On Operators of Weighted Composition on Orlicz Sequence Spaces

Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 40, 1961 - 1968

On Operators of Weighted Composition on Orlicz Sequence Spaces Kuldip Raj School of Mathematics Shri Mata Vaishno Devi University Katra-182320, J&K, India [email protected] B. S. Komal and Vinay Khosla Department of Mathematics University of Jammu, J&K, India

Abstract. In this paper we characterize the weighted composition Operators on Orlicz sequence spaces and we also make an efforts to characterize compactness, closed ranged and invertibility of these operators. Mathematics Subject Classification: Primary 47B33, 47B38, Secondary: 46E30 Keywords: Weighted composition operator, compact operator, invertible operator and Orlicz space

Introduction φ Let (Ω, S, μ) be a σ-finite measure space and let L (μ) = {f |f : Ω → C is measurable such that φ(αf )dμ < ∞, for some α > 0}.Then Lφ (μ) is a Ω

Banach space under the norm, ||f ||φ = inf{∈> 0 :

φ(

|f | )dμ ≤ 1}

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K. Raj, B. S. Komal and V. Khosla

where φ : [0, ∞) → [0, ∞) is a continuous convex function which satisfy the following (i) φ(x) = 0 if and only if x = 0 (ii) lim φ(x) = ∞ x→∞ Such a function φ is known as a Young’s function. Let u : Ω → C be a measurable function and v : Ω → Ω be a non-singular measurable transformation. Then a bounded linear transformation, mu,v : Lφ (μ) → Lφ (μ) defined by (mu,v f )(x) = u(x).f (v(x)) is called a weighted composition operator induced by the pair (u, v) . If we take u(x) = 1, the constant one function on Ω, we write Mu,v as Tv and call it a composition operator induced by v. In case v(x) = x for some x ∈ Ω, we write Mu,v as Mu and call it a multiplication operator induced by u. If v is a non-singular measurable transformation, then the measure μv −1 is absolutely continuous with respect to the measure μ. Hence by Radon Nikodym derivative theorem there exists a positive measurable function w −1 such that μ(v (E)) = wdμ for every E ∈ S. The function w is called the Radon Nikodym derivative of the measure μv −1 with respect to the measure −1 . μ. It is denoted by w = dμv dμ Let (Ω, S, μ) be a σ-finite measure space and S0 ⊂ S be a σ-finite subalgebra. Then the conditional expectation E(.|S0 ) is defined as a linear transformation from certain S-measurable function spaces (i.e.L1 , L2 etc) into their S0 -measurable counterparts. In particular the conditional expectation with respect to the σ-algebra v −1 (S) is a bounded projection from Lp (Ω, S, μ) onto Lp (Ω, v −1 (S), μ). We denote this transformation by E. The transfromation E has the following properties. (i)E(f.gov) = E(f ).(gov) (ii) if f ≥ g almost everywhere, then E(f ) ≥ E(g) almost everywhere (iii) E(1) = 1 (iv) E(f) has the form E(f) = gov for exactly one σ- measurable function g. In particular g = E(f )ov −1 is a well defined measurable function. (v) |E(f g)|2 ≤ (E|f |2)(E|g|2). This is a Cauchy Schwartz inequality for conditional expectation. (vi) For f > 0 almost everywhere, E(f ) > 0 almost everywhere. (vii) If φ is a convex function, then φ(E(f )) ≤ E(φ(f )) μ- almost every where. For deeper study of properties of E see [ 7]

On operators of weighted composition

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In this paper we take Ω = N, the set of natural numbers, S = P (N) and μ is the counting measure defined on P (N), the family of all subsets of N. In this case Lφ (μ) is the space of Orlicz sequences and we shall denote it by φ (N).By B(lφ (N)) we denote the set of all bounded linear operators from lφ (N) into itself. For more study on Orlicz spaces see ([5],[8], [9]) where as the study of weighted composition operators on some function spaces are considered by Carleson ([1], [2]), Kamowitz [6],Singh and Kumar ([10],[11])Komal, B.S. Raj Kuldip and Gupta Sunil [4], N. Four and R. Khalil [3]. In this paper we plan to study the weighted composition operators on Orlicz sequence spaces φ (N). 2. Weighted composition on Orlicz- Sequences spaces Theorem 2.1: Let mu,v : φ (N) → φ (N) be a linear transformation.Then mu,v is bounded if and only if {U(k, φ, n)}∞ n=1 is a bounded sequence, where U(k, φ, n) is defined as |u(k)| U(k, φ, n) =

k∈v−1 (n) 1 w(n)φ−1 ( w(n) )

for n ∈ v(N).

Proof: Suppose mu,v is a bounded operator. Then there exists M > 0 such that ||mu,v en ||φ ≤ M||en ||φ ...(i) For n ∈ v(N ), we have n = v(m) and

|u|en ov )dμ ≤ 1} N |u| = inf { > 0 : φ(E( )ov −1 en )wdμ ≤ 1} N |u| = inf { > 0 : φ(E( )ov −1(n))w(n) ≤ 1} |u| = inf { > 0 : φ(E( )(m))w(v(m))}

||mu,v en ||φ = inf { > 0 :

This on simple computation yields that

||mu,v en ||φ =

φ(

|u(k)|

k∈v−1 (n) 1 w(n)φ−1 ( w(n) )

= U(k, φ, n) Conversely, If the sequence {U(k, φ, n)}∞ n=1 is not bounded, then for every positive integer p, there exists np ∈ N such that U(k, φ, np ) > p[ φ−11(1) ], which

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implies that ||mu,v enp ||φ > p||enp ||φ This contradicts the boundedness of mu,v . Hence {U(k, φ, n)}∞ n=1 must be a bounded sequence. Theorem 2.2: Let mu,v ∈ B(lφ (N)). Then Mu,v has closed range if and only if there exists δ > 0 such that u(m) ≤ δ for all n ∈ s m∈v−1 (n)

where s = support u ∩ v(N ). Proof: Suppose first that the condition is true. Let δ = min(δ, 1δ ) and f ∈ ranmu,v . Then there exists a sequence {fi } ⊂ φ (N) such that mu,v fi → f . Therefore we can find a positive integer io such that ||mu,v fi − mu,v fj ||φ < for all i, j ≥ io . Now δ |fi (k) − fj (k)| φ( ) ||m u,v fi − mu,v fj ||φ k∈s

≤

φ(

m∈v−1 (k)

k∈s

≤

k∈s

φ(

u(m)|fi (k) − fj (k)| ) ||mu,v fi − mu,v fj ||φ

|mu,v fi (k) − mu,v fj (k)| ) ||mu,v fi − mu,v fj ||φ

≤ 1 ...(i) Define fi∼

=

fi (k), 0,

if k ∈ s if k ∈ s.

Then mu,v fi∼ = mu,v fi and therefore it follows from (i), that ∞ k=1

φ(

δ |fi∼ (k) − fj∼ (k)| ) ≤ 1. ||mu,v fi∼ − mu,v fj∼ ||φ

Hence 1 ||mu,v fi − mu,v fj ||φ δ → 0 as i → ∞, j → ∞.

||fi∼ − fj∼ ||φ ≤

Thus {fi∼ } is a Cauchy sequence in φ (N) and in view of completeness of φ (N ), there exists g ∈ φ (N) such that fi∼ → g as i → ∞. In otherwords mu,v fi → mu,v g. Hence f = mu,v g. So that mu,v has closed range. If the

sequence {


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u(m)} is not bounded away from zero, then for every k ∈ N,

m∈v−1 (n)

we can find nk ∈ N such that

u(m)
0 there exists an infinite sequence {np } of positive integer such that U(k, φ, np ) > . Hence ||mu,v enp || > for infinitely many np , ......(ii). But {enp : p ∈ N} is a bounded sequence and enp → 0 weakly. From (ii), it follows that mu,v cannot be compact. Hence conditon (i) must be true. 2 Example 2.5: Let φ : [0, ∞) → [0, ∞) be defined as φ(x) = x2 . Clearly √ φ−1 (x) = 2x. Define v : N → N as v(n) = n+p and u : N → C as u(n) = n12 . Now w(n) = 1 1 n2

and lim sup U(k, φ, n) = lim sup √ . Hence in view of theorem (2.4) mu,v is n→∞ n→∞ 2 a compact operator. Example 2.6: Let E1 , E2 , ..., En , ... be a partition of N such that En = n. Define v : N → N as v(k) = n if k ∈ En . Then w(n) = En = n. Let


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√ 2 u : N → C be defined by u(p) = 2/p if p ∈ En . Define φ(x) = |x|2 . Then φ √ is an Orlicz function and φ−1 (x) = 2x. Now |u(k)| |u(k)| k∈v−1 (n) k∈En lim sup = lim sup 1 1 n→∞ n→∞ w(n)φ−1( w(n) ) w(n)φ−1 ( w(n) ) √ n 2/n = lim sup √ n→∞ n 2/n = 1. Therefore ||mu,v en ||φ does not go to zero as m approches ∞. Hence mu,v is not compact. REFERENCES 1. Carlson, J.W. (1985); Weighted composition operators on 2 , Ph.D thesis, Purdise University. 2. Carlson, J.W. (1990), Hyponormal and quasinormal weighted composition operators on l2 ; Rocky Mountain J. of Mathematics 20, 399-407. 3. Faour, W. and Khalil, R. (1987), On spectrum of weighted shifts, Hauston journal of mathematics 13, 549. 4. Komal, B.S., Raj, Kuldip and Gupta, Sunil, On operators of weighted substitution on the generalized spaces of entire functions-I Math. Today Vol. XV(1997), 3-10. 5. Komal B.S. and Gupta Shally., Multiplication operators between Orlicz spaces, Integr. Eqns. Oper. Theory, 41(2001), 324-330. 6. Kamowitz, H (1981), Compact weighted endomorphism of C(X), Proc. Amer. Math. Soc. 83, 517. 7. Lambert,A.Hypernormal composition operators Bull London Math.Soc.18(1986),395400. 8. Rao, M.M., Linear functionals on Orlics spaces general theory, Pacific J. Math. 25(1968), 553-585.

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9. Rao, M.M and Z.D. Ren, Theory of Orlicz spaces, Marcel-Dekker Inc. New York, 1991. 10. Singh R.K. and Komal B.S., Composition operators on p and its adjoint, Proc. Amer. Math. Soc. 70(1978), 21-25. 11. Singh R.K. and Kumar, Ashok (1977), Multiplication operators with closed ranges, Bull. Aust. Math. Soc. 16, 247-252. Received: April, 2010