MATEMATIQKI VESNIK

59 (2007), 97–111

UDK 517.977

originalni nauqni rad research paper

INVERTIBLE COMPOSITION OPERATORS ON BANACH FUNCTION SPACES

Rajeev Kumar

Abstract. In this paper, we relate composition operators with multiplication operators on the general Banach function spaces on a σ-finite measure space. We use this relation to study the invertibility and Fredholmness properties of composition operators on the Banach function spaces.

1. Introduction

Let Ω = (Ω, Σ, μ) be a σ-finite measure space. By L(μ), we denote the linear space of all equivalence classes of Σ-measurable functions on Ω, where we identify any two functions that are equal μ-a.e. on Ω. Let T: Ω → Ω be a non-singular measurable transformation, that is, T^{−1}(A) ∈ Σ, for each A ∈ Σ and μ(T^{−1}(A)) = 0 for each A ∈ Σ whenever μ(A) = 0. The Radon-Nikodym Theorem ensures the existence of a non-negative locally integrable function f_{T} on Ω so that the measure μ ∘ T^{−1} can be represented as

μ ∘ T^{−1}(A) = ∫_{A} f_{T} (x) dμ(x), for each A ∈ Σ.

Then T defines a well-defined composition transformation C_{T} from L(μ) into itself defined by

C_{T} f(x) = f(T (x)), x ∈ Ω, f ∈ L(μ).

In case C_{T} maps a Banach function space X into itself, then it is called a composition operator on X induced by T.

The study of composition operators on Banach function spaces has been initiated in [16]. For the study of composition operators on L^{p}-spaces and Orlicz spaces (see [7], [18], [25] and the references therein).

AMS Subject Classification: Primary 47B33, 46E30; Secondary 47B07, 46B70.

Keywords and phrases: Banach function spaces, closed range, composition operators, Fredholm operator, invertibility, measurable transformation.

The author is supported by CSIR–grant (sanction no. 9(96)/2002-EMR-I, dated–13-5-2002).

For u ∈ L(μ), we define the multiplication transformation M_{u} on L(μ) as

M_{u}f = uf, for each f ∈ L(μ).

In case M_{u} is continuous on X, we call it a multiplication operator on X. Note that a multiplication operator M_{u} on a Banach function space X is continuous if and only if u ∈ L^{∞}(μ), the Banach space of all essentially bounded measurable functions on Ω.

The study of multiplication operators on the general Banach function spaces has been initiated in [1] and [11]. In [11], the multiplication operators find their applications in semigroup theory in solving the abstract Cauchy problem.

A Banach function space X is a Banach space defined as

X = {f ∈ L(μ) : ∥f∥_{X} < ∞},

where the function norm ∥.∥_{X} on X has the following properties: for each f, g, f_{n} ∈ L(μ), n ≥ 1, we have

(i) 0 ≤| g(x) |≤| f(x) | μ-a.e. x ∈ Ω ⇒ ∥g∥_{X} ≤ ∥f∥_{X}, (ii) 0 ≤| f_{n}(x) |↗| f(x) | μ-a.e. x ∈ Ω ⇒ ∥f_{n}∥_{X} ↗ ∥f∥_{X}, and (iii) E ∈ Σ with μ(E) < ∞ implies that χ_{E} ∈ X, and

∫_{E} |f| dμ ≤ c_{E}∥f∥_{X},

for some constant 0 < c_{E} < ∞ depending on E and the norm ∥.∥_{X} but independent of f. A function f in a Banach function space X is said to have an absolutely continu-

ous norm if ∥fχ_{E_{n}}∥_{X} → 0 for each sequence {E_{n}}_{n=1}^{∞} in Σ satisfying E_{n} → ∅ μ-a.e., that is, χ_{E_{n}} → 0 μ-a.e.

Let X_{a} be the set of all functions in X having absolutely continuous norm. If X = X_{a}, we say that X is a Banach function space having absolutely continuous norm.

Let X_{b} denotes the closure of the set of all μ-simple functions in X. Note that X_{a} ⊆ X_{b} ⊆ X. Throughout this paper, we assume that X = X_{b}, that is, the μ-simple functions are dense in X. In case X has absolutely continuous norm, we have

X_{a} = X_{b} = X

and then its Banach space dual X^{∗} and its associate space X′ coincide, where X′ is also a Banach function space defined as

X′ = {g ∈ L(μ) : ∥g∥_{X′} < ∞},

and

∥g∥_{X′} = sup{∫_{Ω} |f(x)g(x)| dμ(x) : f ∈ X, ∥f∥_{X} ≤ 1}.

The monotone convergence theorem holds in every Banach function space X in the form of the weak Fatou property, by axiom (ii). Note that X_{a} is the largest subspace of X in which the suitable dominated convergence theorem holds (cf., [3, Proposition 3.6]. For details on Banach function spaces, we refer to [3], [19],

[20] and [21].

Note that X = X(N, P (N), μ) indicates a Banach sequence space considered on N with a weight function μ: N → (0, ∞), where P (N) denotes the power set of N. If μ(n) = 1, for each n ∈ N, we write X = X(N).

The separable Banach function spaces form a proper subclass of the absolutely continuous ones. The examples of Banach function spaces having absolutely continuous norm are L^{p}-spaces, Orlicz spaces with Δ_{2}-conditions [23], Lorentz spaces [3], the Orlicz-Lorentz spaces as defined in [10], etc.

By ⟨f, g⟩, we denote the duality pairing between the Banach spaces X and X^{∗}, that is, ⟨f, g⟩ = g(f), for each f ∈ X and g ∈ X^{∗}. In case X is a Banach function space, then for each f ∈ X and g ∈ X′, we have

⟨f, g⟩ = ∫_{Ω} |f(x)g(x)| dμ(x).

We recall from [16, Theorem 2.4] that C_{T} : X → X and C_{T} : X′ → X′ are bounded composition operators on rearrangement invariant spaces X and X′ on a resonant measure space (Ω, Σ, μ) if and only if

μ(T^{−1}(A)) ≤ μ(A), for each A ∈ Σ.

This result forms the base for our assumption that C_{T} maps a Banach function X = X_{b} into itself and also maps the corresponding associate space X′ into itself to prove our main result which is used to prove many important results in the following sections.

Note that the adjoint operator of the composition operator C_{T} on a Banach function space X is C_{T}^{∗} : X^{∗} → X^{∗} such that

⟨f, C_{T}^{∗} g⟩ = ⟨C_{T} f, g⟩,

for each f ∈ X and g ∈ X^{∗}

In section 2, we study composition operators with closed range. In section 3, we discuss the invertibility of composition operators on Banach function spaces and we also give a necessary and sufficient condition for the injectiveness of composition operators on these spaces. We discuss invertible composition operators induced by invertible transformations in section 4. In section 5, we characterise Fredholm composition operators on X and we also give conditions so that the operator C_{T}^{∗} on X^{∗} becomes a composition operator on X′.

2. Composition operators with closed range

In this section, we prove our main result with the help of the following lemma. Note that an operator between Banach spaces has closed range if and only if its adjoint does and the range of the adjoint is closed if and only if it is wk∗-closed.

Lemma 2.1. Suppose u ∈ L(μ) such that M_{u}: X → X. Then M_{u} is one-to-one if and only if u(x) ≠ 0, μ-a.e. x ∈ Ω.

Proof. Suppose that u(x) ≠ 0, μ-a.e. x ∈ Ω. Let g ∈ ker(M_{u}). Then ∥M_{u}g∥_{X} = 0 and by definition, for each E ∈ Σ with μ(E) < ∞, we have

∫_{E} |M_{u}g(x)| dμ(x) ≤ c_{E}∥M_{u}g∥_{X} = 0

which clearly implies that g(x) = 0, μ-a.e. x ∈ Ω, since μ is σ-finite. This proves that g = 0, that is M_{u} is one-to-one.

Let A = {x ∈ Ω : u(x) = 0} have a positive measure. Then, there is a measurable set F ⊆ A with 0 < μ(F) < ∞ so that χ_{F} ∈ X. Clearly, M_{u}χ_{F} = 0 which implies that M_{u} is not injective. This proves the lemma.

Theorem 2.2. (Main result) Let X = X(Ω, Σ, μ) be a Banach function space. Let T : Ω → Ω be a non-singular measurable transformation such that C_{T} : X → X, C_{T} : X′ → X′ and C_{T}^{∗} : X′ → X′ are continuous, ∥f_{T} ∥_{∞} < ∞, and T (Ω) = Ω. Then we have

(i) C_{T}^{∗} C_{T} = M_{f_{T}} on X′. (ii) C_{T} C_{T}^{∗} P = M_{f_{T} ∘T} P on X′, where P : X′ → \overline{range(C_{T})} ⊆ X′ is the projection

operator. (iii) If C_{T} has dense range in X′, then C_{T} C_{T}^{∗} = M_{f_{T} ∘T} on X′. Further, this equality implies that C_{T} has dense range in X.

Proof. (i) For each f ∈ X and g ∈ X′, we have

⟨f, C_{T}^{∗} C_{T} g⟩ = ⟨C_{T} f, C_{T} g⟩ = ∫_{Ω} |f(T (x))g(T (x))| dμ(x)
= ∫_{T (Ω)} |f(y)g(y)|f_{T} (y) dμ(y) = ∫_{Ω} |f(y)(M_{f_{T}} g)(y)| dμ(y)
= ⟨f, M_{f_{T}} g⟩

which implies that C_{T}^{∗} C_{T} = M_{f_{T}} on X′.

(ii) Let P (g) ∈ \overline{range(C_{T})} ⊆ X′, so there exists a sequence \{g_{n}\}_{n≥1} in X′ such that C_{T} g_{n} → P g in norm, that is, ∥C_{T} g_{n} − P g∥_{X′} → 0. Now, we have

C_{T} C_{T}^{∗} P g = lim_{n} C_{T} C_{T}^{∗} (C_{T} g_{n}) = lim_{n} C_{T} (M_{f_{T}} g_{n}) = M_{f_{T} ∘T} lim_{n}(C_{T} g_{n}) = M_{f_{T} ∘T} P g.

Therefore, we have

C_{T} C_{T}^{∗} P = M_{f_{T} ∘T} P on X′.

(iii) Suppose C_{T} has dense range in X′, that is, C_{T} C_{T}^{∗} = M_{f_{T} ∘T}, since then P = I on X′. Now, C_{T} C_{T}^{∗} = M_{f_{T} ∘T} and since f_{T} ∘T ≠ 0 a.e., we see that M_{f_{T} ∘T} is injective so that C_{T} C_{T}^{∗} is also injective. Also, we see that ker(C_{T}^{∗}) ⊆ ker(C_{T} C_{T}^{∗}) = {0}. Therefore, C_{T}^{∗} is also injective.

Thus, we have

\overline{range_{X}(C_{T})} =^{⊥} (ker(C_{T}^{∗})) = X.

Hence C_{T} has dense range in X.

Corollary 2.3. In case C_{T}^{∗} : X → X, then the assertions of the above theorem hold true on X.

Corollary 2.4. If X is a Banach function space such that X = Y′, for some Banach function space Y. Let T be a non-singular measurable transformation such that C_{T} : X → X, C_{T}^{∗} : X → X, then the assertions of the above theorem hold true on X.

The next theorem characterises composition operators with closed range.

Theorem 2.5. Let X be a Banach function space with the associate space X′ and let T be a non-singular measurable transformation such that C_{T} : X′ → X′, C_{T}^{∗} : X′ → X′ and C_{T} and C_{T}^{∗} C_{T} have closed range together in X′. Then C_{T} has closed range in X′ ⊆ X^{∗} if and only if f_{T} is bounded away from 0 on its support.

Proof. By Theorem 2.2, we have C_{T}^{∗} C_{T} = M_{f_{T}} on X′ and so C_{T} has closed range in X′ if and only if M_{f_{T}} has so in X′ if and only if f_{T} is bounded away from 0 a.e. on its support, by [11, Theorem 2.3].

Theorem 2.6. Let T : Ω → Ω be a non-singular measurable transformation satisfying the hypothesis of Theorem 2.5 with Ω = (N, μ). If a = inf μ(n) > 0 and b = sup μ(n) < ∞, then C_{T} has closed range in X′.

Proof. Follows from the above theorem since in this case the Radon-Nikodym function f_{T} is bounded away from 0.

Corollary 2.7. If a > 0, b < ∞ and T is a non-singular transformation on N satisfying the hypothesis of Theorem 2.5 with Ω = (N, μ), then C_{T} has closed range in X = X(N, μ).

Definition 1. A Banach function space X(N, μ) is said to admit composition operators with non-closed range if there exists a measurable transformation T: N → N such that range(C_{T}) is not closed in X(N, μ).

Theorem 2.8. If a > 0 and T is a non-singular transformation on N satisfying the hypothesis of Theorem 2.5 with Ω = (N, μ), then X′(N, μ) admits composition operators with non-closed range if and only if b = ∞.

Proof. Suppose that b < ∞. Then by Theorem 2.6, for every non-singular transformation T such that C_{T} ∈ B(X′(N, μ)), we see that C_{T} has closed range which proves that b = ∞.

Conversely, let b = ∞. Then for each n ∈ N, there exists some m ∈ N such that μ(n)/μ(m) < 1/n. Define a measurable transformation T as T (n) = m. Then, we can choose m such that T is one-to-one and T induces a composition operator on X′(N, μ). Since f_{T} either takes the value μ(n)/μ(m) or 0, for each n ∈ N, so that f_{T} is not bounded below on its support(f_{T}). Hence C_{T} does not have closed range in X′(N, μ).

Theorem 2.9. Let b < ∞ and T be a non-singular transformation on N satisfying the hypothesis of Theorem 2.5 with Ω = (N, μ). Then B(X′(N, μ)) admits composition operators with closed range if and only if a = 0.

Proof. If a > 0, then again by Theorem 2.6, we see that C_{T} has closed range in X′ so that a = 0.

Conversely, if a = 0, choose a subsequence {μ(n_{m})}_{m≥1} of the sequence {μ(n)}_{n≥1} such that ∑_{m≥1} μ(n_{m}) < ∞ and choose another subsequence {μ(n_{m_{i}})}_{i≥1} in {μ(n_{m})}_{m≥1} such that

\frac{μ(n_{m_{i}})}{μ(n_{i})} → 0.

Define T as

T (n) = \{ \begin{matrix}n_{i} & if n = m_{i}, \\ 1 & if n = n_{j}, but n ≠ n_{m_{i}} \\ n & if n ≠ n_{j}. \end{matrix}

Then T induces a composition operator which does not have closed range. This proves the theorem.

3. Invertible composition operators

The next theorem characterizes one-to-one composition operators. For each f ∈ L(μ), we define the essential range of f as

ess ran f = {λ ∈ C : μ(f^{−1}(F)) ≠ 0, for each neighbourhood F of λ}.

Definition 2. A measurable transformation T: Ω → Ω is said to be one-to-one (or left invertible) if there exists a measurable transformation S: Ω → Ω such that

(S ∘ T)(x) = x μ-a.e. x ∈ Ω.

T is said to be onto (or right invertible) if there exists a measurable transformation w: Ω → Ω such that

(T ∘ w)(x) = x μ-a.e. x ∈ Ω.

T is said to be invertible if there exists a measurable transformation S: Ω → Ω such that

(T ∘ S)(x) = (S ∘ T)(x) = x μ-a.e. x ∈ Ω.

Theorem 3.1. Let X be a Banach function space. Let T be a non-singular measurable transformation such that C_{T} : X → X. Then the following are equivalent.

(i) C_{T} is one-to-one on X. (ii) ess ran f = ess ran C_{T} f, for each f ∈ X. (iii) μ ≪ μ ∘ T^{−1}. (iv) f_{T} ≠ 0 a.e.

Proof. The proof of the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) is on similar lines as given in [25, Theorem 2.2.2] for L^{p}-spaces.

(iv) ⇒ (i) Since the simple functions are dense in X, and the simple functions are nothing but the linear combinations of characteristic functions, so it suffices to prove the result for characteristic functions.

Let E ∈ Σ be such that C_{T} χ_{E} = 0 ⇒ χ_{T^{−1}}(E) = 0 a.e. so that μ(T^{−1}(E) ) = 0. Then f_{T} (x) = 0 on E. Therefore, μ(E) = 0 so that χ_{E} = 0, then using the continuity of C_{T} we see that C_{T} is one-to-one. This proves our result.

Corollary 3.2. Let T : Ω → Ω be a non-singular measurable transformation satisfying the hypothesis of Theorem 2.2 on a Banach function space X, then the following are equivalent.

(i) C_{T} is one-to-one on X′. (ii) ess ran f = ess ran C_{T} f, for each f ∈ X′. (iii) μ ≪ μ ∘ T^{−1}. (iv) f_{T} ≠ 0 a.e.

Proof. We prove only (iv) ⇒ (i), since rest of the implications follow from the above theorem. By Theorem 2.2, we have C_{T}^{∗} C_{T} = M_{f_{T}} on X′. Now if f_{T} ≠ 0, then by Lemma 2.1, we see that M_{f_{T}} is one-to-one on X′ which implies that ker C_{T}^{∗} C_{T} = {0}, and ker C_{T} ⊆ ker(C_{T}^{∗} C_{T}). Hence C_{T} is one-to-one on X′.

Corollary 3.3. If T is a non-singular measurable transformation on Ω = (N, P (N), μ), then C_{T} is one-to-one on X = X(N) if and only if T is onto.

Corollary 3.4. Let T be a non-singular measurable transformation on Ω such that ess ran f = ess ran C_{T} f, for each f ∈ X. Then C_{T} f is a characteristic function if and only if f is so.

Theorem 3.5. Let T be a non-singular measurable transformation on Ω such that C_{T} : X → X is continuous. Let T be right invertible (onto) such that the right inverse of T is non-singular. Then C_{T} is one-to-one on X.

Proof. In view of Theorem 2.5, it suffices to prove that μ ≪ μ ∘ T^{−1}. If T is onto, there is a measurable transformation w on Ω such that T ∘ w(x) = x, μ-a.e. x ∈ Ω. Let E ∈ Σ. Then

(T ∘ w)^{−1}(E) = w^{−1}(T^{−1}(E)) = E.

Suppose μ(T^{−1}(E)) = 0, since w is non-singular, we have

μ(w^{−1}(T^{−1}(E))) = 0 ⇒ μ(E) = 0.

This proves that C_{T} is injective on X.

Remark 1. The converse of the above theorem need not be true. For example, let Ω = R, Σ = Borel σ-algebra on R with Lebesgue measure μ. Let T (x) = x^{2}, for each x ∈ R, then T is not onto. But C_{T} is injective on any Banach function space X = X(R), since μ ≪ μ ∘ T^{−1} ≪ μ.

The next theorem characterizes the surjective composition operators. First we prove the next lemma.

Lemma 3.6. Let T be a non-singular measurable transformation on Ω such that C_{T} : X → X is continuous. Then

\overline{range (C_{T})} = X(Ω, T^{−1}(Σ), μ).

Proof. Proof is on the similar lines as in [25, Theorem 2.2.6].

Theorem 3.7. Let X = X(Ω, Σ, μ) be a Banach function space with the associate space X′ and let T be a non-singular measurable transformation on Ω satisfying the hypothesis of Theorem 2.5. Then C_{T} is onto on X′ if and only if there is some c ∈ R such that f_{T} ≥ c > 0 on support(f_{T}) and T^{−1}(Σ) = Σ, where T^{−1}(Σ) = {T^{−1}(E) : E ∈ Σ}.

Proof. Suppose T^{−1}(Σ) = Σ. Then by the above lemma, range(C_{T}) is dense in X. If f_{T} ≥ c > 0 on support(f_{T}), by Theorem 2.5, C_{T} has closed range in X′, which implies that C_{T} is onto on X′.

Conversely, suppose that C_{T} is onto on X′. Then C_{T} has closed range and so by Theorem 2.5, f_{T} is bounded away from 0 on support(f_{T}) ⊆ Ω. The family T^{−1}(Σ) is always a subfamily of Σ. Let E ∈ Σ and μ(E) < ∞, since C_{T} is onto, there is some g ∈ X′ such that C_{T} g = χ_{E}. Let F = {x ∈ Ω : g(x) = 1}. Then

C_{T} χ_{F} = χ_{E} ⇒ T^{−1}(F) = E ⇒ E ∈ T^{−1}(Σ).

Therefore T^{−1}(Σ) = Σ. This proves the result.

Theorem 3.8. Let X be a Banach function space with the associate space X′. Let T be a non-singular measurable transformation on Ω such that C_{T} : X′ → X′ is continuous. Then C_{T} has dense range in X′ if and only if T^{−1}(Σ) = (Σ).

Proof. Let T^{−1}(Σ) = (Σ), then Lemma 3.6 implies that C_{T} has dense range in X′.

Conversely, by Theorem 2.5, if C_{T} has dense range in X′, then f_{T} is bounded way from 0. Also, \overline{C_{T} (X′)} = X′ ⇒ X′(Ω, T^{−1}(Σ), μ) = X′(Ω, Σ, μ) ⇒ T^{−1}(Σ) = Σ. This proves the result.

Corollary 3.9. Under the same hypothesis as in above theorem, if T is one-to-one, then \overline{range(C_{T})} = X′.

Remark 2. The converse of the above theorem is not true in general. For example, let Ω = {a, b, c, d}, Σ = {∅, Ω, {a, b}, {c, d}} and define T: Ω → Ω as

T (a) = a = T (b) and T (c) = c = T (d).

Then, T^{−1}(Σ) = Σ. Thus C_{T} has dense range in X(= l^{2}(N), say) since T^{−1}(Σ) = Σ but T is not one-to-one. Note that here μ ≪ μ ∘ T^{−1} ≪ μ = counting measure.

Theorem 3.10. Let X = X(N, μ) be a Banach function space and let T be a non-singular measurable transformation on Ω such that C_{T} : X → X is continuous. Then C_{T} has dense range in X′ if and only if T is one-to-one.

Proof. The sufficiency follows from Corollary 3.9. To prove the necessary part, suppose T is not one-to-one. Then there exists a set N_{1} containing p (p ≥ 1) elements n_{1}, n_{2}, …, n_{p} such that T (n_{i}) = m for 1 ≤ i ≤ p. We claim that {n_{i}} ∉ T^{−1}(Σ). If {n_{i}} ∈ T^{−1}(Σ), then

{n_{i}} = T^{−1}({n′_{i}}) for some {n′_{i}} ∈ Σ.

Since T ({n_{i}}) = {m}, it follows that {n′_{i}} = {m}. Hence T^{−1}({n′_{i}}) = T^{−1}({m}) or in other words

{n_{i}} = {n_{1}, n_{2}, …, n_{p}},

which is a contradiction. Thus T^{−1}(Σ) ≠ Σ. Hence C_{T} dose not have dense range. This completes the proof of the theorem.

Corollary 3.11. Let X = X(N, μ) be a Banach function space. If a > 0 and b < ∞ and T is a non-singular measurable transformation on N such that C_{T} is continuous on X. Then C_{T} is surjective on X′ if and only if T is one-to-one.

The next theorem characterizes invertible composition operators on the associate space of a Banach function space X.

Theorem 3.12. Let X be a Banach function space with the associate space X′ and T is a non-singular measurable transformation on Ω such that C_{T} is continuous on X and X′ and T (Ω) = Ω. Then C_{T} is invertible on X′ if and only if f_{T} ≥ c > 0 a.e. on Ω and T^{−1}(Σ) = Σ.

Proof. Using Theorem 2.2 and Theorem 3.7, we get the result.

Remark 3. If Σ = {∅, Ω}, then every composition operator on X′ is invertible, for some Banach function space X such that C_{T} : X′ → X′.

Theorem 3.13. Let X be a Banach function space with the associate space X′ and T be a non-singular measurable transformation on Ω such that C_{T} is continuous on X′ and T (Ω) = Ω. Then C_{T} is an injection on X′ if and only if f_{T} ≠ 0 a.e. and T^{−1}(Σ) = Σ.

Proof. Using Theorem 2.2 and Theorem 3.8, we get the result.

Corollary 3.14. Let X = X(N, μ) be a Banach function space and T is a non-singular measurable transformation on N such that C_{T} is continuous on X′. Then C_{T} is an injection on X′ with dense range if and only if T is invertible.

Corollary 3.15. Let X = X(N, μ) be a Banach function space and T is a non-singular measurable transformation on N such that C_{T} is continuous on X′. If a > 0 and b < ∞, then C_{T} is an invertible composition operator on X′ if and only if T is so.

Remark 4. From the above results, we conclude that there are invertible composition operators induced by non-invertible transformations T on Ω.

The next theorem generalizes the result for Orlicz spaces given in [18, Proposition 3.3] on a σ-finite Borel measure space.

Definition 3. A measurable transformation T: Ω → Ω is said to be essentially surjective if μ(Ω \ T (Ω)) = 0.

Theorem 3.16. Let X be a Banach function space. Let T be a non-singular measurable transformation on Ω such C_{T} is continuous on X. Then C_{T} is one-to-one if and only if T is essentially surjective.

Proof. Suppose that C_{T} is injective, that is, ker(C_{T}) = {0}. Let Ω_{∘} = {x ∈ Ω : f_{T} (x) = 0} and X(Ω_{∘}) = {f ∈ X : f(x) = 0 a.e. x ∈ Ω \ Ω_{∘}}

and support(f) = {f ∈ Ω : f(x) ≠ 0}, for each f ∈ X. Then, we have

X(Ω_{∘}) = {f ∈ X : support(f) ⊆ Ω_{∘} a.e.} = {f ∈ X : f_{T} |_{support(f)}= 0}. We claim that X(Ω_{∘}) = {0}. Let f ∈ X(Ω_{∘}) ⊆ X(Ω) ⇒ ∥f∥_{X} < ∞. Then, using the axiom (iii) in the definition of Banach function spaces, we see that

∫_{Ω} |(C_{T} f)(x)| dμ(x) = ∫_{T (Ω)} |f(y)|f_{T} (y) dμ(y)
≤ ∫_{Ω} |f(y)|f_{T} (y) dμ(y)
= ∫_{Ω\Ω_{∘}} |f(y)|f_{T} (y) dμ(y) + ∫_{Ω_{∘}} |f(y)|f_{T} (y) dμ(y)
= ∫_{Ω\Ω_{∘}} |f(y)|f_{T} (y) dμ(y) = 0,

which implies that

C_{T} f(x) = 0 a.e. x ∈ Ω ⇒ f ∈ ker (C_{T}).

Therefore X(Ω_{∘}) ⊆ ker (C_{T}) = {0}. This shows that X(Ω_{∘}) = {0}. Thus μ(Ω_{∘}) = 0. To complete the proof, it suffices to show that Ω \ Ω_{∘} = T (Ω). Now Ω \ Ω_{∘} = support(f_{T}) ⊆ T (Ω), since E ⊆ Ω \ T (Ω). Therefore, we see that

0 = μ(T^{−1}(E)) = ∫_{E} f_{T} (x) dμ(x)

so that f_{T} |_{E}= 0 ⇒ E ⊆ Ω_{∘}. Thus Ω \ T (Ω) ⊆ Ω_{∘}. Therefore, we have

μ(Ω_{∘}) = 0 ⇒ μ(Ω \ T (Ω)) = 0.

This proves that T is essentially surjective.

Conversely, assume that T is essentially surjective so that Ω = T (Ω) ∪ B whenever μ(B) = 0. Then, clearly, we have

ker(C_{T}) = {f ∈ X : C_{T} f = 0} = {f ∈ X : f |_{T (Ω)}= 0} = {0}.

Corollary 3.17. If (Ω, Σ, μ) is a non-atomic measure space, then the nullity of C_{T} on X is either 0 or ∞.

Proof. Follows from X(Ω_{∘}) ⊆ ker(C_{T}) and Ω_{∘} = {x ∈ Ω : f_{T} (x) = 0}.

4. Invertible composition operators induced by invertible transformations

The underlying σ-algebra of measurable sets plays an important role in the invertibility of C_{T}.

Definition 4. A topological space (Y, τ) is said to be an absolute Borel space if it is homeomorphic to a Borel subset of a Hilbert cube (see [22, p. 52] for the definition of a Hilbert cube). If the measure μ is σ-finite on the Borel subsets of an absolute Borel space (Y, τ), then (Y, τ, μ) is called an absolute measure space.

Definition 5. Let (Ω_{1}, Σ_{1}, μ_{1}) and (Ω_{2}, Σ_{2}, μ_{2}) be two measure spaces. Then a transformation h: Ω_{1} → Ω_{2} is said to be a homomorphism if it satisfies the following properties:

h(A ∪ B) = h(A) ∪ h(B), h(A ∩ B) = h(A) ∩ h(B), and h(Ω_{1} \ A) = Ω_{2} \ h(A).

Also, if h(⋃_{i≥1} A_{i}) = ⋃_{i≥1} h(A_{i}), we say that h is a σ-homomorphism. If h is a one-to-one homomorphism, then h is called an automorphism.

Now we need a lemma to prove the main result of this section.

Lemma 4.1. If T is a non-singular measurable transformation on Ω such that C_{T} is an invertible composition operator on a Banach function space X = X_{b}, then C_{T}^{−1} takes characteristic functions into characteristic functions.

Proof. Let E ∈ Σ be such that χ_{E} ∈ X. Then, since C_{T} is onto, there exists a function g ∈ X such that C_{T} g = χ_{E}. Since C_{T} is one-to-one, by Corollary 2.5, we have g = χ_{F} for some F ∈ Σ. Then, we have

C_{T}^{−1}χ_{E} = C_{T}^{−1}C_{T} χ_{F} = χ_{F} .

This proves our assertion.

Theorem 4.2. Let Ω be an absolute measure space and C_{T} is bounded on a Banach function space X = X_{b}. Then C_{T} is an invertible composition operator on X if and only if T is so with non-singular inverse and T^{−1} induces a composition operator.

Proof. Proof is on similar lines as for L^{2}-spaces, see [25, p. 31].

Corollary 4.3. If C_{T} is an invertible composition operator on a Banach function space X, then C_{T}^{−1} = C_{T^{−1}}.

Remark 5. The above theorem is valid for each Banach function space X where every automorphism, or σ-homomorphism of σ-algebra of underlying measure space (Ω, Σ, μ) is induced by a unique point mapping. For examples of such measure space, we refer to [4, p. 123].

5. Fredholm composition operators

The study of Fredholm composition operators has been initiated by Cima, Thomson and Wogen on H^{2}(D)-spaces in [5], they have proved that a composition operator on H^{2}(D) is Fredholm if and only if T is a conformal automorphism of the unit disc D = {z ∈ C : |z| = 1} in the complex plane C. This study has been initiated on L^{2}(μ)-spaces in [14] and [26]. In this section, we generalize this result to the general Banach function spaces.

Definition 6. An operator A ∈ B(Y, Z), the space of all bounded linear operators from Y into Z, where Y and Z are Banach spaces, is said to be a Fredholm operator if range(A) is closed in Z, dim(ker(A)) < ∞ and codim(A) < ∞.

Definition 7. A standard Borel space Ω is a Borel subset of a complete metric space (S, d), where d is metric on a set S. The class Σ will consist of all sets of the form Ω ∩ E, where E is a Borel subset of S.

We recall the next result from [28, p. 6] here.

Proposition 5.1. Let Ω_{1} and Ω_{2} be two standard Borel spaces and T a Borel map of Ω_{1} into Ω_{2}. Let λ be a finite measure on Ω_{1} and q be the measure E ↦ λ(T^{−1}(E)) on Ω_{2}. Then the range T (Ω_{1}) of T is q-measurable and its complement has q-measure 0. Moreover, there exist Borel sets A and Z such that

(i) A ⊆ Ω_{1}, Z ⊆ T (Ω_{1}), (ii) q(Ω_{2} \ Z) = 0, (iii) A is a section for T over Z, that is, T is one-to-one on A and maps A onto Z.

Theorem 5.2. Let X = X(Ω, Σ, μ) be a Banach function space with the associate space X′. Let T : Ω → Ω be a non-singular measurable transformation such that C_{T} is continuous on X. If C_{T} is invertible, then C_{T} is a Fredholm composition operator.

Conversely, if (Ω, Σ, μ) is a σ-finite non-atomic standard Borel measure space, then C_{T} is a Fredholm composition operator on X′.

Proof. If C_{T} is invertible, then clearly C_{T} is a Fredholm operator on X.

Conversely, using Corollary 3.9 and Proposition 5.1, the rest of the proof is on similar lines as in [14, Theorem 1].

Theorem 5.3. Let T be a non-singular measurable transformation such that C_{T} is continuous on a Banach function space X = X(N, μ), then C_{T} is a Fredholm operator on X if and only if range(C_{T}) contains all but finitely many elements of N and restriction of T to the complement of some finite set is one-to-one.

Proof. Using Theorem 2.2, proof is on similar lines as in [14, Theorem 2]. Remark 3. The adjoint of the unilateral shift on l^{2}(N) is an example of a

Fredholm composition operator.

The next theorem shows when does the adjoint of a composition operator become a composition operator.

Theorem 5.4. Let (Ω, Σ, μ) be an absolute measure space and T : Ω → Ω be a non-singular measurable transformation satisfying the hypothesis of Theorem 2.2 such that C_{T} is a composition operator on X and X′. Then the following statements are equivalent.

(i) C_{T} C_{T}^{∗} = I on X′ and C_{T} is invertible on X′. (ii) f_{T} (x) = 1 μ-a.e. and T is one-to-one. (iii) f_{T} (x) = 1 μ-a.e. and C_{T} is invertible on X′. (iv) C_{T}^{∗} is a composition operator on X′.

Proof. (i) ⇒ (ii). By (i), we have

C_{T} C_{T}^{∗} = M_{f_{T}} = I on X′.

Since C_{T} is invertible, we see that

C_{T}^{∗} = C_{T}^{−1}f_{T} = C_{T}^{−1} on X′.  (5.1)

This implies that f_{T} ∘ T^{−1}(x) = T^{−1}(x) μ-a.e. x ∈ Ω and so f_{T} (x) = 1 μ-a.e. x ∈ Ω, using Theorem 4.2.

(i) ⇒ (iv). Using (5.1), we have C_{T}^{∗} = C_{T}^{−1} on X′. Also by Corollary 4.3, we have C_{T}^{∗} = C_{T^{−1}} on X′. This proves that C_{T}^{∗} is a composition operator on X′.

(iv) ⇒ (i). Suppose that C_{T}^{∗} is a composition operator on X′. Then there exists a non-singular measurable transformation S such that C_{T}^{∗} = C_{S}. Since C_{T}^{∗} C_{T} = M_{f_{T}}, we get

C_{S}C_{T} = C_{T ∘S} = M_{f_{T}} .

Let Ω = ⋃_{n=1}^{∞} E_{n}, where μ(E_{n}) < ∞ for every n and E_{n} ⊆ E_{m} if m > n. Let f_{n} = χ_{E_{n}}. Then C_{T ∘S}f_{n} = f_{T} f_{n}, for all n. Equivalently χ_{(T ∘S)^{−1}(E_{n})} = f_{T} χ_{E_{n}} for all n. Since Ω = ⋃_{n=1}^{∞}(T ∘ S)^{−1}(E_{n}), we conclude that f_{T} = 1 a.e. In view of the Theorem 3.12, it is enough to show that T^{−1}(Σ) = Σ to complete the proof. Obviously T^{−1}(Σ) ⊆ Σ. For reverse inclusion, let E ∈ Σ such that μ(E) < ∞. Then, if χ_{E} is in the range of C_{T}, we have χ_{E} = C_{T} h for some h ∈ X. Since C_{T} is one-to-one, by Corollary 3.4, we have h = χ_{F}, for some F ∈ Σ. Hence χ_{E} = C_{T} χ_{F} = χ_{T^{−1}(F)}, which gives E = T^{−1}(F). This shows that E ∈ T^{−1}(Σ). In case χ_{E} does not belong to the range of C_{T}, we can write

χ_{E} = f + g, for some f ∈ X − Ran(C_{T}) and g ∈ Ran(C_{T}).

Let g = C_{T} g_{1} for some g_{1} ∈ X. Then, since C_{T}^{∗} is a composition operator, we have

C_{T}^{∗} χ_{E} = C_{T}^{∗} f + C_{T}^{∗} g = C_{T}^{∗} g = C_{T}^{∗} C_{T} g_{1} = M_{f_{T}} g1 = g_{1}

is a characteristic function. Thus g is a characteristic function. Let g = χ_{G}. Then f = χ_{E} − χ_{G} = χ_{E\G} − χ_{G\E}. Since

⟨f, g⟩ = ⟨χ_{E\G} − χ_{G\E}, χ_{G}⟩ = −⟨χ_{G\E}, χ_{G}⟩ = 0,

so it follows that G ⊆ E. Let F_{1} = T^{−1}(F_{2}) for some F_{2} ∈ Σ, that is, χ_{F_{1}} ∈ Ran(C_{T}), such that E \ G ⊆ F_{1}. Thus

μ((E \ G) ∩ F_{1}) = ⟨f, χ_{F_{1}}⟩ = 0.

This implies that E ⊆ G. Thus we have χ_{E} = χ_{G} = g so that E ∈ T^{−1}(Σ). Now, if E ∈ Σ is of infinite measure, then we can write E = ⋃_{i=1}^{∞} E_{i} where {E_{i}} is a disjoint sequence of measurable sets of finite measure. For each E_{i}, there exists a set F_{i} such that E_{i} = T^{−1}(F_{i}). Therefore

E = ⋃_{i=1}^{∞} E_{i} = ⋃_{i=1}^{∞} T^{−1}(F_{i}) = T^{−1}(⋃_{i=1}^{∞} F_{i}) = T^{−1}(F),

where F = ⋃_{i=1}^{∞} F_{i}. Thus T^{−1}(Σ) = Σ. This completes the proof.

(ii) ⇒ (i). Clearly, by Theorem 2.2 and by (ii), we have C_{T} C_{T}^{∗} = I on X′. By the equivalence of (i) and (iv), we see that C_{T}^{∗} is a composition operator. Therefore, there is a measurable transformation S on Ω such that C_{T}^{∗} = C_{S} is a composition operator on X′. Thus, we see that C_{T} C_{T}^{∗} = I on X′ which implies that T ∘ S(x) = x μ-a.e. x ∈ Ω.

Also, since T is one-to-one, we get a measurable transformation U on Ω such that

(U ∘ T)(x) = x μ-a.e. x ∈ Ω.

Thus, we conclude that T is an invertible transformation on Ω. By Theorem 4.2, we see that C_{T} is invertible on X′.

The equivalence of (i) and (iii) is obvious.

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(received 02.06.2006, in revised form 29.09.2007)

Rajeev Kumar, 73, Vidhata Nagar, Near Bathindi Morh, Jammu–180 006, INDIA.

E-mail: raj1k2@yahoo.co.in