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Kuldip Raj, Charu Sharma and Suruchi Pandoh ... For example Singh and Kumar [21] proved that ... of all bounded linear operators from Cesϕ(N) into itself. ..... new difference sequence spaces, Linear Algebra Appl., 436(2012), 41-52.
FASCICULI

MATHEMAT ICI

Nr 57

2016 DOI:10.1515/fascmath-2016-0021

Kuldip Raj, Charu Sharma and Suruchi Pandoh ` MULTIPLICATION OPERATORS ON CESARO-ORLICZ SEQUENCE SPACES

Abstract. In this paper, we characterize the compact, invertible, Fredholm and closed range multiplication operators on Ces` aro-Orlicz sequence spaces. Key words: multiplication operator, Fredholm multiplication operator, invertible operator, compact operator, isometry, Ces`aro sequence space, Ces` aro-Orlicz sequence space. AMS Mathematics Subject Classification: 47B38, 46A06.

1. Introduction and preliminaries Ces`aro sequence spaces Cesp , 1 ≤ p < ∞, appeared for the first time in 1968 as a problem of finding their duals [1]. Some basic properties of these spaces were studied in the early seventies by Shiue [23] and Leibowitz [13]. In 1974 Jagers [9] found the dual space of Cesp (see also [12]). In the late nineties mathematicians became interested in geometric properties of these spaces. Cui and Pluciennik studied Local Uniform Nonsquareness [6] and Banach-Saks Property and Property β [7], Cui and Hudzik studied Fixed Point Property [5] and obtained the Packing Constant [3]. An Orlicz function ϕ : [0, ∞) → [0, ∞) is a continuous, non-decreasing and convex such that ϕ(0) = 0, ϕ(x) > 0 for x > 0 and ϕ(x) → ∞ as x → ∞. An Orlicz function is said to satisfy the δ2 − condition if their exists K > 0 such that ϕ(Lx) ≤ KLϕ(x), for all x ≥ 0 and for L > 1. By w we shall denote the space of all complex sequences. The Ces`aro-Orlicz sequence space Cesϕ (N) is defined as ∞ m  o n 1 X X |λxk | < ∞ . Cesϕ (N) = x = (xk ) ∈ w : ϕ m m=1

k=1

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The space Cesϕ (N) is a Banach space under the norm ∞ n  1 Pm |x |  o X k kxk = inf λ > 0 : ϕ m k=1 ≤ 1 (see [17]). λ m=1

The Ces`aro-Orlicz sequence spaces Cesϕ (N) appeared for the first time in 1988, when Lim and Lee [11] found their dual spaces. Recently, Cui et al. [4] obtained important properties of the spaces. In 2007 Maligranda et al. [14] showed that Cesϕ (N) is not B-convex, if ϕ ∈ δ2 and Cesϕ (N) 6= {0}. The extreme points and strong U -points of Cesϕ (N) have been characterized by Foralewski et al. in [8]. Although the spaces Cesϕ (N) have been studied by several mathematicians, some essential and basic properties remain still unknown but recently some of its properties have been discussed by Damian in [10]. In the case when ϕ(x) = |x|p , (p > 1) the Ces`aro-Orlicz sequence space Cesϕ (N) becomes the Ces` aro sequence space Cesp . Let u : N → C be a function such that u.f ∈ Cesϕ (N) for every f ∈ Cesϕ (N), then we can define a multiplication transformation Mu : Cesϕ (N) → Cesϕ (N) by Mu f = uf ,

∀ f ∈ Cesϕ (N).

If Mu is continuous, we call it a multiplication operator induced by u. These operators received considerable attention over the past several decades especially on Lp -spaces and Bergman spaces. From the recent literature available in Operator theory we find that multiplication operators are very much intimately connected with the composition operators as most of the properties of composition operators on Lp -spaces can be stated in terms of properties of multiplication operators. For example Singh and Kumar [21] proved that a composition operator on Lp (X, C) is compact if and only if the multi−1 plication operator Mu is compact, where u = dµT dµ , the Radon-Nikodym derivative of the measure µT −1 with respect to the measure µ. Infact the multiplication operators play an important role in the theory of Hilbert space operators. One of the main application is that every normal operator on a separable Hilbert space is unitarily equivalent to a multiplication operator. Moreover, multiplication operators has its roots in the spectral theory and is being pursued today in such guises as the theory of subnormal operators and the theory of Toeplitz operators. For more details on multiplication operators we refer to ([2], [18], [19], [20], [22], [24]) and refrences therein. Moreover, Compact operators on sequence spaces have recently been studied by Mursaleen and Noman in ([15], [16]). By B(Cesϕ (N)) we denote the set of all bounded linear operators from Cesϕ (N) into itself. Unauthenticated Download Date | 1/17/17 2:44 AM

` ro-Orlicz . . . Multiplication operators on Cesa

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A bounded linear operator A : E → E (where E is a Banach space) is called compact if A(B1 ) has compact closure, where B1 denotes the closed unit ball of E. A bounded linear operator A : E → E is called Fredholm if A has closed range, dim(kerA) and co-dim(ranA) are finite. The main purpose of this paper is to characterize the boundedness, compactness, closed range and Fredholmness of multiplication operators on Ces`aro-Orlicz sequence spaces.

2. Multiplication operators Theorem 1. Let u : N → C be a mapping. Then Mu : Cesϕ (N) → Cesϕ (N) is bounded if and only if u is a bounded function. Proof. Suppose u is a bounded function. Then there exists M > 0 such that |un | ≤ M, ∀ n ∈ N. Let x ∈ Cesϕ (N), we have ! ∞ 1 Pm X |(ux) | k kMu xk = ϕ m k=1 λ m=1 ! P ∞ m 1 X k=1 |uk ||xk | m = ϕ λ m=1 ! P ∞ m 1 X k=1 |xk | m ≤ M ϕ λ m=1

= M kxk. Thus, kMu xk ≤ M kxk, ∀ x ∈ Cesϕ (N) which implies that Mu is a bounded operator. Conversely, suppose that Mu is a bounded operator. We show that u is a bounded function. For, if u is not a bounded function, then for every n ∈ N, there exists some pn ∈ N such that |u(pn )| > n. Now kepn k = ∞ X 1 and mλϕ−1 (1) m=pn ! ∞ X |u | m kMu epn k = mλϕ−1 (1) m=pn ! ∞ X 1 > n mλϕ−1 (1) m=p n

= nkepn k. This shows that Mu is not a bounded operator. Hence, u must be a bounded function. 

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Theorem 2. Mu is an isometry if and only if |un | = 1, ∀ n ∈ N. Proof. Suppose first that |un | = 1, ∀ n ∈ N. Then ! ∞ 1 Pm X |u x | k k kMu xk = ϕ m k=1 λ m=1 ! P ∞ m 1 X k=1 |xk | m = ϕ λ m=1

= kxk. Therefore, kMu xk = kxk, ∀ x ∈ Cesϕ (N) and hence Mu is an isometry. Conversely, suppose that |un | 6= 1 for some n = n0 . Then ken0 k = ∞ X 1 . Suppose |un0 | > 1. Then mλϕ−1 (1) m=n 0

kMu en0 k = >

∞ X m=n0 ∞ X m=n0 n0

|un0 | mλϕ−1 (1)

!

1 mλϕ−1 (1)

= ke k. Similarly if |un0 | < 1, then we can show that kMu en0 k < ken0 k. In both cases, when |un0 | < 1 or |un0 | > 1, we get contradiction. Hence, |un | = 1 ∀ n ∈ N. 

3. Compact multiplication operators Theorem 3. Let Mu ∈ B(Cesϕ (N)). Then Mu is a compact operator if and only if u(n) → 0 as n → ∞. Proof. We first assume that Mu is a compact operator. We show that u(n) → 0 as n → ∞. For if this were not true, then there exists  > 0 such that the set A = {k ∈ N : |uk | ≥ } is an infinite set. Let p1 , p2 , ..., pn , .. be in A . Then {epn : pn ∈ A } is an infinite bounded set in Cesϕ (N). Consider ! ∞ 1 Pm pn ps X k=1 |u(k)e (k) − u(k)e (k)| pn ps m kMu e − Mu e k = ϕ λ m=1 ! ∞ 1 Pm pn (k) − eps (k)| X |u(k)||e k=1 = ϕ m λ m=1

≥ kepn − eps k, ∀ pn , ps ∈ A . Unauthenticated Download Date | 1/17/17 2:44 AM

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` ro-Orlicz . . . Multiplication operators on Cesa

This proves that {epn : pn ∈ A } is a bounded sequence which cannot have a convergent subsequence under Mu . This shows that Mu cannot be a compact operator, which is a contradiction. Hence, u(n) → 0 as n → ∞. Conversely, suppose u(n) → 0 as n → ∞. Then for every  > 0, the set A = {n ∈ N : |u(n)| ≥ } is a finite set. Then Cesϕ (A ) is a finite dimensional space for each  > 0. Therefore, Mu |Cesϕ (A ) is a compact operator. For each n ∈ N, define un : N → C by ( un (m) =

u(m), ∀m ∈ A 1 n 0, ∀m ∈ / A1. n

Clearly, Mun is a compact operator as the space Cesϕ (A 1 ) is finite dimenn sional for each n ∈ N. Now k(Mun − Mu )xk = =

∞ X

1 m

ϕ

m=1 ∞ X

Pm

k=1 |un (k)xk

− u(k)xk |

!

λ 1 m

ϕ

Pm

k=1 |un (k)xk − u(k)xk | λ

m∈A 1

!

n

+

∞ X

ϕ

1 m

Pm

k=1 |un (k)xk − u(k)xk | λ

0 m∈A 1 n

=

∞ X

1 m

ϕ

Pm

k=1 |u(k)xk |

!

!

λ

0

m∈A 1

n

1 < n ≤

∞ X 0 m∈A 1 n

ϕ

1 m

Pm

k=1 |xk |

!

λ

1 kxk n

or k(Mun − Mu )(x)k ≤ n1 kxk. This proves that k(Mun − Mu )k ≤ n1 and that Mu is a limit of compact operators and hence, Mu is a compact operator.  Theorem 4. Let Mu ∈ B(Cesϕ (N)). Then Mu has closed range if and only if u is bounded away from zero on N \ keru = S. Proof. Suppose u is bounded away from zero on S. Then there exists  > 0 such that |un | ≥  ∀ n ∈ N \ keru. We have to prove that ranMu Unauthenticated Download Date | 1/17/17 2:44 AM

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is closed. Let z be a limit point of ranMu . Then there exists a sequence Mu xn → z. Clearly, the sequence {Mu xn } is a Cauchy sequence. Now, ! ∞ 1 Pm n − u xm | X |u x k k k k kMu xn − Mu xm k = ϕ m k=1 λ m=1 1 Pm n m ! ∞ k=1 |uk ||xk − xk | X m k∈S = ϕ λ m=1 P m 1 n m ! ∞ k=1 |xk − xk | X m k∈S ≥  ϕ λ m=1 ! ∞ 1 Pm ˜n − x˜m | X | x k =  ϕ m k=1 k λ = where x˜nk =

m=1 ||x˜n −

x˜m ||,

xnk , if k ∈ S 0, if k ∈ / S.



This proves that {x˜n } is a Cauchy sequence in Cesϕ (N). But Cesϕ (N) is complete. Therefore, there exists x ∈ Cesϕ (N) such that x˜n → x as n → ∞. Inview of continuity of Mu , Mu x˜n → Mu x. But Mu xn = Mu x˜n → z. Therefore, Mu x = z. Hence, z ∈ ranMu . This proves that Mu has closed range. Conversely, suppose that Mu has closed range. Then Mu is bounded away from zero on (kerMu )⊥ = Cesϕ (N \ keru). That is, there exists  > 0 such that (1)

kMu xk ≥ kxk ∀ x ∈ Cesϕ (N \ keru).

Let G = {k ∈ N \ keru : |uk | < 2 }. If G 6= φ, then for n0 ∈ G, we have ! ∞ 1 Pm n0 (k)| X |u(k)e k=1 kMu en0 k = ϕ m λ m=1 ! ∞ X |u(n0 )| = mλϕ−1 (1) m=n 0

< 

∞ X m=n0 n0

1 mλϕ−1 (1)

= ke k. That is, kMu en0 k < ken0 k which contradicts (1). Hence, G = φ so that |uk | ≥ , ∀ k ∈ N \ keru. This proves the theorem. 

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4. Invertible and Fredholm multiplication operators Theorem 5. Let u : N → C be a mapping. Then Mu : Cesϕ (N) → Cesϕ (N) is invertible if and only if there exist m > 0 and M > 0 such that m < un < M, ∀ n ∈ N. Proof. Suppose that the condition is true. Define γ : N → C by γn = u1n . Then Mu and Mγ are bounded linear operators in view of Theorem 1. Also Mu .Mγ = Mγ .Mu = I. Hence, Mγ is the inverse of Mu . Conversely, suppose that Mu is invertible. Then ranMu = Cesϕ (N). Therefore, ranMu is closed. Hence, by Theorem 4, there exists  > 0 such that |un | ≥  ∀ n ∈ N \ keru. Now keru = φ, otherwise un0 = 0, for some n0 ∈ N, in which case en0 ∈ kerMu which is a contradiction, since kerMu is trivial. Hence, |un | ≥  ∀ n ∈ N. Since Mu is bounded, so by Theorem 1, there exists M > 0 such that |un | ≤ M, ∀ n ∈ N. Thus, we have proved that  ≤ |un | ≤ M, ∀ n ∈ N.  Theorem 6. Let Mu : Cesϕ (N) → Cesϕ (N) be a bounded operator. Then Mu is Fredholm operator if and only if (i) keru is a finite subset of N. (ii) |un | ≥ , ∀ n ∈ N \ keru. Proof. Suppose Mu is Fredholm. If keru is an infinite subset of N, then ∈ kerMu ∀ n ∈ keru. But en ’s are linearly independent, which shows that kerMu is an infinite dimensional which is a contradiction. Hence, keru must be a finite subset of N. The condition (ii) follows from Theorem 4. Conversely, If the conditions (i) and (ii) are true, then we prove that Mu is Fredholm. In view of Theorem 4, the condition (ii) implies that Mu has closed range. The condition (i) implies that kerMu and kerMu? are finite dimensional. This proves that Mu is Fredholm.  en

References [1] Programma van Jaarlijkse Prijsvragen, (Annual Problem Section), Nieuw Arch. Wiskd., 16(1968), 47–51. [2] Abrahmse M.B., Multiplication operators, Lecture Notes in Mathematics, 693 Springer Verlag (1978), 17-36. [3] Cui Y., Hudzik H., Packing constant for Ces`aro sequence spaces, Nonlinear Anal., 47(2001), 2695-2702. [4] Cui Y., Hudzik H., Petrot N., Szymaszkiewicz A., Basic topological and geometric properties of Ces` aro-Orlicz spaces, Proc. Indian Acad. Sci. Math. Sci., 115(2005), 461-476. [5] Cui Y., Hudzik H., Some geometric properties related to fixed point theory in Ces` aro spaces, Collect. Math., 50(1999), 277-288.

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[6] Cui Y., R.Pluciennik, Local uniform nonsquareness in Ces`aro sequence spaces, Comment. Math. Prace Mat., 37(1997), 47-58. [7] Cui Y., R.Pluciennik, Banach-Saks property and property β in Ces`aro sequence spaces, Southeast Asian Bull. Math., 24(2000), 201-210. [8] Foralewski P., Hudzik H., Szymaszkiewicz A., Local rotundity structure of Ces` aro-Orlicz spaces, J. Math. Anal. Appl., 345(2008), 410-419. [9] Jagers A.A., A note on Ces` aro sequence spaces, Nieuw Arch. Wiskund, 22 (1974), 113-124. [10] Kubiak D.M., A note on Ces` aro-Orlicz spaces, J. Math. Anal. Appl., 349 (2009), 291-296. [11] Lim S.K., Lee P.Y., An Orlicz extension of Ces`aro sequence spaces, Comment. Math. Prace Mat., 28(1988), 117-128. [12] Lee P.Y., Ces` aro sequence spaces, Math. Chronicle, New Zealand, 13(1984), 29-45. [13] Leibowitz G.M., A note on Ces` aro sequence spaces, Tamkang J. Math., 2 (1971), 151-157. [14] Maligranda L., Petrot N., Suantai S., On the james constants and B-convexity of Ces` aro and Ces` aro-Orlicz sequence spaces, J. Math. Anal. Appl., 326(2007), 312-326. [15] Mursaleen M., Noman A.K., Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73(2010), 2541-2557. [16] Mursaleen M., Noman A.K., Compactness of matrix operators on some new difference sequence spaces, Linear Algebra Appl., 436(2012), 41-52. [17] Petrot N. Suantai S., Some geometric properties in Orlicz-Ces`aro spaces, Science Asia, 31(2005), 173-177. [18] Raj K., Sharma S.K., Kumar A., Multiplication operator on MusielakOrlicz spaces of Bochner type, J. Adv. Stud. Topol., 3(2012), 1-7. [19] Sharma A., Raj K., Sharma S.K., Products of multiplication composition and differentiation operators from H ∞ to weighted Bloch spaces, Indian J. Math., 54(2012), 159-179. [20] Singh R.K., Manhas J.S., Composition Operators on Function Spaces, North Holland, 1993. [21] Singh R.K., Kumar A., Multiplication operators with closed range, Bull. Aust. Math. Soc., 16(1977), 247-252. [22] Singh R.K., Manhas J.S., Multiplication operators and composition operators with closed ranges, Bull. Aust. Math. Soc., 16(1977), 247-252. [23] Shiue J.S., On the Ces` aro sequence spaces, Tamkang J. Math., 1(1970), 19-25. [24] Takagi H., Yokouchi K., Multiplication and composition operators between two Lp -spaces, Contemp. Math., 232(1999), 321-338.

Kuldip Raj Department of Mathematics Shri Mata Vaishno Devi University Katra-182320, J & K, India e-mail: [email protected]

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145

Charu Sharma Department of Mathematics Shri Mata Vaishno Devi University Katra-182320, J & K, India e-mail: [email protected] Suruchi Pandoh Department of Mathematics Shri Mata Vaishno Devi University Katra-182320, J & K, India e-mail: [email protected] Received on 07.09.2016 and, in revised form, on 24.11.2016.

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