MATEMATIQKI VESNIK

60 (2008), 47–51

UDK 517.977

originalni nauqni rad research paper

ASCENT AND DESCENT OF WEIGHTED COMPOSITION OPERATORS ON L^{p}-SPACES

Rajeev Kumar

Abstract. In this paper, we study weighted composition operators on L^{p}-spaces with finite ascent and descent. We also characterize the injective weighted composition operators.

1. Introduction

Let Ω = (Ω, Σ, μ) be a σ-finite measure space. Let L(μ) denotes 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 ν be another measure on the measurable space (Ω, Σ) such that ν(A) = 0 for each A ∈ Σ whenever μ(A) = 0. Then we say that the measure ν is absolutely continuous with respect to the measure μ and we write ν ≪ μ. By Radon-Nikodym Theorem, there exists a non-negative locally integrable function f_{ν} on Ω so that the measure ν can be represented as

ν(A) = ∫_{A} f_{ν}(x) dμ(x), for each A ∈ Σ.

The function f_{ν} is called the Radon Nikodym derivative of the measure ν with respect to the measure μ.

Let T: Ω → Ω be a non-singular measurable transformation, that is, μ∘T^{−1} ≪ μ. Let u: Ω → C be an essentially bounded measurable function. We assume that the support u is the domain of T. Then the linear transformation W = W_{u,T} : L(μ) → L(μ) is defined as

W f = W_{u,T} f = u.f ∘ T, for each f ∈ L(μ),

In case W maps L^{p}(μ) into itself, for p ∈ [1, ∞), we call W = W_{u,T} a weighted composition operator on L^{p}(μ) induced by the pair (u, T).

Note that the pair (u, T) induces a weighted composition operator while T may fail to induce a composition operator on L^{p}(μ). For example if u(y) = 0, for each y ∈ Ω, then W_{u,T} induces a weighted composition operator whether T induces the corresponding composition operator or not.

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

Keywords and phrases: Ascent, descent, measurable transformation, weighted composition operators.

Now, we define a measure μ_{u,T}^{1} on Σ as

μ_{u,T}^{1} = ∫_{T^{−1}(A)} |u(x)|^{q} dμ(x), for each A ∈ Σ.

Clearly μ_{u,T}^{1} ≪ μ ∘ T^{−1} ≪ μ. Let f_{u,T}^{1} denotes the Radon-Nikodym derivative of μ_{u,T}^{1} with respect to μ and let h_{1} = (f_{u,T}^{1} )^{\frac{1}{p}} : Ω → C.

Note that W is a continuous weighted composition operator on L^{p}(μ), for p ∈ [1, ∞) if and only if h_{1} ∈ L^{∞}(μ). For details on the study of weighted composition operators on L^{p}-spaces, see [5, p. 51]. The study of weighted composition operators between two L^{p}-spaces has been initiated in [3]. The interesting study of composition operators on Banach function spaces with finite ascent and finite descent has been initiated in [2].

We also define a measure μ_{u,T}^{2} on Σ as

μ_{u,T}^{2} = ∫_{T^{−1}(A)} |u(x)|^{q} dμ_{u,T}^{1} (x), for each A ∈ Σ.

Clearly μ_{u,T}^{2} ≪ μ_{u,T}^{1} ∘ T^{−1} ≪ μ. Let f_{u,T}^{2} denotes the Radon-Nikodym derivative of μ_{u,T}^{2} with respect to μ and let h_{2} = (f_{u,T}^{2} )^{\frac{1}{p}} : Ω → C.

Definition 1. For a bounded operator A: F → F on a Banach space F, the ascent α(A) of A is the least non-negative integer such that ker(A^{k}) = ker(A^{k+1}) and the descent d(A) of A is the least non-negative integer such that \overline{Ran(A^{k})} = \overline{Ran(A^{k+1})}.

Note that ker(A^{k}) ⊆ ker(A^{k+1}) and Ran(A^{k+1}) ⊆ Ran(A^{k}), for each k ≥ 0. In case α(A) < ∞ and d(A) < ∞, then d = α(A) = d(A) on L^{p}(μ)-spaces, for p ∈ [1, ∞).

We also note that if d = α(A) = d(A) < ∞, then V = ker(A^{d}) and W = Ran(A^{d}), is the only reducing pair for the operator A such that A is nilpotent on V and invertible on W, see [1, p. 81]. In particular, we take A = W = W_{u,T}, a weighted composition operator induced by the pair (u, T).

Definition 2. A standard Borel space Ω is a Borel subset of a complete metric space (S, d), where d is a 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.

In this paper, we give a necessary and sufficient condition for weighted composition operators with ascent 1 and descent 1. We also give a necessary and sufficient condition for the injective weighted composition operators.

Ascent and descent of weighted composition operators on L^{p}-spaces 49

2. Main results

In this section, we prove our main result with the help of the following lemma.

Lemma 2.1. Let W = W_{u,T} be a continuous weighted composition operator on L^{p}(μ), for p ∈ [1, ∞). Then, we have ker(W) = L^{p}(Ω_{∘}), where Ω_{∘} = {x ∈ Ω : f_{u,T}^{1} (x) = 0} and

L^{p}(Ω_{∘}) = { f ∈ X : f(x) = 0 a.e. x ∈ Ω \ Ω_{∘} }.

Proof. For f ∈ L^{p}(μ), the support of f is supp(f) = {x ∈ Ω : f(x) ≠ 0}. Clearly, we have

L^{p}(Ω_{∘}) = { f ∈ L^{p}(μ) : supp(f) ⊆ Ω_{∘} a.e. } = { f ∈ L^{p}(μ) : f_{u,T}^{1} |_{supp(f)} = 0 }.

For f ∈ L^{p}(Ω_{∘}), we have

∥W f∥_{p}^{p} = ∫_{Ω} |W f(x)|^{p} dμ(x) = ∫_{Ω} |f(x)|^{p}f_{u,T}^{1} (x) dμ(x)
= ∫_{Ω\Ω_{∘}} |f(y)|^{p}f_{u,T}^{1} (x) dμ(y) + ∫_{Ω_{∘}} |f(y)|^{p}f_{u,T}^{1} (x) dμ(y) = 0.

Thus f ∈ ker(W) so that L^{p}(Ω_{∘}) ⊆ ker(W).

Conversely, let f ∈ ker(W). Then u.f ∘ T = 0 a.e.. We have

0 = ∫_{Ω} |u(x)|^{p}|f(T (x))|^{p} dμ(x) = ∫_{Ω} |f(x)|^{p}f_{u,T}^{1} (x) dμ(x)

which implies that f_{u,T}^{1} |_{supp(f)} = 0 a.e., so that f ∈ L^{p}(Ω_{∘}). This proves the reverse inclusion.

The next result characterizes the injective weighted composition operators. For this we need the following definition.

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

Theorem 2.2. Let W = W_{u,T} be a continuous weighted composition operator on L^{p}(μ), for 1 ≤ p < ∞. Then W is injective if and only if T is essentially surjective.

Proof. If W is injective, then using Lemma 2.1, we see that L^{p}(Ω_{∘}) = {0}. Thus f_{u,T}^{1} (x) ≠ 0 a.e.. This implies that μ(Ω_{∘}) = 0. Therefore T is essentially surjective.

Now we show that Ω \ Ω_{∘} = T (Ω). Clearly, Ω \ Ω_{∘} = supp(f_{u,T}^{1} ) ⊇ T (Ω). Also, for each E ∈ Σ such that E ⊆ Ω \ T (Ω), we have

0 = μ_{u,T}^{1} (E) = ∫_{E} f_{u,T}^{1} (x) dμ(x),

which implies that f_{u,T}^{1} |_{E} = 0 ⇒ E ⊆ Ω_{∘}. This shows that Ω \ T (Ω) ⊆ Ω_{∘} ⇒ Ω \ Ω_{∘} ⊆ T (Ω). This proves that T (Ω) = Ω \ Ω_{∘}.

Note that we have used the fact that μ_{u,T}^{1} ≪ μ ∘ T^{−1}.

Corollary 2.3. If (Ω, Σ, μ) is a non-atomic measure space, then the nullity of W is either zero or infinite.

Remark. The above results in this section has been proved for composition operators on Orlicz spaces in [4].

The next theorem characterises weighted composition operators with ascent 1.

Theorem 2.4. Let W = W_{u,T} be a continuous weighted composition operator on L^{p}(μ). Then W has ascent 1 if and only if the measures μ_{u,T}^{1} and μ_{u,T}^{2} are equivalent.

Proof. Since W is bounded, we have μ_{u,T}^{2} ≪ μ_{u,T}^{1} ∘ T^{−1} ≪ μ. Then, we have

μ_{u,T}^{2} = ∫_{E} f_{u,T}^{2} (x) dμ(x) = ∫_{E} |u(x)|^{p} dμ_{u,T}^{1} (x), for each E ∈ Σ.

Now, suppose μ_{u,T}^{1} ≪ μ_{u,T}^{2} ≪ μ_{u,T}^{1} . Then, we see that

Ω_{∘} = { x ∈ Ω : f_{u,T}^{1} (x) = 0 } = { x ∈ Ω : f_{u,T}^{2} (x) = 0 }.

Then, by using Lemma 2.1, we have

ker(W) = ker M_{f_{u,T}^{1}} = L^{p}(Ω_{∘}) = ker M_{f_{u,T}^{2}} = ker(W^{2}).

This shows that W is a weighted composition operator with ascent 1.

Conversely, suppose ker(W) = ker(W^{2}). Since ker(W) = L^{p}(Ω_{∘}), where Ω_{∘} = {x ∈ Ω : f_{u,T}^{1} (x) = 0} and ker(W^{2}) = L^{p}(Ω′_{∘}), where Ω′_{∘} = {x ∈ Ω : f_{u,T}^{2} = 0}. We conclude that Ω_{∘} = Ω′_{∘}. Since μ_{u,T}^{1} = ∫_{E} f_{u,T}^{1} (x) dμ(x) and μ_{u,T}^{2} = ∫_{E} f_{u,T}^{2} (x) dμ(x) for each E ∈ Σ. Thus, we have μ_{u,T}^{1} ≪ μ_{u,T}^{2} ≪ μ_{u,T}^{1} . This proves the theorem.

Theorem 2.5. Let (Ω, Σ, μ) be a σ-finite standard Borel space and W is a bounded operator on L^{p}(μ), for p ∈ [1, ∞). Then the operator W has ascent 1 if and only if T [Ω_{1}] ⊇ Ω_{1}, where Ω_{1} = Ω \ Ω_{∘} and Ω_{∘} = {x ∈ Ω : f_{u,T}^{1} (x) = 0}.

Proof. Suppose T [Ω_{1}] ⊇ Ω_{1}. By Lemma 2.1, we have ker(W) = L^{p}(Ω_{∘}). Then L^{p}(Ω) = L^{p}(Ω_{∘}) ⊕ L^{p}(Ω_{1}). Thus each f ∈ ker(W^{2}) can be written as f = f_{1} + g_{1}, where f_{1} ∈ ker(W) and g_{1} ∈ L^{p}(Ω_{1}). Since

0 = W^{2}f = W^{2}(f_{1} + g_{1}) = W^{2}g_{1} = u.u ∘ T.g_{1} ∘ T^{2}

and T [Ω_{1}] ⊇ Ω_{1}, we see that g_{1} = 0 a.e. on Ω_{1}. Then f = f_{1}. Thus ker(W^{2}) ⊆ ker(W). Therefore, we have ker(W) = ker(W^{2}). This implies that W has ascent 1.

Conversely, suppose that T [Ω_{1}] ⊉ Ω_{1}. Suppose E ∈ Σ with E ⊆ Ω_{1} \ T [Ω_{1}] of non zero finite measure such that W^{2}χ_{E} = 0. Since E ⊆ Ω_{1}, we have W χ_{E} ≠ 0, which contradicts the fact that W has ascent 1.

Corollary 2.6. Let W be as above. Then W is of ascent 1 if and only if (T ∘ T)[N] = T (N), where T [N] is the range of N.

The following theorem characterises composition operators with descent 1.

Theorem 2.7. Let W = W_{u,T} be a continuous weighted composition operator on L^{p}(μ), for 1 ≤ p < ∞. Then W has descent 1 if and only if the measures μ_{u,T}^{1} and μ_{u,T}^{2} are equivalent.

Proof. Using Theorem 2.4 and the arguments following the definition 1, the proof is through.

Remark. For the examples of composition operators on L^{p} spaces with finite ascent and finite descent, see [2].

REFERENCES

[1] Y. A. Abramovich and C. D. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics 50, American Mathematical Society, 2002.

[2] Rajeev Kumar, Ascent and descent of composition operators on Banach function spaces, preprint.

[3] Rajeev Kumar, Weighted composition operators between two L^{p}-spaces, preprint.

[4] R. Kumar, Composition operators on Orlicz spaces, Integral Equations and Operator Theory 29 (1997), 17–22.

[5] R. K. Singh and J. S. Manhas, Composition Operators on Function Spaces, North Holland Math. Studies 179, Amsterdam 1993.

[6] H. Takagi, Compact weighted composition operators on L^{p}, Proc. Amer. Math. Soc. 116, 2 (1992), 505–511.

[7] V. S. Varadarajan, Geometry of quantum theory, vol. II, University Series in Higher Mathematics, Van Nostrand Reinhold, New York, 1970.

(received 29.03.2007)

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E-mail: raj1k2@yahoo.co.in, rajeev32002@yahoo.co.in