MULTIPLIERS OF BANACH VALUED WEIGHTED FUNCTION SPACES

We generalize Banach valued spaces to Banach valuedweighted function spaces and study the multipliers space of these spaces. We also show the relationship between multipliers and tensor product of Banach valued weighted function spaces.


Introduction.
We generalize Banach valued function spaces and its multipliers to the Banach valued weighted function spaces and its multipliers.We also get some elementary results.
Throughout we let G be a locally compact abelian group with Haar measure, X be a Banach space and A be a commutative Banach algebra with identity of norm 1.A weight, Ψ , is a measurable and strictly positive function on G. L p Ψ (G, X) denote the space of equivalence classes of X-valued strongly measurable functions f on G such that Ψ (t)f (t) ∈ L p (G, X).The space L p Ψ (G, X) is a Banach space normed by and its dual space of , where 1/p + 1/p = 1 if and only if X * , the dual space of X, has the wide RNP (Radon-Nikodym Property) (see Lai [2] and Yoshikawa [6]).We assume that our weight function Ψ will satisfy locally bounded and w(x) ≥ 1, also X * and X * * have the wide RNP.It is known that every weight function is equivalent to a continuous weight function and we assume continuity (see Spector [5]).Let C c (G, X) denote the space of all X-valued continuous functions on G with compact support and C o (G, X) the space of all X-valued continuous functions vanishing at infinity of G.For the weight function Ψ , let C Ψ −1 (G, X) denote the space of all functions f such that f Ψ −1 ∈ C o (G, X) and define the norm as A Banach space X is said to be Banach A-module for a Banach algebra A if X is an A-module in the algebraic sense and satisfies ax X ≤ a x X , for all a ∈ A, x ∈ X. (1.4) It is known that if X is a Banach module, then X * , the dual of X, is also a Banach module under the adjoint action of A. If V and W are Banach A-modules, then the A-module tensor product V ⊗ A W is defined to be (V ⊗ γ W )/K, where K is the closed linear subspace in the projective tensor product V ⊗ γ W generated by the elements of the form (1.5) (see Rieffel [4]), had shown that under which a linear functional F on V ⊗ A W corresponds to an operator T ∈ Hom A (V , W * ) such that The operator T , in fact, satisfies so is an A-module homomorphism which is a continuous linear transform from V to W * .We call this A-module homomorphism a multiplier (operator) from V to W * .

Some main properties of
Let G be a locally compact abelian group with Haar measure and X be a Banach A-module.Then the A-valued functions acting on the X-valued functions are well defined.In the X-valued functions, we know that [2]).Now we will show that these properties are satisfied in the X-valued weighted functions.
This completes the proof.
Proposition 2.2.Let A be a commutative Banach algebra with identity of norm 1.
) is a commutative Banach algebra with the convolution we have Using the inequality (1.2) we can obtain easily Using the inequality we obtain the module inequality (2.6)
Analogous to the scalar function case, we can obtain the following proposition. ) Thus by Proposition 3.1 we can define a bilinear map, b, from Definition 3.2.The range of B, with the quotient norm, will be denoted by A p,q Ψ (G, X).Thus A p,q Ψ (G, X) is a Banach space of functions on G which can be viewed as a linear submanifold in L r Ψ (G, X) or C ∞,Ψ −1 (G, X).In view of the fact that every element of where we see that A p,q Ψ (G, X) consists of exactly those functions, h, on G which have at least one expansion of the form where with the expansion converging in the norm of L r Ψ (G, X) or C ∞,Ψ −1 (G, X).As before , we let K denote the closed subspace of From the relations (3.12) We finally obtain .
(3.13) Definition 3.4.We shall say that ) can be approximated in the ultraweak operator topology by operators of the form T φ , φ ∈ C c (G, X).Theorem 3.5.Let 1/p + 1/q ≥ 1.Then the following statements are equivalent: Proof.Suppose now that {L p Ψ (G, X), L q Ψ −1 (G, X * )} satisfies the approximation hypothesis.It is easy to see that K is always contained in the kernel of B using the definition of B.
To show that the kernel of B is contained in K it suffices to show that K ⊥ ⊂ (Ker B) ⊥ .Let F ∈ K ⊥ be given.Because of the isometric isomorphism where h ∈ Ker B, Also it is easy to see that fi * g i ,φ j . (3.17) Using the Hölder inequality, we obtain Ψ (G, X).Suppose conversely that Ker B = K.We will show that the set it is sufficient to show that the corresponding functionals are dense in ] * in the weak*-topology.Let M be the set of the linear functionals which corresponds to the operators in Conversely, if we use (3.17 Corollary 3.6.Let 1/p + 1/q ≥ 1, 1/p + 1/q − 1 = 1/r , and 1/q + 1/q = 1.If {L p Ψ (G, X), L q Ψ −1 (G, X * )} satisfies the approximation hypothesis, then we have the identification (3.23) Remark 3.7.For particular cases of Theorem 3.5, if weight function Ψ = 1 then one can obtain the space of multipliers from L p (G, X) to the space L q (G, X * ).
If Ψ = 1, X = K (K = R or C) the space of multipliers from L p (G) to L q (G), G is a locally compact group, was studied in [4].
If X = K (K = R or C) the space of multipliers from L p Ψ (G) to L q Ψ −1 (G) was studied (for Ψ is a Beurling weight function and G is unimodular group) in [3].

) and b ≤ 1 .
Then b lifts to a linear map, B, from ) and(3.19) we obtain that∞ i=1 fi * g i ,φ = h, F = 0. (3.22)Since φ ∈ C c (G, X) and hence can be viewed as an element of L rΨ −1 (G, X * ) and fi * g i ∈ L r Ψ (G, X) this implies that ∞ i=1 fi * g i = 0. Therefore M ⊥ ⊂ Ker B.Thus M ⊥ = Ker B. This proves the assertion.