∞} and \overline{b} ∈ \overline{H^{∞}}. From (2.5) and (2.4) we deduce that, respectively,

P_{+}M\overline{b}P_{−}F = 0, P_{−}M_{b}P_{+}F = 0. (3.6)

Then from (3.6) and the definition of 𝔍^{+}(𝔸) and 𝔍^{−}(𝔸) we get P_{−}F ∈ 𝔍^{+}(𝔸) and P_{+}F ∈ 𝔍^{−}(𝔸), respectively. Part (a) is proved.

(b) From (3.6) it follows that

P_{+}M\overline{b}F = P_{+}M\overline{b}P_{+}F, P_{−}M_{b}F = P_{−}M_{b}P_{−}F. (3.7)

From (3.7) and the definition of 𝔍^{±}(𝔸) we infer that F ∈ 𝔍^{±}(𝔸) if and only if P_{±}F ∈ 𝔍^{±}(𝔸). Part (b) is proved.

(c) Necessity. By Part (a), P_{−}F ∈ 𝔍^{+}(𝔸) and P_{+}F ∈ 𝔍^{−}(𝔸). Due to Part (b), if F ∈ 𝔍(𝔸) = 𝔍^{−}(𝔸) ∩ 𝔍^{+}(𝔸), then P_{+}F ∈ 𝔍^{+}(𝔸) and P_{−}F ∈ 𝔍^{−}(𝔸). Thus, P_{−}F and P_{+}F belong to 𝔍(𝔸). Necessity of (c) is proved.

Sufficiency. If P_{−}F and P_{+}F belong to 𝔍(𝔸), then by Part (b), F ∈ 𝔍^{−}(𝔸) and F ∈ 𝔍^{+}(𝔸). Thus, F ∈ 𝔍(𝔸). Part (c) is proved.

(d) By Part (c), if F ∈ 𝔍(𝔸), then P_{+}F and P_{−}F belong to 𝔍(𝔸). Hence, SF = P_{+}F − P_{−}F ∈ 𝔍(𝔸). □

Lemma 3.5. Let 𝔸 be a Douglas algebra.

(a) If f ∈ H_{∓}^{∞} and F ∈ 𝔍^{±}(𝔸), then M_{f}F ∈ 𝔍^{±}(𝔸).

(b) Suppose f ∈ 𝔅_{𝔸}. If F ∈ 𝔍^{−}(𝔸) (resp. F ∈ 𝔍^{+}(𝔸)), then M\overline{f}F ∈ 𝔍^{−}(𝔸) (resp. M_{f}F ∈ 𝔍^{+}(𝔸)).

(c) If f ∈ 𝔸_{∓} and F ∈ 𝔍^{±}(𝔸), then M_{f}F ∈ 𝔍^{±}(𝔸).

Proof. (a) If f(τ) = 0 a.e. on 𝕋 and F ∈ 𝔍^{±}(𝔸), then M_{f}F = 0 ∈ 𝔍^{±}(𝔸).

Suppose f ∈ H_{−}^{∞} \ {0} and F ∈ 𝔍^{+}(𝔸). Then for any ε > 0 there exists b ∈ 𝔅_{𝔸} such that

∥P_{+}M\overline{b}F ∥_{L(X)} < \frac{ε}{∥P_{+}∥_{L(X)}∥f∥_{∞}} . (3.8)

Since f ∈ H_{−}^{∞} = \overline{H^{∞}}, from (2.5) we get P_{+}M_{f}P_{−} = 0. Therefore,

P_{+}M\overline{b}M_{f}F = P_{+}M_{f}(P_{+} + P_{−})M\overline{b}F = P_{+}M_{f}P_{+}M\overline{b}F. (3.9)

From (3.8), (3.9), and (2.1) we get

∥P_{+}M\overline{b}M_{f}F ∥_{L(X)} ≤ ∥P_{+}∥_{L(X)}∥M_{f}∥_{L(X)}∥P_{+}M\overline{b}F ∥_{L(X)} < ε.

Since ε > 0 is arbitrary, the latter inequality means that M_{f}F ∈ 𝔍^{+}(𝔸).

Analogously, one can prove that f ∈ H_{+}^{∞} and F ∈ 𝔍^{−}(𝔸) imply M_{f}F ∈ 𝔍^{−}(𝔸). Part (a) is proved.

(b) Suppose f ∈ 𝔅_{𝔸} and F ∈ 𝔍^{+}(𝔸). Then for any ε > 0 there exists b ∈ 𝔅_{𝔸} such that ∥P_{+}M\overline{b}F ∥_{L(X)} < ε. Since b ∈ 𝔅_{𝔸} and f ∈ 𝔅_{𝔸}, we have bf ∈ 𝔅_{𝔸}. Therefore, for F_{1} = M_{f}F and any ε > 0 there exists b_{1} = bf ∈ 𝔅_{𝔸} such that

∥P_{+}M\overline{b_{1}}F_{1}∥_{L(X)} = ∥P_{−}M\overline{bf}M_{f}F ∥_{L(X)} = ∥P_{+}M\overline{b}F ∥_{L(X)} < ε.

Since ε is arbitrary, the latter inequality means that F_{1} = M_{f}F ∈ 𝔍^{+}(𝔸).

Analogously one can show that f ∈ 𝔅_{𝔸} and F ∈ 𝔍^{−}(𝔸) imply M\overline{f}F ∈ 𝔍^{−}(𝔸). Part (b) is proved.

(c) Suppose f ∈ 𝔸_{−} = \overline{𝔸} and F ∈ 𝔍^{+}(𝔸). By Lemma 3.1, for every ε > 0 there exists b ∈ 𝔅_{𝔸} and h ∈ H_{−}^{∞} such that

∥\overline{f} − \overline{bh}∥_{∞} = ∥f − hb∥_{∞} < ε. (3.10)

In view of Part (a), since h ∈ H_{−}^{∞}, we have M_{h}F ∈ 𝔍^{+}(𝔸). Further, by Part (b), M_{bh}F = M_{b}(M_{h}F) ∈ 𝔍^{+}(𝔸). From (3.10) and (2.1) it follows that

∥M_{f}F − M_{bh}F ∥_{L(X)} ≤ ∥f − bh∥_{∞}∥F ∥_{L(X)} < ε∥F ∥_{L(X)}. (3.11)

Since M_{bh}F ∈ 𝔍^{+}(𝔸), Lemma 3.3 and (3.11) imply that M_{f}F ∈ 𝔍^{+}(𝔸).

Analogously, one can show that f ∈ 𝔸_{+} and F ∈ 𝔍^{−}(𝔸) imply M_{f}F ∈ 𝔍^{−}(𝔸). Part (c) and the lemma are proved. □

Lemma 3.6. Let 𝔸 be a Douglas algebra.

(a) If φ ∈ 𝔸_{±}, then H_{φ}^{±} ∈ 𝔍(𝔸).

(b) If φ ∈ Q_{𝔸}, then H_{φ}^{±} ∈ 𝔍(𝔸).

Proof. (a) Let φ ∈ 𝔸_{−} = \overline{𝔸}. By Lemma 3.1, for any ε > 0 there exist h ∈ H_{−}^{∞} and b ∈ 𝔅_{𝔸} such that ∥\overline{φ} − \overline{h}\overline{b}∥_{∞} = ∥φ − hb∥_{∞} < ε. In view of (3.7),

P_{+}M\overline{b}H_{φ}^{−} = P_{+}M\overline{b}P_{+}M_{φ}P_{−} = P_{+}M\overline{b}M_{φ}P_{−} = H_{\overline{b}φ}^{−}. (3.12)

From (3.12) and (2.5) it follows that

∥P_{+}M\overline{b}H_{φ}^{−}∥_{L(X)} = ∥H_{\overline{b}φ}^{−}∥_{L(X)} ≤ γ^{2} inf_{ψ∈H_{−}^{∞}} ∥\overline{b}φ − ψ∥_{∞}. (3.13)

Since |b(τ)| = 1 a.e. on 𝕋, we have

inf_{ψ∈H_{−}^{∞}} ∥\overline{b}φ − ψ∥_{∞} = inf_{ψ∈H_{−}^{∞}} ∥φ − bψ∥_{∞} ≤ ∥φ − hb∥_{∞} < ε. (3.14)

Combining (3.13) and (3.14), we infer that for any ε > 0 there exists b ∈ 𝔅_{𝔸} such that ∥P_{+}M\overline{b}H_{φ}^{−}∥_{L(X)} < γ^{2}ε. This means that

inf_{b∈𝔅_{𝔸}} ∥P_{+}M\overline{b}H_{φ}^{−}∥_{L(X)} = 0,

that is, H_{φ}^{−} ∈ 𝔍^{+}(𝔸). On the other hand, applying Lemma 3.4(a) to F = M_{φ}P_{−}, we obtain H_{φ}^{−} = P_{+}(M_{φ}P_{−}) ∈ 𝔍^{−}(𝔸). Thus, H_{φ}^{−} ∈ 𝔍^{−}(𝔸) ∩ 𝔍^{+}(𝔸) = 𝔍(𝔸).

The proof for φ ∈ 𝔸_{+} is similar. Part (a) is proved.

Statement (b) is a direct consequence of (a) because Q_{𝔸} = 𝔸_{−} ∩ 𝔸_{+}. □

Corollary 3.7. For every φ ∈ L^{∞}, we have M_{φ}P_{∓} ∈ 𝔍^{±}(L^{∞}).

Proof. From the definitions of T_{φ}^{±} and H_{φ}^{±} it follows that

M_{φ}P_{∓} = T_{φ}^{∓} + H_{φ}^{∓}. (3.15)

In view of Example 3.2(a), L^{∞} = Q_{L^{∞}}. Then, by Lemma 3.6(a),

H_{φ}^{±} ∈ 𝔍(L^{∞}) = 𝔍^{−}(L^{∞}) ∩ 𝔍^{+}(L^{∞}). (3.16)

Applying Lemma 3.4(a) to F_{1} = M_{φ}P_{−} and to F_{2} = M_{φ}P_{+}, we get

T_{φ}^{−} = P_{−}F_{1} = P_{−}M_{φ}P_{−} ∈ 𝔍^{+}(L^{∞}), T_{φ}^{+} = P_{+}F_{2} = P_{+}M_{φ}P_{+} ∈ 𝔍^{−}(L^{∞}), (3.17)

respectively. Combining (3.15)–(3.17), we obtain M_{φ}P_{∓} ∈ 𝔍^{±}(L^{∞}). □

Theorem 3.8. Let Ω be a Banach subalgebra of L^{∞} and let 𝔸 be a Douglas algebra. If Ω ⊂ Q_{𝔸}, then Alg (Ω, S) ∩ 𝔍(𝔸) is a closed two-sided ideal of Alg (Ω, S).

Proof. From Lemma 3.3 it follows that Alg (Q_{𝔸}, S) ∩ 𝔍(𝔸) is a closed right ideal of Alg (Q_{𝔸}, S). On the other hand, if F ∈ 𝔍(𝔸), then, by Lemma 3.4(b), SF ∈ 𝔍(𝔸). If f ∈ Q_{𝔸} = 𝔸_{−} ∩ 𝔸_{+} and F ∈ 𝔍(𝔸) = 𝔍^{−}(𝔸) ∩ 𝔍^{+}(𝔸), then, due to Lemma 3.5(c), M_{f}F ∈ 𝔍^{−}(𝔸)∩𝔍^{+}(𝔸). This means that for every F ∈ 𝔍(𝔸) and every generator B of Alg (Q_{𝔸}, S) we have BF ∈ 𝔍(𝔸). Therefore, for every F ∈ Alg (Q_{𝔸}, S)∩𝔍(𝔸) and every C ∈ Alg (Q_{𝔸}, S) we have CF ∈ Alg (Q_{𝔸}, S)∩𝔍(𝔸), that is, Alg (Q_{𝔸}, S)∩𝔍(𝔸) is also a left ideal of Alg (Q_{𝔸}, S). Thus, Alg (Q_{𝔸}, S) ∩ 𝔍(𝔸) is a closed two-sided ideal of Alg (Q_{𝔸}, S).

By Lemma 3.3, Alg (Ω, S) ∩ 𝔍(𝔸) is a closed right ideal of Alg (Ω, S).

On the other hand, let F_{1} ∈ Alg (Ω, S) and F_{2} ∈ Alg (Ω, S) ∩ 𝔍(𝔸). Then, obviously, F_{1}F_{2} ∈ Alg (Ω, S). Since Ω ⊂ Q_{𝔸} and Alg (Q_{𝔸}, S) ∩ 𝔍(𝔸) is a closed two-sided ideal of Alg (Q_{𝔸}, S), we have F_{1}F_{2} ∈ Alg (Q_{𝔸}, S) ∩ 𝔍(𝔸). Therefore,

F_{1}F_{2} ∈ Alg (Ω, S) ∩ (Alg (Q_{𝔸}, S) ∩ 𝔍(𝔸)) = Alg (Ω, S) ∩ 𝔍(𝔸),

that is, Alg (Ω, S) ∩ 𝔍(𝔸) is a left ideal of Alg (Ω, S). Thus, Alg (Ω, S) ∩ 𝔍(𝔸) is a closed two-sided ideal of Alg (Ω, S). □

From Example 3.2(b) and the definition of 𝔍(H^{∞}) one can straightforwardly deduce that 𝔍(H^{∞}) = {0}. A more interesting example is the ideal 𝔍(H^{∞} + C).

Lemma 3.9. We have

𝔍(H^{∞} + C) = K(X). (3.18)

Proof. First, we show that

𝔍(H^{∞} + C) ⊂ K(X). (3.19)

Let F ∈ 𝔍(H^{∞} + C). By Lemma 3.4(c), P_{+}F ∈ 𝔍(H^{∞} + C) ⊂ 𝔍^{+}(H^{∞} + C). Therefore, by the definition of 𝔍^{+}(H^{∞} + C) and Example 3.2(c), for an arbitrary ε > 0 there exists a finite Blaschke product b such that

∥P_{+}M\overline{b}P_{+}F ∥_{L(X)} < ε. (3.20)

On the other hand,

P_{+}F − M_{b}H+ ; \overline{b}F = M_{b}(M\overline{b} − P_{−}M\overline{b})P_{+}F = M_{b}P_{+}M\overline{b}P_{+}F. (3.21)

From (3.20), (3.21), and (2.1) it follows that

∥P_{+}F − M_{b}H+ ; \overline{b}F ∥_{L(X)} ≤ ∥M_{b}∥_{L(X)}∥P_{+}M\overline{b}P_{+}F ∥_{L(X)} < ε. (3.22)

Since the finite Blaschke product b is continuous on 𝕋, by [13, Lemma 6.4], the operator M\overline{b}S − SM\overline{b} is compact on X. Hence, the operator

M_{b}H+ ; \overline{b}F = M_{b}P_{−}(M\overline{b}P_{+} − P_{+}M\overline{b})F

is compact on X. From this and (3.22), taking into account that ε is arbitrary, we obtain P_{+}F ∈ K(X). Analogously one can show that P_{−}F ∈ K(X). Thus, F = P_{−}F + P_{+}F ∈ K(X), and we have proved (3.19).

Let F(X) be the ideal of all operators of finite rank on X. Let us show that

F(X) ⊂ 𝔍(H^{∞} + C). (3.23)

Every operator K ∈ F(X) has the form

(Kf)(t) = ∑_{j=1}^{m} a_{j}(t) ∫_{𝕋} b_{j}(τ)f(τ) dτ, t ∈ 𝕋, (3.24)

where a_{j} ∈ X and b_{j} ∈ X′ for j ∈ {1, …, m}. Since X is reflexive, the set P of all trigonometric polynomials is dense in X (see, e.g., [16, Corollary 3.2]). Hence, every operator of the form (3.24) can be approximated in the operator norm by the operators of the form (3.24) with a_{j} ∈ P. This means that it is sufficient to prove that the operator of the form

(K_{i}f)(t) = χ_{i}(t) ∫_{𝕋} f(τ)g(τ) dτ, g ∈ X′, t ∈ 𝕋,

belongs to 𝔍(H^{∞} + C) for every i ∈ ℤ.

Obviously, χ_{j} ∈ {f ∈ C : |f| = 1} ⊂ 𝔅_{H^{∞}+C} for each j ∈ ℤ. For every i ∈ ℤ, we take j_{1}, j_{2} ∈ ℤ such that j_{1} < i < j_{2}. Then P_{−}(χ_{j_{1}}χ_{i}) = 0 and P_{+}(\overline{χ_{j_{2}}}χ_{i}) = 0. Therefore, P_{−}M_{χ_{j_{1}}} K_{i} = 0 and P_{+}M\overline{χ_{j_{2}}} K_{i} = 0. This means that

K_{i} ∈ 𝔍^{−}(H^{∞} + C) ∩ 𝔍^{+}(H^{∞} + C) = 𝔍(H^{∞} + C) for every i ∈ ℤ.

Thus, we have proved (3.23).

Since X is reflexive and its Boyd indices are nontrivial, Corollary 6.11 of [1, Ch. 3] says that every function in X can be approximated in the norm of X by the partial sums of its Fourier series. That is, there exists a sequence of finite-rank operators on X converging strongly to the identity operator. Consequently, every

Algebras of Singular Integral Operators 225

operator in K(X) can be approximated in the operator norm by operators in F(X). On the other hand, by Lemma 3.3, 𝔍(H^{∞}+C) is a closed ideal. Thus, (3.23) implies K(X) ⊂ 𝔍(H^{∞} + C). (3.25) Combining (3.19) and (3.25), we arrive at (3.18). □ 3.3. Estimates for quotient norms. Let N be a closed subspace of L(X). We denote by |F |_{N} the quotient norm of F ∈ L(X) modulo N, that is, the norm of the image of F in the quotient algebra L(X)/N. In other words,

|F |_{N} := inf_{N∈N} ∥F − N∥_{L(X)}.

Theorem 3.10. Let 𝔸 be a Douglas algebra. If φ ∈ L^{∞}, then:

\frac{1}{γ} inf_{ψ∈𝔸_{+}} ∥φ − ψ∥_{∞} ≤ |H_{φ}^{+}|_{𝔍^{−}(𝔸)} ≤ |H_{φ}^{+}|_{𝔍(𝔸)} ≤ γ^{2} inf_{ψ∈𝔸_{+}} ∥φ − ψ∥_{∞}, (3.26)

\frac{1}{γ} inf_{ψ∈𝔸_{−}} ∥φ − ψ∥_{∞} ≤ |H_{φ}^{−}|_{𝔍^{+}(𝔸)} ≤ |H_{φ}^{−}|_{𝔍(𝔸)} ≤ γ^{2} inf_{ψ∈𝔸_{−}} ∥φ − ψ∥_{∞}. (3.27)

Proof. Since 𝔍(𝔸) = 𝔍^{−}(𝔸) ∩ 𝔍^{+}(𝔸), we immediately get |H_{φ}^{+}|_{𝔍^{−}(𝔸)} ≤ |H_{φ}^{+}|_{𝔍(𝔸)}, |H_{φ}^{−}|_{𝔍^{+}(𝔸)} ≤ |H_{φ}^{−}|_{𝔍(𝔸)}. (3.28) By Lemma 3.6(a), if ψ ∈ 𝔸_{−}, then H_{ψ}^{−} ∈ 𝔍(𝔸). Hence, taking into account (2.1),

(3.29) |H_{φ}^{−}|_{𝔍(𝔸)} = inf_{F ∈𝔍(𝔸)} ∥H_{φ}^{−} − F ∥_{L(X)} ≤ inf_{ψ∈𝔸_{−}} ∥H_{φ}^{−} − H_{ψ}^{−}∥_{L(X)}

= inf_{ψ∈𝔸_{−}} ∥H_{φ−ψ}^{−}∥_{L(X)} ≤ inf_{ψ∈𝔸_{−}} (∥P_{+}∥_{L(X)}∥M_{φ−ψ}∥_{L(X)}∥P_{−}∥_{L(X)})

≤ γ^{2} inf_{ψ∈𝔸_{−}} ∥φ − ψ∥_{∞}.

Let us prove that

inf_{ψ∈𝔸_{−}} ∥φ − ψ∥_{∞} ≤ γ|H_{φ}^{−}|_{𝔍^{+}(𝔸)}. (3.30)

For any F ∈ 𝔍^{+}(𝔸) and b ∈ 𝔅_{𝔸} from (3.12) we deduce that P_{+}M\overline{b}(H_{φ}^{−} − F) = H_{\overline{b}φ}^{−} − P_{+}M\overline{b}F. (3.31) Then, taking into account (2.1), from (3.31) we get

(3.32)  γ∥H_{φ}^{−} − F ∥_{L(X)} ≥ ∥P_{+}∥_{L(X)}∥M\overline{b}∥_{L(X)}∥H_{φ}^{−} − F ∥_{L(X)}
≥ ∥H_{\overline{b}f}^{−} − P_{+}M\overline{b}F ∥_{L(X)}
≥ ∥H_{\overline{b}f}^{−}∥_{L(X)} − ∥P_{+}M\overline{b}F ∥_{L(X)}.

Since \overline{H^{∞}} ⊂ \overline{𝔸} = 𝔸_{−} and \overline{b} ∈ 𝔸 for any Douglas algebra 𝔸, from (2.5) we get

(3.33) ∥H_{\overline{b}φ}^{−}∥_{L(X)} ≥ inf_{ψ∈\overline{H^{∞}}} ∥\overline{b}φ − ψ∥_{∞} = inf_{ψ∈\overline{H^{∞}}} ∥φ − bψ∥_{∞} ≥ inf_{ψ∈𝔸_{−}} ∥φ − ψ∥_{∞}.

From (3.32) and (3.33) we obtain for any F ∈ 𝔍^{+}(𝔸) and b ∈ 𝔅_{𝔸},

inf_{ψ∈𝔸_{−}} ∥φ − ψ∥_{∞} ≤ γ∥H_{φ}^{−} − F ∥_{L(X)} + ∥P_{+}M\overline{b}F ∥_{L(X)}. (3.34)

226 Alexei Yu. Karlovich Then (3.34) and (3.2) imply

inf_{ψ∈𝔸_{−}} ∥φ − ψ∥_{∞} ≤ inf_{F ∈𝔍^{+}(𝔸)} ( inf_{b∈𝔅_{𝔸}} (γ∥H_{φ}^{−} − F ∥_{L(X)} + ∥P_{+}M\overline{b}F ∥_{L(X)}))

= inf_{F ∈𝔍^{+}(𝔸)} (γ∥H_{φ}^{−} − F ∥_{L(X)} + inf_{b∈𝔅_{𝔸}} ∥P_{+}M\overline{b}F ∥_{L(X)})

= γ inf_{F ∈𝔍^{+}(𝔸)} ∥H_{φ}^{−} − F ∥_{L(X)} = γ|H_{φ}^{−}|_{𝔍^{+}(𝔸)}.

So, we have proved (3.30).

Combining (3.28)–(3.30), we arrive at (3.27). Inequalities (3.26) are proved similarly to (3.27). □ Theorem 3.11. If φ, ψ ∈ L^{∞}, then

|M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍(L^{∞})} ≥ \frac{1}{γ} max\{∥φ∥_{∞}, ∥ψ∥_{∞}\}. (3.35)

Proof. Let us prove that

|M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍(L^{∞})} ≥ \frac{∥ψ∥_{∞}}{γ} . (3.36)

Let F ∈ 𝔍^{−}(L^{∞}) and b ∈ 𝔅_{L^{∞}}. Then, taking into account (2.1), we get

(3.37)  γ∥M_{ψ}P_{−} − F ∥_{L(X)} ≥ ∥P_{−}∥_{L(X)}∥M_{b}∥_{L(X)}∥M_{ψ}P_{−} − F ∥_{L(X)}
≥ ∥P_{−}M_{b}(M_{ψ}P_{−} − F)∥_{L(X)}
≥ ∥P_{−}M_{b}M_{ψ}P_{−}∥_{L(X)} − ∥P_{−}M_{b}F ∥_{L(X)}
= ∥T_{bψ}^{−}∥_{L(X)} − ∥P_{−}M_{b}F ∥_{L(X)}.

By (2.3), taking into account that |b(τ)| = 1 a.e. on 𝕋, ∥T_{bψ}^{−}∥_{L(X)} ≥ ∥bψ∥_{∞} = ∥ψ∥_{∞}. (3.38) From (3.37) and (3.38), for every F ∈ 𝔍^{−}(L^{∞}) and every b ∈ 𝔅_{L^{∞}}, we get

\frac{∥ψ∥_{∞}}{γ} ≤ ∥M_{ψ}P_{−} − F ∥_{L(X)} + \frac{1}{γ} ∥P_{−}M_{b}F ∥_{L(X)}. (3.39)

From (3.39) and (3.1) we deduce that

(3.40) \frac{∥ψ∥_{∞}}{γ} ≤ inf_{F ∈𝔍^{−}(L^{∞})} ( inf_{b∈𝔅_{L^{∞}}} (∥M_{ψ}P_{−} − F ∥_{L(X)} + \frac{1}{γ} ∥P_{−}M_{b}F ∥_{L(X)}))

= inf_{F ∈𝔍^{−}(L^{∞})} (∥M_{ψ}P_{−} − F ∥_{L(X)} + \frac{1}{γ} inf_{b∈𝔅_{L^{∞}}} ∥P_{−}M_{b}F ∥_{L(X)})

= inf_{F ∈𝔍^{−}(L^{∞})} ∥M_{ψ}P_{−} − F ∥_{L(X)} = |M_{ψ}P_{−}|_{𝔍^{−}(L^{∞})}.

By Corollary 3.7, M_{φ}P_{+} ∈ 𝔍^{−}(L^{∞}). Therefore, |M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍^{−}(L^{∞})} = |M_{ψ}P_{−}|_{𝔍^{−}(L^{∞})}. (3.41) Since 𝔍(L^{∞}) ⊂ 𝔍^{−}(L^{∞}), we have |M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍(L^{∞})} ≥ |M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍^{−}(L^{∞})}. (3.42) Combining (3.40)–(3.42), we arrive at (3.36).

Analogously one can prove that

|M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍(L^{∞})} ≥ \frac{∥φ∥_{∞}}{γ} . (3.43)

From (3.36) and (3.43) we obtain (3.35). □

4. The presymbol of the algebra Alg (Ω, S)

4.1. The construction of a presymbol. For a Banach subalgebra Ω of L^{∞}, we denote by H(Ω) the closed two-sided ideal of Alg (Ω, S) generated by all Hankel operators H_{φ}^{+} and H_{ψ}^{−} with φ, ψ ∈ Ω.

Lemma 4.1. If Ω is a Banach subalgebra of L^{∞}, then H(Ω) = Com Alg (Ω, S).

This lemma follows from the straightforwardly checked identities

M_{φ}M_{ψ} = M_{ψ}M_{φ}, 2(H_{φ}^{+} − H_{φ}^{−}) = M_{φ}S − SM_{φ}, (M_{φ}S − SM_{φ})P_{±} = ±2H_{φ}^{±}.

Lemma 4.2. If Ω is a Banach subalgebra of L^{∞}, then

Com Alg (Ω, S) ⊂ Alg (Ω, S) ∩ 𝔍(L^{∞}).

Proof. In view of Example 3.2(a), we have Ω ⊂ L^{∞} = Q_{L^{∞}}. Due to Lemma 3.6(b), if φ, ψ ∈ Ω, then H_{φ}^{+} ∈ 𝔍(L^{∞}) and H_{ψ}^{−} ∈ 𝔍(L^{∞}). On the other hand, obviously, H_{φ}^{+}, H_{ψ}^{−} belong to Alg (Ω, S). Thus, H(Ω) ⊂ Alg (Ω, S) ∩ 𝔍(L^{∞}). From the latter imbedding and Lemma 4.1 it follows that Com Alg (Ω, S) ⊂ Alg (Ω, S)∩𝔍(L^{∞}). □

Lemma 4.3. Let Ω be a Banach subalgebra of L^{∞}. For any φ, ψ ∈ Ω we have

(4.1) |M_{φ}P_{+} + M_{ψ}P_{−}|_{Com Alg (Ω,S)} ≥ \frac{1}{γ} max\{∥φ∥_{∞}, ∥ψ∥_{∞}\}.

Proof. From Lemma 4.2 it follows that

Com Alg (Ω, S) ⊂ Alg (Ω, S) ∩ 𝔍(L^{∞}) ⊂ 𝔍(L^{∞}).

Therefore,

|M_{φ}P_{+} + M_{ψ}P_{−}|_{Com Alg (Ω,S)} ≥ |M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍(L^{∞})}. (4.2)

On the other hand, by Theorem 3.11,

|M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍(L^{∞})} ≥ \frac{1}{γ} max\{∥φ∥_{∞}, ∥ψ∥_{∞}\}. (4.3)

From (4.2) and (4.3) we get (4.1). □

Let Alg_{0}(Ω, S) denote the linear subspace of Alg (Ω, S) consisting of all operators of the form

F = M_{φ}P_{+} + M_{ψ}P_{−} + K, (4.4)

where φ, ψ ∈ Ω and K ∈ Com Alg (Ω, S).

Lemma 4.4. If Ω is a Banach subalgebra of L^{∞}, then Alg_{0}(Ω, S) is a Banach subalgebra of Alg (Ω, S).

Proof. Let a, b, c, d ∈ Ω and K_{1}, K_{2} ∈ Com Alg (Ω, S). Then

F_{1} := M_{a}P_{+} + M_{b}P_{−} + K_{1}, F_{2} := M_{c}P_{+} + M_{d}P_{−} + K_{2}

belong to Alg_{0}(Ω, S) and

F_{1}F_{2} = M_{ac}P_{+} + M_{bd}P_{−} + K, (4.5)

where

K = (M_{a}P_{+} + M_{b}P_{−})K_{2} + K_{1}(M_{c}P_{+} + M_{d}P_{−})
+ K_{1}K_{2} + M_{a}(H_{d}^{−} − H_{c}^{+}) + M_{b}(H_{c}^{+} − H_{d}^{−}).

From the properties of the two-sided ideal Com Alg (Ω, S) and Lemma 4.1 it follows that K ∈ Com Alg (Ω, S). Therefore, F_{1}F_{2} ∈ Alg_{0}(Ω, S), that is, Alg_{0}(Ω, S) is a subalgebra of Alg (Ω, S).

Now we show that Alg_{0}(Ω, S) is closed in Alg (Ω, S). Suppose that a sequence {F_{n}}_{n=1}^{∞} converges to F ∈ Alg (Ω, S), where

(4.6) F_{n} = M_{φ_{n}}P_{+} + M_{ψ_{n}}P_{−} + K_{n}, K_{n} ∈ Com Alg (Ω, S), φ_{n}, ψ_{n} ∈ Ω.

By Lemma 4.3, for any m, n ∈ ℕ, we have

(4.7)  ∥F_{m} − F_{n}∥_{L(X)} ≥ |M_{φ_{m}−φ_{n}}P_{+} + M_{ψ_{m}−ψ_{n}}P_{−}|_{Com Alg (Ω,S)}

≥ \frac{1}{γ} max\{∥φ_{m} − φ_{n}∥_{∞}, ∥ψ_{m} − ψ_{n}∥_{∞}\}.

Since {F_{n}}_{n=1}^{∞} is a Cauchy sequence in L(X), from (4.7) it follows that {φ_{n}}_{n=1}^{∞} and {ψ_{n}}_{n=1}^{∞} are Cauchy sequences in L^{∞}. But Ω is closed in L^{∞}, therefore, the limits

φ := lim_{n→∞} φ_{n}, ψ := lim_{n→∞} ψ_{n}

belong to Ω and

M_{φ}P_{+} = lim_{n→∞} M_{φ_{n}}P_{+}, M_{ψ}P_{−} = lim_{n→∞} M_{ψ_{n}}P_{−}. (4.8)

Put K := F − (M_{φ}P_{+} + M_{ψ}P_{−}). From (4.6) and (4.8) we get

K = lim_{n→∞} [F_{n} − (M_{φ_{n}}P_{+} + M_{ψ_{n}}P_{−})] = lim_{n→∞} K_{n}. (4.9)

Since Com Alg (Ω, S) is a closed two-sided ideal in Alg (Ω, S), we infer from (4.9) that K ∈ Com Alg (Ω, S). Thus, F ∈ Alg_{0}(Ω, S), which proves that Alg_{0}(Ω, S) is closed in Alg (Ω, S). □

Now we are in a position to prove the main result of this paper.

Theorem 4.5. Let Ω be a Banach subalgebra of L^{∞}. An operator F ∈ Alg (Ω, S) admits a unique representation (4.4), where φ, ψ ∈ Ω and K ∈ Com Alg (Ω, S). The mapping F ↦ (φ, ψ) defines a Banach algebra homomorphism (presymbol)

μ_{Ω} : Alg (Ω, S) → Ω ⊕ Ω.

The kernel ker μ_{Ω} of this homomorphism coincides with Com Alg (Ω, S) and the norm of this homomorphism satisfies the inequality ∥μ_{Ω}∥ ≤ γ.

Proof. Since 0 ∈ Com Alg (Ω, S) and ±1 ∈ Ω, we have

M_{φ} = M_{φ}P_{+} + M_{φ}P_{−} ∈ Alg_{0}(Ω, S), S = P_{+} − P_{−} ∈ Alg_{0}(Ω, S).

So, the generators of Alg (Ω, S) lie in Alg_{0}(Ω, S). Then, in view of Lemma 4.4, Alg_{0}(Ω, S) = Alg (Ω, S). Consequently, every operator F belonging to Alg (Ω, S) admits a representation of the form (4.4).

This representation is unique. Indeed, assume the contrary. Then there exist φ_{1}, φ_{2}, ψ_{1}, ψ_{2} ∈ Ω and K_{1}, K_{2} ∈ Com Alg (Ω, S) such that the triple {φ_{1}, ψ_{1}, K_{1}} does not coincide with the triple {φ_{2}, ψ_{2}, K_{2}} and

F = M_{φ_{1}}P_{+} + M_{ψ_{1}}P_{−} + K_{1} = M_{φ_{2}}P_{+} + M_{ψ_{2}}P_{−} + K_{2}.

Clearly, the situation φ_{1} = φ_{2}, ψ_{1} = ψ_{2}, K_{1} ≠ K_{2} is impossible. Therefore, φ_{1} ≠ φ_{2} or ψ_{1} ≠ ψ_{2}. From Lemma 4.3 it follows that

0 = ∥(M_{φ_{1}} − M_{φ_{2}})P_{+} + (M_{ψ_{1}} − M_{ψ_{2}})P_{−} + K_{1} − K_{2}∥_{L(X)}

≥ |M_{φ_{1}−φ_{2}}P_{+} + M_{ψ_{1}−ψ_{2}}P_{−}|_{Com Alg (Ω,S)}

≥ \frac{1}{γ} max\{∥φ_{1} − φ_{2}∥_{∞}, ∥ψ_{1} − ψ_{2}∥_{∞}\}.

Hence, φ_{1} = φ_{2} and ψ_{1} = ψ_{2}, so we arrive at a contradiction.

The fact that μ_{Ω} is a homomorphism with kernel Com Alg (Ω, S) follows from the first statement of the theorem and the relation (4.5).

Let F ∈ Alg (Ω, S) and F ≠ 0. Then, by just proved, F admits a unique representation F = M_{φ}P_{+} + M_{ψ}P_{−} + K with (φ, ψ) ∈ Ω ⊕ Ω and K ∈ Com Alg (Ω, S). In view of Lemma 4.3,

∥F ∥_{L(X)} ≥ |M_{φ}P_{+} + M_{ψ}P_{−}|_{Com Alg (Ω,S)} ≥ \frac{1}{γ} max\{|φ∥_{∞}, ∥ψ∥_{∞}\}.

Then

\frac{∥μ_{Ω}(F)∥_{Ω⊕Ω}}{∥F ∥_{L(X)}} ≤ \frac{max\{∥φ∥_{∞}, ∥ψ∥_{∞}\}}{\frac{1}{γ} max\{∥φ∥_{∞}, ∥ψ∥_{∞}\}} = γ.

Taking the supremum over all F ≠ 0 in the latter inequality, we get ∥μ_{Ω}∥ ≤ γ. □

Remark 4.6. Theorem 4.1 in [9] contains wrong estimate for ∥μ_{Ω}∥.

4.2. Necessary conditions for Fredholmness. In this subsection we obtain necessary conditions for Fredholmness of an operator F ∈ Alg (Ω, S) in terms of its presymbol.

It is well-known that an operator A is Fredholm if and only if there exists a regularizer R of A, that is, AR−I and RA−I are compact operators (see, e.g., [10, Vol. 1, Section 4.7]). Recall that a Banach subalgebra Ω of L^{∞} is said to be inverse closed in L^{∞} if for every f ∈ Ω satisfying ess inf_{τ∈𝕋} |f(τ)| > 0, we have 1/f ∈ Ω. The

group of invertible elements of an algebra A is denoted by GA.

Theorem 4.7. (a) If F ∈ Alg (L^{∞}, S) is Fredholm, then

μ_{L^{∞}}(F) ∈ G (L^{∞} ⊕ L^{∞}) .

(b) Suppose Ω is an inverse closed subalgebra of L^{∞}. If F ∈ Alg (Ω, S) is Fredholm, then

μ_{Ω}(F) ∈ G (Ω ⊕ Ω) .

Proof. (a) Assume the contrary, that is F ∈ Alg (L^{∞}, S) is Fredholm, but its presymbol μ_{L^{∞}}(F) = (φ, ψ) is not invertible in L^{∞} ⊕ L^{∞}. Then there exists a regularizer R of F and

ess inf_{τ∈𝕋} |φ(τ)| = 0, ess inf_{τ∈𝕋} |ψ(τ)| = 0.

Assume for definiteness that ψ ∉ GL^{∞}, then there exists a set of positive measure on 𝕋 with the characteristic function χ satisfying the inequality

∥χψ∥_{∞} < \frac{1}{γ^{3}∥R∥_{L(X)}} . (4.10)

By the definition of regularizer, we have

F R = I + K_{1} (4.11)

with K_{1} ∈ K(X). In view of Theorem 4.5,

F = M_{φ}P_{+} + M_{ψ}P_{−} + K_{2}, φ, ψ ∈ L^{∞}, K_{2} ∈ Com Alg (L^{∞}, S). (4.12)

Multiplying (from the left) both sides of (4.11) by P_{−}M_{χ} and taking into account (4.12), we get

P_{−}M_{χ}(M_{φ}P_{+} + M_{ψ}P_{−} + K_{2})R = P_{−}M_{χ} + P_{−}M_{χ}K_{1}.

This equality can be rewritten in the form

T_{χψ}^{−}R = M_{χ}P_{−} + K, (4.13)

where

K = P_{−}M_{χ}K_{1} − P_{−}M_{χ}K_{2}R − H_{χφ}^{+}R + (P_{−}M_{χ} − M_{χ}P_{−}).

In view of Lemma 3.9, K(X) = 𝔍(H^{∞} + C). Since H^{∞} + C ⊂ L^{∞}, we have 𝔍(H^{∞} + C) ⊂ 𝔍(L^{∞}). Therefore, K(X) ⊂ 𝔍(L^{∞}). Since K_{1} ∈ K(X), we deduce that

P_{−}M_{χ}K_{1} ∈ K(X) ⊂ 𝔍(L^{∞}). (4.14)

Further, K_{2} belongs to the closed two-sided ideal Com Alg (L^{∞}, S). Consequently, −P_{−}M_{χ}K_{2}R belongs to Com Alg (L^{∞}, S) too. Taking into account that χ, φ ∈ L^{∞}, from Lemma 4.1 we obtain H_{χφ}^{+} ∈ H(L^{∞}) = Com Alg (L^{∞}, S), whence, −H_{χφ}^{+}R ∈ Com Alg (L^{∞}, S). By the definition of the ideal Com Alg (L^{∞}, S), we get P_{−}M_{χ} − M_{χ}P_{−} ∈ Com Alg (L^{∞}, S). Thus,

−P_{−}M_{χ}K_{2}R − H_{χφ}^{+}R + (P_{−}M_{χ} − M_{χ}P_{−}) ∈ Com Alg (L^{∞}, S). (4.15)

Due to Lemma 4.2, Com Alg (L^{∞}, S) ⊂ 𝔍(L^{∞}). Then from (4.14) and (4.15) we obtain K ∈ 𝔍(L^{∞}). Applying Theorem 3.11, we get

∥M_{χ}P_{−} + K∥_{L(X)} ≥ |M_{χ}P_{−}|_{𝔍(L^{∞})} ≥ \frac{1}{γ} ∥χ∥_{∞} = \frac{1}{γ} . (4.16)

On the other hand, from (4.13), (4.10), and (2.1) we obtain

(4.17)  ∥M_{χ}P_{−} + K∥_{L(X)} = ∥T_{χψ}^{−}R∥_{L(X)} = ∥P_{−}M_{χψ}P_{−}R∥_{L(X)}

≤ γ^{2}∥χφ∥_{∞}∥R∥_{L(X)} < \frac{γ^{2}∥R∥_{L(X)}}{γ^{3}∥R∥_{L(X)}} = \frac{1}{γ} .

Combining (4.16) and (4.17), we arrive at the wrong inequality 1/γ < 1/γ. Therefore, ψ ∈ GL^{∞}. Analogously one can prove that φ ∈ GL^{∞}. Part (a) is proved.

(b) If Ω is inverse closed in L^{∞}, then Ω⊕Ω is inverse closed in L^{∞} ⊕L^{∞}. Clearly, F ∈ Alg (Ω, S) ⊂ Alg (L^{∞}, S) and, by Theorem 4.5, μ_{Ω}(F) = μ_{L^{∞}}(F). Due to Part (a), if F is Fredholm, then μ_{L^{∞}}(F) ∈ G (L^{∞} ⊕ L^{∞}). Since Ω ⊕ Ω is inverse closed in L^{∞} ⊕ L^{∞}, the inclusion μ_{Ω}(F) = μ_{L^{∞}}(F) ∈ G (L^{∞} ⊕ L^{∞}) implies the inclusion μ_{Ω}(F) ∈ G (Ω ⊕ Ω). Part (b) and the theorem is proved. □

4.3. The commutator ideal of the algebra Alg (Ω, S). In this subsection we describe the commutator ideal of the algebra Alg (Ω, S).

Theorem 4.8. Let Ω be a Banach subalgebra of L^{∞} and let 𝔸 be a Douglas algebra. Then Ω ⊂ Q_{𝔸} if and only if

(4.18) Alg (Ω, S) ∩ 𝔍(𝔸) = Com Alg (Ω, S).

Proof. Necessity. By Theorem 3.8, the set Alg (Ω, S) ∩ 𝔍(𝔸) is a closed two-sided ideal of Alg (Ω, S). Moreover, by Lemma 3.6(b), this ideal contains all operators H_{φ}^{+} and H_{ψ}^{−} with φ, ψ ∈ Ω. But Lemma 4.1 states that Com Alg (Ω, S) is the smallest closed two-sided ideal containing all such operators. This means that

(4.19) Com Alg (Ω, S) ⊂ Alg (Ω, S) ∩ 𝔍(𝔸).

To prove the reverse inclusion, assume that F ∈ Alg (Ω, S)∩𝔍(𝔸). By Theorem 4.5, there exist φ, ψ ∈ Ω and K ∈ Com Alg (Ω, S) such that F = M_{φ}P_{+} + M_{ψ}P_{−} + K. Therefore, in view of Lemma 4.2, K ∈ Alg (Ω, S) ∩ 𝔍(L^{∞}). On the other hand, since 𝔸 ⊂ L^{∞}, we have 𝔍(𝔸) ⊂ 𝔍(L^{∞}). Then F ∈ Alg (Ω, S) ∩ 𝔍(L^{∞}). Thus,

M_{φ}P_{+} + M_{ψ}P_{−} = F − K ∈ Alg (Ω, S) ∩ 𝔍(L^{∞}). (4.20)

From Example 3.2(a) and Theorem 3.8 it follows that Alg (Ω, S)∩𝔍(L^{∞}) is a closed two-sided ideal of Alg (Ω, S). Since Alg (Ω, S) ∩ 𝔍(L^{∞}) ⊂ 𝔍(L^{∞}), we have

|M_{φ}P_{+} + M_{ψ}P_{−}|_{Alg (Ω,S)∩𝔍(L^{∞})} ≥ |M_{φ}P_{+} + M_{ψ}P_{−}|_{𝔍(L^{∞})}.

From here and Theorem 3.11 it follows that

|M_{φ}P_{+} + M_{ψ}P_{−}|_{Alg (Ω,S)∩𝔍(L^{∞})} ≥ \frac{1}{γ} max\{∥φ∥_{∞}, ∥ψ∥_{∞}\}. (4.21)

From (4.20) and (4.21) we get φ = ψ = 0 a.e. on 𝕋. Thus, F = K ∈ Com Alg (Ω, S). This means that

(4.22) Alg (Ω, S) ∩ 𝔍(𝔸) ⊂ Com Alg (Ω, S).

Combining (4.19) and (4.22), we arrive at (4.18). Necessity is proved.

Sufficiency. Let φ ∈ Ω. From (4.18) and Lemma 4.1 it follows that

(4.23) H_{φ}^{±} ∈ H(Ω) = Com Alg (Ω, S) = Alg (Ω, S) ∩ 𝔍(𝔸) ⊂ 𝔍(𝔸).

By Theorem 3.10,

inf_{ψ∈𝔸_{+}} ∥φ − ψ∥_{∞} ≤ γ|H_{φ}^{+}|_{𝔍(𝔸)}, inf_{ψ∈𝔸_{−}} ∥φ − ψ∥_{∞} ≤ γ|H_{φ}^{−}|_{𝔍(𝔸)}. (4.24)

From (4.23) it follows that |H_{φ}^{+}|_{𝔍(𝔸)} = |H_{φ}^{−}|_{𝔍(𝔸)} = 0. Therefore, (4.24) implies φ ∈ 𝔸_{+} ∩ 𝔸_{−} = Q_{𝔸}. Thus, Ω ⊂ Q_{𝔸}. □

Corollary 4.9. Let Ω be a Banach subalgebra of L^{∞} and let 𝔸 be a Douglas algebra. If Ω ⊂ Q_{𝔸}, then

(4.25) Com Alg (Ω, S) = Alg (Ω, S) ∩ Com Alg (Q_{𝔸}, S).

Proof. Since Ω ⊂ Q_{𝔸}, we have

(4.26) Alg (Ω, S) = Alg (Ω, S) ∩ Alg (Q_{𝔸}, S).

By Theorem 4.8 and (4.26),

(4.27) Com Alg (Ω, S) = Alg (Ω, S) ∩ 𝔍(𝔸) = (Alg (Ω, S) ∩ Alg (Q_{𝔸}, S)) ∩ 𝔍(𝔸)
= Alg (Ω, S) ∩ (Alg (Q_{𝔸}, S) ∩ 𝔍(𝔸)).

Applying Theorem 4.8 to the trivial situation Q_{𝔸} ⊂ Q_{𝔸}, we get

(4.28) Alg (Q_{𝔸}, S) ∩ 𝔍(𝔸) = Com Alg (Q_{𝔸}, S).

Combining (4.27) and (4.28), we arrive at (4.25). □

4.4. Singular integral operators with quasicontinuous coefficients. In the following lemma we characterize the commutator ideal of Banach algebras Alg (Ω, S) for Ω being a subalgebra between C and QC.

Lemma 4.10.

(a) For a Banach subalgebra Ω of L^{∞} which lies between C and QC,

Com Alg (Ω, S) = K(X). (4.29)

(b) For every Banach subalgebra Ω of L^{∞} such that C ⊂ Ω and QC \ Ω ≠ ∅, the ideal of compact operators K(X) is properly contained in the commutator ideal Com Alg (Ω, S).

Proof. (a) Repeating the proof of [12, Lemma 9.1] (see also [3, Lemma 8.23]), one can show that

K(X) ⊂ Alg (C, S). (4.30)

On the other hand, consider the Douglas algebra 𝔸 = H^{∞} + C. Then Q_{𝔸} = QC (see Example 3.2(c)). By Lemma 3.9, 𝔍(H^{∞} + C) = K(X). Combining these facts with Theorem 4.8, we get

(4.31) Com Alg (Ω, S) = Alg (Ω, S) ∩ 𝔍(H^{∞} + C) = Alg (Ω, S) ∩ K(X).

Since C ⊂ Ω ⊂ QC, we have

(4.32) Alg (C, S) ⊂ Alg (Ω, S) ⊂ Alg (QC, S).

From (4.30)–(4.32) we get

K(X) ⊂ Alg (C, S) ∩ K(X) ⊂ Alg (Ω, S) ∩ K(X) = Com Alg (Ω, S) ⊂ K(X).

Thus, we arrive at (4.29).

(b) Since C ⊂ Ω, we have K(X) = Com Alg (C, S) ⊂ Com Alg (Ω, S). On the other hand, by [17, Theorem 4.1], for a function φ ∈ L^{∞}, the commutator M_{φ}S − SM_{φ} is compact on X if and only if φ ∈ QC. Thus, for φ ∈ Ω \ QC, we have M_{φ}S − SM_{φ} ∈ Com Alg (Ω, S) \ K(X). □

Lemma 4.10(b) shows that if the set QC \ Ω is nonempty, then Theorem 4.7 gives only necessary conditions for Fredholmness of an operator F ∈ Alg (Ω, S). However, in view of Lemma 4.10(a), they become sufficient if Ω lies between C and QC and it is inverse closed in L^{∞}. More precisely, the following criterion is true.

Algebras of Singular Integral Operators 233

Theorem 4.11. Suppose a Banach subalgebra Ω of L^{∞} is inverse closed in L^{∞} and lies between C and QC. An operator F ∈ Alg (Ω, S) admits a unique representation of the form (4.4) with φ, ψ ∈ Ω and K ∈ K(X), and it is Fredholm if and only if

(4.33)  ess inf_{τ∈𝕋} |φ(τ)| > 0, ess inf_{τ∈𝕋} |ψ(τ)| > 0.

Proof. From Theorem 4.5 and Lemma 4.10(a) it follows that F ∈ Alg (Ω, S) admits a unique representation of the form (4.4) with φ, ψ ∈ Ω and K ∈ K(X) = Com Alg (Ω, S). Theorem 4.7 implies that conditions (4.33) are necessary for Fredholmness of F.

Let us prove that these conditions are also sufficient. Since Ω is inverse closed in L^{∞}, conditions (4.33) imply 1/φ, 1/ψ ∈ Ω. Then, taking into account that K(X) = Com Alg (Ω, S), it is not difficult to show that R := M_{1/φ}P_{+} + M_{1/ψ}P_{−} is a regularizer for F. Hence, F is Fredholm. □

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Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049– 001 Lisboa, Portugal

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