The Semi Orlicz Spaces

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 32, The Semi Orlicz Spaces N. Subramanian Department of Mathematics, SASTRA University Tanjore , India K. S. Ravichandran School of Computing, SASTRA University Tanjore , India R. Babu Department of Mathematics Shanmugha Polytechnic College Tanjore , India babu Abstract The aim of this paper is to introduce a new class of sequence spaces, namely the semi-orlicz spaces. It is shown that the intersection of all semi-orlicz spaces is semi-orlicz. The intersection of all semi-orlicz spaces is l M and l M is smallest semi-orlicz space. Mathematics Subject Classification: 40A05, 40C05, 40D05 Keywords: Orlicz function, semi Orlicz, entire sequence 1 Introduction A complex sequence, whose th terms is x is denoted by {x } or simply x. Let w be the set of all sequences x =(x ) and φ be the set of all finite sequences. Let l,c,c 0 be the sequence spaces of bounded, convergent and null sequences x =(x ) respectively. In respect of l,c,c 0 we have x = sup x, where x = (x ) c 0 c l. Orlicz [11] used the idea of Orlicz function to construct the space (L M ). Lindenstrauss and Tzafriri

2 1552 N. Subramanian, K. S. Ravichandran and R. Babu [7] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space l M contains a subspace isomorphic to l p (1 p< ). Subsequently different classes of sequence spaces defined by Parashar and Choudhary[12], Mursaleen et al.[8], Betas and Altin[1], Tripathy et al.[16], Rao and subramanian[3] and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in Ref[5]. Recall([5],[11]) an Orlicz function is a function M :[0, ) [o, ) which is continuous, non-decreasing and convex with M(0) = 0,M(x) > 0, for x>0 and M(x) as x. If convexity of Orlicz function M is replaced by M(x + y) M(x) + M(y) then this function is called modulus function, introduced by Naano[10] and further discussed by Rucle[13] and Maddox[9] and many others. An Orlicz function M is said to satisfy Δ 2 condition for all values of u, if there exists a constant K>0, such that M(2u) KM(u)(u 0). The Δ 2 condition is equivalent to M(lu) KlM(u), for all values of u and for l>1. Lindenstrauss and Tzafriri[7] used the idea of Orlicz function to construct Orlicz sequence space { ( ) } x l M = x w : M <, forsomeρ > 0. (1) ρ =1 The space l M with the norm { x = inf ρ>0: ( ) } x M 1 ρ =1 (2) becomes a Banach space which is called an Orlicz sequence space. For M(t) = t p, 1 p<, the space l M coincide with the classical sequence space l p Given a sequence x = {x } its n th section is the sequence x (n) = {x 1,x 2,..., x n, 0, 0,...} δ (n) =(0, 0,..., 1, 0, 0,...), 1 in the n th place and zero s else where. An FKspace (Frechet coordinate space) is a Frechet space which is made up of numerical sequences and has the property that the coordinate functionals p (x) =x ( =1, 2, 3,...) are continuous. We recall the following definitions [see [17]]. An FK-space is a locally convex Frechet space which is made up of sequences and has the property that coordinate projections are continuous. An FK-space X is said to have AK (or sectional convergence) if x [n] x as n holds for every x X. The space is said to have AD (or) be an AD space if φ is dense in X. We note that AK implies AD by [2]. If X is a sequence space, we define (i)x = the continuous dual of X. (ii)x α = {a =(a ): =1 a x <, foreachx X} ;

3 The semi Orlicz spaces 1553 (iii)x β = {a =(a ): { =1 a x isconvergent, foreachx X} ; (iv)x γ = a =(a ): sup n } n =1 a x <, foreachx X ; (v)let X be an FK-space φ. Then X f = { } f(δ (n) ):f X. X α,x β,x γ are called the α (or Kö the-t öeplitz)dual of X, β (or generalized Kö the-t öeplitz)dual of X, γ dual of X. Note that X α X β X γ. If X Y then Y μ X μ, for μ = α, β, or γ. 1.1 Lemma (See(17, T heorem7.27)). Let X be an FK-space φ. Then (i)x γ X f. (ii)if X has AK, X β = X f. (iii)if X has AD, X β = X γ. Because of the historical roots of summability in convergence, conservative space and matrices play a special role in its theory. However, the results seem mainly to depend on a weaer assumption, that the spaces be semi conservative. (See[17]). Snyder and Wilansy[14] introduced the concept of semi conservative spaces. Snyder[15] studied the properties of semi conservative spaces. Later on, in the year 1996 the semi replete spaces were introduced by Chandrasehara Rao and Srinivasalu[4]. In a similar way, in this result we define semi-orlicz spaces, and show that l M is smallest semi-orlicz space. 2 Results: Properties of semi-orlicz spaces 2.1 Definition An FK-space X is called semi-orlicz if its dual X f l N. In other words X is semi-orlicz if f ( δ ()) l N f X for each fixed. 2.2 Example l M is semi-orlicz. Indeed if l M is the space of all Orlicz sequence, then by Lemma 2.3 (l M ) f = l N. 2.3 Lemma (l M ) f = l N. Proof: (l M ) β = l N in Kamthan and Gupta[6]. But l M has AK in Kamthan and Gupta[6]. Hence (l M ) β =(l M ) f. Therefore (l M ) f = l N. This completes the proof.

4 1554 N. Subramanian, K. S. Ravichandran and R. Babu 2.4 Proposition Let Y be any FK-space φ. Then Y l M if and only if the sequence { δ ()} is a member of the complementary Orlicz space. Proof: Y l M y f (l M ) f Y f l N, since (l M ) f = l N [Kamthan and Gupta[6]] ( δ ()) is a member of the complementary Orlicz space. We recall 2.5 Lemma (See 17, Theorem 4.3.7) Let z be a sequence. Then ( z β,p ) is an AK space with P =(P : =0, 1, 2,...), where P 0 (x) = sup m m =1 z x,p n (x) = x n. For any such that z 0,P may be omitted. If z φ, p 0 may be omitted. 2.6 Proposition Let z be a sequence z β is semi-orlicz if and only if z l N. Proof: Step 1. Suppose that z β is semi-orlicz. z β has AK by Lemma 2.5. Therefore Z ββ = ( z β) f by Theorem of Wilansy [17]. So Z β is semi- Orlicz if and only if z ββ l N. But then z z ββ l N. Step2: Conversely, suppose that z l N. Then z β {l N } β and z ββ {l N } ββ =(l M ) β = l N, because (l M ) β = l N. But ( z β)f = z ββ. Hence ( z β )f l N. Therefore z β is semi-orlicz. This completes the proof. 2.7 Proposition Every semi-orlicz space contains l M. Proof: Let X be any semi-orlicz space. Hence X f l N. Therefore f ( δ ()) l N f X. So, { δ ()} is a member of the complementary Orlicz space with respecdt to X. Hence X l M by Proposition 2.4. This completes the proof. 2.8 Proposition The intersection of all semi-orlicz spaces {X n : n =1, 2,...} is semi-orlicz. Proof: Let X = n=1 X n. Then X is an FK-space which contains φ. Also every f X can be written as f = g 1 + g g m, where g X n for some n and for 1 m. But then f ( δ ) ( = g ) ( 1 δ + g ) ( 2 δ g ) m δ (. Since X n (n =1, 2,...) are semi-orlicz spaces, it follows that g ) i δ l N for all i =1, 2,...m. Therefore f ( δ ) l N. Hence X is semi-orlicz. This completes the proof.

5 The semi Orlicz spaces Proposition The intersection of all semi-orlicz space is l M. Proof: Let I be the intersection of all semi-orlicz spaces. Then the intersection I { z β : z l N },(by Proposition 2.6) ={ln } β = l M (3) By Proposition 2.8 it follows that I is semi Orlicz. Consequently l M I(byP roposition2.7) (4) From (3) and (4) we get I = l M. This completes the proof Corollary The smallest semi-orlicz space is l M. References [1] C.Betas and Y.Altin, The sequence space l M (p, q, s) on seminormed spaces, Indian J. Pure Appl. Math., 34(4) (2003), [2] H.I.Brown, The summability field of a perfect l l method of summation, J. Anal. Math., 20(1967), [3] K.Chandrasehara Rao and N.Subramanian, The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68(2004), [4] K.Chandrasehara Rao and T.G.Srinivasalu, The Hahn sequence space-ii, Journal of Faculty of Education, 1(2)(1996), [5] M.A.Krasnoselsii and Y.B.Ruticii, Convex functions and Orlicz spaces, Gorningen, Netherlands, [6] P.K.Kamthan and M.Gupta, Sequence spaces and series. Lecture Notes in Pure and Applied Mathematics, Marcel Deer Inc. New Yor, 65(1981). [7] J.Lindenstrauss and L.Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), [8] M.Mursaleen,M.A.Khan and Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio Math., Vol. XXXII (1999),

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