IS S N 1 3 0 3 — 5 9 9 1
ON A NEW SEQUENCE SPACE DEFINED BY ORLICZ FUNCTIONS
VAKEEL A. KHAN
Abstract. The sequence space EY was introduced and studied by Mursaleen
[9]. In this paper we extendEY toEY (P> s> u) and study some properties
and inclusion relations on this space.
1. Introduction
Let o4 and f denote the Banach spaces of bounded and convergent sequences { = ({n)4n=1 respectively. Let be an injection of the set of positive integers L
Q into itself having no finite orbits and W be the operator defined on o4 by W(({q)4q=1) = ({ (q))4q=1.
A positive linear functional! , with||!|| = 1 , is called a - mean or an invariant mean if!({) = !(W {) for all { 5 o4.
A sequence{ is said to be - convergent , denoted by { 5 Y , if!({) takes the same value, called lim {, for all - means !. We have (see Schaefer [14])
Y = ( {= ({q) : 4 X p=1 wp>q({) = O uniformly in n> O = lim { ) > where forp 0, q A 0 wp>q({) = {q+ { (q)+ · · · + { p(q) p+ 1 > and w1>q= 0=
where p(q) denotes the m th iterate of at q. In particular, if is the translation, a - mean is often called a Banach limit and Y reduces to f , the set of almost
Received by the editors Nov. 01, 2007; Accepted: Sept. 05, 2008. 2000 Mathematics Subject Classification. Primary 40F05, 40C05, 46A45. Key words and phrases. Invariant mean, Paranorm, Orlicz function, Solid space .
c
°2008 A nkara U niversity
- convergent sequences (see Lorentz [5]). Subsequently invariant means have been studied by Ahmad and Mursaleen [1] , Mursaleen [8], Raimi [12] and many others. The concept of paranorm is closely related to linear metric spaces. It is a gener-alization of that of absolute value. Let[ be a linear space. A function j : [ $ LU is called paranorm, if
[P1] j({) 0> for all { 5 [> [P2] j({) = j({)> for all { 5 [>
[P3] j({ + |) j({) + j(|)> for all {> | 5 [>
[P4] If(q) is a sequence of scalars with q$ (q $ 4) and ({q) is a sequence of vectors withj({q {) $ 0 (q $ 4)> then j(q{q {) $ 0 (q $ 4)= A paranormj for which j({) = 0 implies { = 0 is called a total paranorm on [, and the pair([> j) is called a totally paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (cf. [15, Theorem 10.4.2, p. 183]).
A map P : LU $ [0> +4] is said to be an Orlicz function if P is even, convex, left continuous onUL+, continuous at zero, P(0) = 0 and P(x) $ 4 as x $ 4. If P takes value zero only at zero we will write P A0 and if P takes only finite values we will writeP ? 4. [2,3,6,7,10,13].
W.Orlicz [11] used the idea of orlicz function to construct the space (OP) . Lindendstrauss and Tzafriri [4] used the idea of Orlicz function to define orlicz sequence space cP := ( { 5 $: 4 X n=1 P μ |{n| ¶ ? 4 for some A0 )
in more detail . cP is a Banach space with the norm ||{|| := inf{ A 0 : 4 X n=1 P μ |{n| ¶ 1}
The spaceoP is closely related to the space os , which is an Orlicz sequence space withP({) = {s i ru1 s ? 4=
The42 - condition is equivalent to
P(O{) NOP({)> for all values of { 0> and for O A 1= An Orlicz functionP can always be represented in the following integral form
P({) = Z {
0 (w)gw>
where is known as the kernel of P, is right di erentiable for w 0> (0) = 0> (w) A 0> is non-decreasing and (w) $ 4 as w $ 4.Note that an Orlicz function
satisfies the inequality
P({) P({) for all with 0 ? ? 1=
LetH be a sequence space . Then H is called
(i) A sequence spaceH is said to be symmetric if({q) 5 H implies ({ (q)) 5 H, where (q) is a permutation of the elements of the elements of LQ.
(ii) Solid (or normal), if (n{n) 5 H, whenever ({n) 5 H for all sequences of scalars (n) with |n| 1 for all n 5 LQ.
Lemma 1.1. . A sequence space H is solid implies H is monotone. Mursaleen [9] defined the sequence space
EY = ( { 5 o4: X p |!p>q({)| ? 4> uniformly in n ) > where !p>q({) = wp>q({) wp1>q({) assuming that wp>q({) = 0> for m = 1=
A straightforward calculation shows that
!p>q({) = ; A A ? A A = 1 p(p+1) p P m=1m({ m(q) { m1(q)) (p 1) {q> (p = 0)
Note that for any sequence {> | and scalar we have
2. Main Results.
LetP be an Orlicz function, s= (sp) be any sequence of strictly positive real numbers and u 0. Now we define the sequence space as follows :
EY (P> s> u) = ; ? = {= ({n) : 4 P p=1 1 pu h P³|!p>q({)| ´isp ? 4> uniformly in n and for some A0
< @ >= ForP({) = { we get EY (s> u) = ( {= ({n) : 4 X p=1 1 pu|!p>q({)| sp ? 4> uniformly in n ) = Forsp= 1, for all p, we get
EY (P> u) = ; ? = {= ({n) : 4 P p=1 1 pu h P³|!p>q({)| ´i ? 4> uniformly in n and for some A0
< @ >= Foru= 0 we get EY (P> s) = ; ? = {= ({n) : 4 P p=1 h P³|!p>q({)| ´isp ? 4> uniformly in n and for some A0
< @ >= ForP({) = { and u = 0 we get
EY (s) = ( {= ({n) : 4 X p=1 |!p>q({)|sp ? 4> uniformly in n ) = Forsp= 1, for all p and u = 0 we get
EY (P) = ; ? = {= ({n) : 4 P p=1 h P³|!p>q({)| ´i ? 4> u 0> uniformly in n and for some A0
< @ >= ForP({) = {, sp= 1, for all p, and u = 0 we get
EY = ( {= ({n) : 4 X p=1 |!p>q({)| ? 4> uniformly in n ) =
Theorem 2.1. The sequence space EY (P> s> u) is a linear space over the field LF of complex numbers.
Proof. . Let {> | 5 EY (P> s> u) and > 5 LF. Then there exist positive numbers 1 and 2such that
4 X p=1 1 pu Pμ|!p>q({)| 1 ¶¸sp ? 4 and 4 X p=1 1 pu Pμ|!p>q(|)| 2 ¶¸sp ? 4> uniformly in n=
Define 3= max(2|| 1>2|| 2). Since P is nondecreasing and convex we have 4 X p=1 1 pu Pμ|!p>q({) + !p>q(|)| 3 ¶¸sp 4 X p=1 1 pu Pμ|!p>q({)| 3 + |!p>q(|)| 3 ¶¸sp 4 X p=1 1 pu 1 2 P μ! p>q({) 1 ¶ + P μ! p>q(|) 2 ¶¸ ? 4> uniformly in n= This proves thatEY (P> s> u) is a linear space over the field LF of complex numbers.
¤ Theorem 2.2. For any Orlicz function P and a bounded sequence s = (sp) of strictly positive real numbers, EY (P> s> u) is a paranormed(need not be total paranormed) space with
j({) = inf q1 ; ? = sq N : à 4 X p=1 1 pu Pμ|!p>q({)| ¶¸sp! 1 N 1> uniformly in n < @ >= whereN= max(1> sup sp).
Proof. It is clear thatj({) = j({). Since P(0) = 0, we get infn sqN
o
= 0> for { = 0= By using Theorem 1, for= = 1> we get
j({ + |) j({) + j(|)=
For the continuity of scalar multiplication let o 6= 0 be any complex number. Then by the definition we have
j(o{) = inf q1 ; ? = sq N : à 4 X p=1 1 pu Pμ|!p>q(o{)| ¶¸sp!N1 1> uniformly in n < @ >
j(o{) = inf q1 ; ? = (|o|v) sq N : μP4 p=1 1 pu h P³|!p>q(o{)| v|o| ´isp¶ 1 N 1> uniformly in n < @ > where v=|o| . Since|o|sqmax(1> |o|K), we have
j(o{) max(1> |o|K) inf q1 ; ? = vsqN : μP4 p=1 1 pu h P³|!p>q({)| v ´isp¶ 1 N 1> uniformly in n < @ > = max(1> |o|K)j({)
and thereforej(o{) converges to zero when j({) converges to zero in EY (P> s> u) . ¤ Now let{ be fixed element in EY (P> s> u) . There exists A 0 such that j({) = inf q1 ; ? = sq N : à 4 X p=1 1 pu Pμ|!p>q({)| ¶¸sp!N1 1> uniformly in n < @ >= Now j(o{) = inf q1 ; ? = sq N : à 4 X p=1 1 pu Pμ|!p>q(o{)| ¶¸sp!N1 1> uniformly in n < @ >$0> aso $ 0=
This completes the proof.
Theorem 2.3. Suppose that 0 ? sp wp? 4 for each p 5 LQ and u 0. Then (l) EY (P> s) EY (P> w)>
(ll) EY (P) EY (P> u).
Proof. . [i] Suppose that { 5 EYh (P> s). This implies that P³|!l>q({)|
´isp
1for su!ciently large values of l , say l p0for some fixed p05 LQ . Since P is non decreasing, we have
4 X p=p0 Pμ|!l>q({)| ¶¸wp 4 X p=p0 Pμ|!l>q({)| ¶¸sp ? 4= Hence { 5 EY (P> w).
The proof of [ii] is trivial.
The following result is a consequence of the above result.
Corollary 1. If 0 ? sp1 for each p , then EY (P> s) EY (P). Ifsp1 for all p , then EY (P) EY (P> s).
Theorem 2.4. . The sequence space EY (P> s> u) is solid. Proof. Let{ 5 EY (P> s> u). This implies that
4 X p=1 pu Pμ|!n>q({)| ¶¸sp ? 4=
Let(p) be sequence of scalars such that |p| 1 for all p 5 LQ. Then the result follows from the following inequality
4 X p=1 pu Pμ|p!n>q({)| ¶¸sp 4 X p=1 pu Pμ|!n>q({)| ¶¸sp ? 4= Hence{ 5 EY (P> s> u) for all sequences of scalars (p) with |p| 1 for all p 5 L
Q whenever { 5 EY (P> s> u).
¤ From Theorem 4 and Lemma we have :
Corollary 2. . The sequence space EY (P> s> u) is monotone.
Theorem 2.5. . Let P1, P2 be Orlicz functions satisfying 42 - condition and u> u1> u20. Then we have
(l) If u A 1 then EY (P1> s> u) EY (P0P1> s> u), (ll) EY (P1> s> u) _ EY (P2> s> u) EY (P1+ P2> s> u), (lll) If u1 u2 thenEY (P> s> u1) EY (P> s> u2).
Proof. [i] SinceP is continuous at 0 from right , for A 0 there exists 0 ? ? 1 such that0 f implies P(f) ? . If we define
L1= ½ p 5 LQ : P1μ|!p>q({)| ¶ for some A0 ¾ >
L2= ½ p 5 LQ : P1μ|!p>q({)| ¶ A for some A0 ¾ > then , whenP1 ³|! p>q({)| ´ A we get P μ P1μ|!p>q({)| ¶¶ {2P(1)@}P1μ|!p>q({)| ¶ = Hence for { 5 EY (P1> s> u) and u A 1
4 P p=1p uhP0P 1 ³|! p>q({)| ´isp = P p5L1 puhP0P 1 ³|! p>q({)| ´isp + P p5L2 puhP0P 1 ³|! p>q({)| ´isp P p5L1 pu[]sp + P p5L2 puh{2P(1)@}P 1 ³|! p>q({)| ´isp max(k> K) P4 p=1p u + max¡{2P(1)@}k>{2P(1)@}K¢ (where 0 ? k = inf sp sp K= sup
p sp? 4)=
[ii] The proof follows from the following inequality pu (P1+ P2)μ|!p>q ({)| ¶¸sp Fpu P1μ|!p>q ({)| ¶¸sp +Fpu P2μ|!p>q ({)| ¶¸sp = [iii ]The proof is straightforward.
¤ Corollary 3. . Let P be an Orlicz function satisfying 42 - condition. Then we have
(1) Ifu A1, then EY (s> u) EY (P> s> u), (2) EY (P> s) EY (P> s> u),
(3) EY (s) EY (s> u), (4) EY (P) EY (P> u), The proof is straightforward.
ÖZET: EY dizi uzayı, Mursaleem tarafından tanımlanmı¸s ve in-celenmi¸stir[9]. Bu çalı¸smada EY uzayını,EY (P> s> u) uzayına geni¸sleterek bu uzaya ili¸skin bazı özelikleri ve kapsama ba˘ gın-tılarını elde ettik.
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Current address : Department of Mathematics, A. M. U. Aligarh-202002 INDIA, E-mail address : vakhan@math.com,
