1. Introduction
Let be a σfinite measure space. A measurable transformation T is said to be nonsingular if whenever for every .
If T is nonsingular, then we say that is absolutely continuous with respect to μ. Hence, by RadonNikodym theorem there exists a unique nonnegative essentially bounded function f_{T} such that for .
Let f be any complexvalued measurable function. For s ≥ 0, the distribution function μ_{f} of f is defined as
The nonincreasing 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 nondecreasing function and is equivalent to a nonincreasing 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 LorentzKaramata 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 LorentzKaramataBochner space is a rearrangement invariantBochner space for where the norm is given as
The LorentzKaramata space is a Banach space and we still have the density of simple functions in it and its dual is
where has the RadonNikodym 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 P_{T} defined as
is called Frobenius Perron and is the RadonNikodym derivative of with respect to μ. It satisfies the property
Let T be a nonsingular measurable transformation on Ω then the composition operator C_{T} 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 LorentzKaramataBochner space and study its various properties like boundedness, closedness and compactness.
2. Composition Operators
Theorem 2.1. A nonsingular transformation induces the composition operator C_{T} 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 x_{0} 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, C_{T} is a bounded operator on .
Corollary 2.2. A measurable transformation T induces the composition operator C_{T} on if and only if is absolutely continuous with respect to μ and belongs to .
Theorem 2.3. If C_{T} is the composition operator on . Then C_{T} is measure preserving if and only if C_{T} is an isometry.
Proof. Suppose that T is measure preserving them
for all .
The distribution function of C_{T} 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 nonsingular transformation on Ω which is not measure preserving. Hence C_{T} is not an isometry on .
Theorem 2.4. If C_{T} is a composition operator on . Then C_{T} 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, C_{T} has closed range.
Conversely, C_{T} 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 C_{T} has dense range in .
Proof. We will consider two cases:
Case 1. When . Then so we can obtain B ∈ such that
Thus C_{T} belong to the range of C_{T} and hence all simple functions of belong to where denotes the range of C_{T}. Hence, range of C_{T} is dense in .
Case 2. When . Let then there is a sequence of functions in converging to g in . Since
Clearly, each g_{n} 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 σfinite 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 C_{T} on is a surjection if and only if is bounded away from zero on its support and
Proof. Suppose C_{T} is a surjection. Then from the last theorem C_{T} has closed range if and only if is bounded away from zero on its support. Let be of finite measure. Since C_{T} is a surjection there exist such that . Let
Then
Hence,. Thus Thus Converse is obvious.
Corollary 2.7. A composition operator C_{T} 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 C_{T} is P_{T}.
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 C_{T} is invertible if and only if C_{T} is Fredholm.
Proof. If C_{T} is invertible then C_{T} is Fredholm. Conversely, let C_{T} be Fredholm then and are both finite dimensional and are of zero dimension. Therefore C_{T} is injective and has dense range. Since is closed, therefore C_{T} is surjective. Thus C_{T} 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 C_{T} is a composition operator on , then the following are equivalent:
(i) C_{T} 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.
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