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Composition Operators on Lorentz-Karamata-Bochner Spaces

ANURADHA GUPTA, NEHA BHATIA*

Department of Mathematics, University of Delhi, Delhi -110007, India

Abstract: In this paper, study of the composition operators on Lorentz-Karamata-Bochner spaces and characterization of the properties like boundedness, closedness and essential range of these operators on the space has been made.

Keywords: slowly varying function, Lorentz-Karamata-Bochner spaces, composition operators, closed range, essential range

Received January 10, 2015; Revised February 10, 2015; Accepted February 27, 2015

Citation: ANURADHA GUPTA, NEHA BHATIA, Composition Operators on Lorentz-Karamata-Bochner Spaces. American Journal of Mathematical Analysis, 3 (1). 21-25. January 2015. doi:10.12691/ajma-3-1-5

1. Introduction

Let be a σ-finite measure space. A measurable transformation T is said to be non-singular if whenever for every .

If T is non-singular, then we say that is absolutely continuous with respect to μ. Hence, by Radon-Nikodym theorem there exists a unique non-negative essentially bounded function fT such that for .

Let f be any complex-valued measurable function. For s ≥ 0, the distribution function μf of f is defined as

The non-increasing rearrangement of f is defined as

The maximal (average) operator is given by

One can refer to [4] for the properties of these functions.

Definition 1. A positive and Lebesgue measurable function b is said to be slowly varying (s.v.) on (0, ∞) if, for each , is equivalent to a non-decreasing function and is equivalent to a non-increasing function on (0, ∞).

Given a s.v. function b on (0, ∞), we denote by the positive function defined by

For various properties of slowly varying function we can refer to [4,10].

For 1 < p < ∞, 1 ≤ q < ∞ and for a measurable function f on Ω, define

The Lorentz-Karamata space introduced in [4] is the set of all measurable functions f on Ω such that

Let be a strongly measurable function on a Banach space X. Define a function as

for all . Then the Lorentz-Karamata-Bochner space is a rearrangement invariant-Bochner space for where the norm is given as

The Lorentz-Karamata space is a Banach space and we still have the density of simple functions in it and its dual is

where has the Radon-Nikodym property. For every , we can find a bounded linear functional defined as

for all . For each , there exists a unique measurable function E(g) such that

for each measurable function f for which the left integral exists. E(g) is called the conditional expectation [11] of g with respect to . The operator PT defined as

is called Frobenius Perron and is the Radon-Nikodym derivative of with respect to μ. It satisfies the property

Let T be a non-singular measurable transformation on Ω then the composition operator CT from into the space of strongly measurable functions is given by

for all . An operator T is called Fredholm if R(T) is closed, dim N(T) < ∞ and dim N(T*) < ∞ where R(T), N(T) and N(T*) denote the range, kernel and cokernel of T. B(X) denotes the space of all bounded linear operators on X. Multiplication operators on this space are already studied in [5] and on different spaces in [1,2,3,6,7,8,9,12]. In this paper, we discuss about the composition operators on the Lorentz-Karamata-Bochner space and study its various properties like boundedness, closedness and compactness.

2. Composition Operators

Theorem 2.1. A non-singular transformation induces the composition operator CT if and only if for some k > 0,

for .

Proof. Suppose that the composition operator is bounded on . Then there exists K > 0 such that

Let x₀ be the fixed element of X with . Define the characteristic function for each measurable subset A of by,

Then we find that

and

This gives

and

Thus, we get

Conversely, suppose the given condition holds. Then

and

Thus

Also

Therefore

Thus, CT is a bounded operator on .

Corollary 2.2. A measurable transformation T induces the composition operator CT on if and only if is absolutely continuous with respect to μ and belongs to .

Theorem 2.3. If CT is the composition operator on . Then CT is measure preserving if and only if CT is an isometry.

Proof. Suppose that T is measure preserving them

for all .

The distribution function of CT becomes

and

Also

This gives

Converse of the theorem is obvious.

Example 1. Let with Lebesgue measure and X be any Banach space. Define

Then T is a non-singular transformation on Ω which is not measure preserving. Hence CT is not an isometry on .

Theorem 2.4. If CT is a composition operator on . Then CT has closed range if and only if there exists such that for almost all , the support of .

Proof. Suppose is bounded away from zero then there exists a positive real number , such that

for almost all .

where

and

and

This gives

Thus, CT has closed range.

Conversely, CT has closed range then there exists such that

for all .

Choose a natural number n such that . Let if possible, where . Then . Then

and

This gives

which is a contradiction. Hence is bounded away from zero.

Theorem 2.5. If is a composition operator on . Then CT has dense range in .

Proof. We will consider two cases:

Case 1. When . Then so we can obtain B ∈ such that

Thus CT belong to the range of CT and hence all simple functions of belong to where denotes the range of CT. Hence, range of CT is dense in .

Case 2. When . Let then there is a sequence of functions in converging to g in . Since

Clearly, each gn is measurable and hence g is also measurable. Now suppose . By adjusting f on a set of measure zero, suppose for some . Since is a σ-finite space

where for each n and is an increasing sequence of measurable sets.

This gives

which converges to zero. Thus is dense in

Theorem 2.6. T inducing the composition operator CT on is a surjection if and only if is bounded away from zero on its support and

Proof. Suppose CT is a surjection. Then from the last theorem CT has closed range if and only if is bounded away from zero on its support. Let be of finite measure. Since CT is a surjection there exist such that . Let

Then

Hence,. Thus Thus Converse is obvious.

Corollary 2.7. A composition operator CT on B(Ω, X), has dense range if and only if .

Theorem 2.8. If T induces a composition operator on (Ω, X), then , the adjoint of CT is PT.

Proof. Let be such that . Then for

By identifying with the functional , we get

Theorem 2.9. If T induces a composition operator on B(Ω, X), then is either zero dimensional or infinite dimensional.

Proof. Suppose and . Let

then . Let be a sequence of disjoint measurable subsets of A such that

where . For each , let . For each n,

Therefore is a linearly independent subset of . Hence, if is not zero dimensional, it is infinite dimensional.

Theorem 2.10. If T induces a composition operator on B(Ω, X). Then CT is invertible if and only if CT is Fredholm.

Proof. If CT is invertible then CT is Fredholm. Conversely, let CT be Fredholm then and are both finite dimensional and are of zero dimension. Therefore CT is injective and has dense range. Since is closed, therefore CT is surjective. Thus CT is invertible.

Definition 2. For a strongly measurable function B(X), the set

is called the essential range of f.

Theorem 2.11. [11] If CT is a composition operator on , then the following are equivalent:

(i) CT is injective.

(ii) f and have the same essential ranges for every .

(iii) μ is absolutely continuous with respect to .

(iv) is different from zero almost everywhere.

References

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