The sequence space bv(u,p) has been defined and the classes (bv(u,p):l?), (bv(u,p):c),and&nbsp;(bv(u,p):c0) of infinite matrices have been characterized by Ba?ar, Altay and Mursaleen ( see, [2] ). The main purposes of the present paper is to characterize the classes (bv(u,p):ƒ?),(bv(u, p):ƒ), and&nbsp;(bv(u,p):ƒ0),&nbsp;where ƒ?, ƒ, and ƒ0 denotes the spaces of almost bounded sequences, almost convergent sequences and almost convergent null sequences, respectively, with real or complex terms.
Kathmandu University Journal of Science, Engineering and Technology Vol. 8, No. II, December, 2012, 89-9

Ganie, Ab. Hamid

Sheikh, Neyaz Ahmad

English

Kathmandu University Open Journal Systems

KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 8, No. II, DECEMBER, 2012, pp 89-92 89  SOME MATRIX TRANSFORMATIONS AND ALMOST CONVERGENCE  Neyaz Ahmad Sheikh, Ab. Hamid Ganie  Department of Mathematics, National Institute of Technology Srinagar - 190006.  Corresponding address: neyaznit@yahoo.co.in Received 16 May, 2012; Revised 9 August, 2012  ABSTRACT  The sequence space 𝑏𝑣(𝑢, 𝑝) has been defined and the classes  𝑏𝑣 𝑢, 𝑝 : 𝑙∞ ,  𝑏𝑣 𝑢, 𝑝 : 𝑐  and  𝑏𝑣 𝑢, 𝑝 : 𝑐0  of infinite matrices have been characterized by  Başar, Altay and Mursaleen ( see, [2] ). The main purposes of the present paper is to characterize the classes 𝑏𝑣 𝑢, 𝑝 : 𝑓∞ ,  𝑏𝑣 𝑢, 𝑝 : 𝑓  and 𝑏𝑣 𝑢, 𝑝 : 𝑓0 , where 𝑓∞ , 𝑓 and 𝑓0 denotes the spaces of almost bounded sequences, almost convergent sequences and almost convergent null sequences, respectively, with real or complex terms.  2010 AMS Mathematical Subject Classification: 46A45; 46B45; 40C05.  Keywords and Phrases: Sequence space of non-absolute type, almost bounded sequences,     β-duals and Matrix mappings.  1 . INTRODUCTION, BACKGROUND AND PRELIMINARIES A sequence space is defined to be a linear space with real or complex sequences. Throughout the paper ℕ, ℝ and ℂ denotes the set of non-negative integers, the set of real numbers and the set of complex numbers, respectively. Let ℓ∞ , 𝑐 and 𝑐0 respectively be  Banach spaces of  bounded, convergent and null sequences 𝑥 =  𝑥𝑛 𝑛=0∞  normed by  𝑥 = 𝑠𝑢𝑝𝑛≥0|𝑥 𝑛 |;  also, by 𝑐𝑠 we denote the sequence of all convergent series(see, [7]).  Let  𝑋 and 𝑌 be two non-empty subsets of the space 𝜔 of real or complex sequences. Let  𝐴 = 𝑎𝑛𝑘  ,(𝑛, 𝑘 ∈ ℕ), be an infinite matrix of real or complex numbers. We write  𝐴𝑥 𝑛 =𝐴𝑛 𝑥 =  𝑎𝑛𝑘 𝑥𝑘  𝑘 . Then 𝐴𝑥 =  𝐴𝑛 𝑥   is called the 𝐴-transform of 𝑥, whenever 𝐴𝑛 𝑥 = 𝑎𝑛𝑘 𝑥𝑘  𝑘  converges for each 𝑛 ∈ ℕ. We write lim𝑛 𝐴𝑥 = lim𝑛 𝐴𝑛  𝑥 . If 𝑥 ∈ 𝑋  implies 𝐴𝑥 ∈ 𝑌, we say that 𝐴 defines a (matrix) transformation from 𝑋 into 𝑌 and we denote it by 𝐴: 𝑋 → 𝑌. By (𝑋: 𝑌), we mean the class of all matrices 𝐴 such that 𝐴: 𝑋 → 𝑌.  Let 𝐷 denote the shift operator on 𝜔, that is , 𝐷𝑥 =  𝑥 𝑛  𝑛=1∞ , 𝐷2𝑥 =  𝑥 𝑛  𝑛=2∞  and so on. Obviously, 𝐷 is a bounded linear operator on 𝑙∞  onto itself. A Banach limit 𝐿 is a non-negative linear functional on 𝑙∞  such that 𝐿 is invariant under the shift operator that is, 𝐿 𝑆𝑥 = 𝐿(𝑥) and that  𝐿 𝑒 = 1, where  𝑒 = {1, 1, … } (see, [1]). A sequence space is said to be almost convergent (see, [3]) to the generalized limit 𝛼 if all Banach limits of 𝑥 are 𝛼. We denote the set of almost convergent sequences by 𝑓. It was proved by Lorentz (see, [3]) that   𝑓 =  𝑥 ∈ 𝑙∞ ∶ lim𝑚 𝜏𝑚𝑛  𝑥 = 𝛼 , 𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦 𝑖𝑛 𝑛 ,  where,        𝜏𝑚𝑛  𝑥 =1𝑚+1 𝑥𝑗 +𝑛𝑚𝑗=0  ,  𝜏−1,𝑛 = 0 and  𝛼 = 𝑓-lim⁡𝑥. Nanda [6] has defined a new set of sequences 𝑓∞as follows:             𝑓∞  =  𝑥 ∈ 𝑙∞ ∶  lim𝑚  𝜏𝑚𝑛  𝑥  < ∞ .    KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 8, No. II, DECEMBER, 2012, pp 89-92 90  We call 𝑓∞  the set of all almost bounded sequences.  We denote by  𝑋𝛽 ,  the  𝛽-dual of a sequence space 𝑋 and mean the set of all these  sequences 𝑥 =  𝑥𝑘  such that 𝑥𝑦 =  𝑥𝑘𝑦𝑘 ∈ 𝑐𝑠  for all  𝑦 =  𝑦𝑘 ∈ 𝑋.  The approach of constructing a new sequence space by means of matrix domain of a particular limitation method has been studied by several authors viz., ( [2, 4, 5] ).  The sequence space 𝑏𝑣(𝑢, 𝑝) has been defined and the various classes  𝑏𝑣 𝑢, 𝑝 :   𝑙∞   𝑏𝑣 𝑢, 𝑝 : 𝑐  and  𝑏𝑣 𝑢, 𝑝 : 𝑐0  have been characterized (see, [2]). In the present paper, we characterize the classes  𝑏𝑣 𝑢, 𝑝 : 𝑓∞ ,   𝑏𝑣 𝑢, 𝑝 : 𝑓  and  𝑏𝑣 𝑢, 𝑝 : 𝑓0  , where 𝑢 =  𝑢𝑘  is a sequence such that 𝑢𝑘 ≠ 0 for all 𝑘 ∈ ℕ.  The space 𝑏𝑣 𝑢, 𝑝  is defined (see, [2]) as                𝑏𝑣 𝑢, 𝑝 =   𝑥 =  𝑥𝑘 ∈ 𝜔 ∶    𝑢𝑘∆𝑥𝑘  𝑝𝑘 < ∞ 𝑘  , where, ∆𝑥𝑘 = 𝑥𝑘 − ∆𝑥𝑘−1.  2. MAIN RESULTS   Define the sequence 𝑦 = (𝑦𝑘) which will be used as the 𝐴𝑢 -transform of a sequence 𝑥 =(𝑥𝑘), 𝑖. 𝑒.,                            𝑦𝑘 = 𝑢𝑘∆𝑥𝑘  ;  𝑘 ∈ ℕ.      (2.1)    For brevity in notation, we write                          𝑡𝑚𝑛  𝑥 =1𝑚+1 𝐴𝑛+𝑗  𝑥 =𝑚𝑗=0  𝑎 𝑛, 𝑘, 𝑚 𝑥𝑘 ,𝑘   where ,  𝑎 𝑛, 𝑘, 𝑚 =1𝑚+1 𝑎𝑛+𝑗 ,𝑘𝑚𝑗 =0   ;  𝑛, 𝑘, 𝑚 ∈ ℕ  Also,   𝑎 𝑛, 𝑘, 𝑚 =  𝑎(𝑛 ,𝑘 ,𝑚)𝑢𝑘  ; 𝑛, 𝑘, 𝑚 ∈ ℕ . Now, we give the following lemmas which will be needed in proving the main  Theorems.  Lemma 2.1 [2] :  Define the sets 𝐷1 𝑝 and 𝐷2 𝑝  as follows:                         𝐷1 𝑝 =  𝑎 =  𝑎𝑘 ∈ 𝜔 ∶ sup𝑛    𝑎𝑗𝑢𝑘𝑛𝑗 =𝑘  𝑝𝑘< ∞𝑘  ,    𝐷2 𝑝 =   𝑎 =  𝑎𝑘 ∈ 𝜔 ∶ sup𝑛    𝑎𝑗𝑢𝑘𝐵−1𝑛𝑗 =𝑘  𝑝𝑘′< ∞𝑛𝑘=0  𝐵>1 .  Then ,   𝑏𝑣(𝑢, 𝑝) 𝛽 = 𝐷1 𝑝  𝑐𝑠 ; ( 0 < 𝑝𝑘 ≤ 1) and   𝑏𝑣(𝑢, 𝑝) 𝛽 = 𝐷2 𝑝  𝑐𝑠 ; ( 1 < 𝑝𝑘 < ∞).  Lemma 2.2 [6]: 𝑓 ⊂ 𝑓∞ .  We consider only the case 1 < 𝑝𝑘 ≤ 𝑀 < ∞ and the case  0 < 𝑝𝑘 ≤ 1 may be proved in a similar fashion.   Theorem 2.3: (a) Let 1 < 𝑝𝑘 ≤ 𝑀 < ∞ for every 𝑘 ∈ ℕ. Then 𝐴 ∈  𝑏𝑣 𝑢, 𝑝 : 𝑓∞  if and only if    sup𝑛 ,𝑚   𝑎  𝑛, 𝑘, 𝑚 𝐵−1 𝑝𝑘′< ∞𝑘      (2.2) and   𝑎𝑛𝑘  ∈ 𝐷2 𝑝  𝑐𝑠.       (2.3) KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 8, No. II, DECEMBER, 2012, pp 89-92 91   (b) Let 0 < 𝑝𝑘 ≤ 1 for every 𝑘 ∈ ℕ. Then 𝐴 ∈  𝑏𝑣 𝑢, 𝑝 : 𝑓∞  if and only if    sup𝑛 ,𝑚   𝑎  𝑛, 𝑘, 𝑚  𝑝𝑘 < ∞𝑘      (2.4) and   𝑎𝑛𝑘  ∈ 𝐷1 𝑝  𝑐𝑠.       (2.5)  Proof : Sufficiency: Suppose the conditions  holds and 𝑥 ∈ 𝑏𝑣(𝑢, 𝑝).Using the inequality which holds for any  𝐶 > 0  and any two complex numbers 𝑎, 𝑏    𝑎𝑏 ≤ 𝐶  𝑎𝐶−1 𝑞 + 𝑏𝑝 , where,  𝑝 > 1 and 𝑝−1 + 𝑞−1 = 1  (see, [3]), we have                              𝑡𝑚𝑛 (𝐴𝑥) =   𝑎 𝑛, 𝑘, 𝑚 𝑥𝑘𝑘  =   𝑎  𝑛, 𝑘, 𝑚 𝑦𝑘𝑘                                                 ≤  𝐵   𝑎  𝑛, 𝑘, 𝑚 𝐵−1 𝑝𝑘′+  𝑦𝑘  𝑝𝑘  𝑘   Now, taking 𝑠𝑢𝑝 over 𝑚, 𝑛 on both sides to the above inequality, we get 𝐴𝑥 ∈ 𝑓∞  for every  𝑥 ∈ 𝑏𝑣(𝑢, 𝑝), 𝑖. 𝑒., 𝐴 ∈  𝑏𝑣 𝑢, 𝑝 : 𝑓∞ .  Necessity: Suppose that  𝐴 ∈  𝑏𝑣 𝑢, 𝑝 : 𝑓∞ . Then 𝐴𝑥 exists for every 𝑥 ∈ 𝑏𝑣(𝑢, 𝑝), and this implies that  𝑎𝑛 ,𝑘 𝑘∈ℕ ∈ 𝑏𝑣(𝑢, 𝑝) 𝛽  for every  𝑛 ∈ ℕ,  the necessity of  (2.3) is immediate.  Now,  𝑎 𝑛, 𝑘, 𝑚 𝑥𝑘𝑘  exists for each 𝑚, 𝑛 and  𝑥 ∈ 𝑏𝑣 ( 𝑢, 𝑝), the sequences  𝑎(𝑛, 𝑘, 𝑚) 𝑘∈ℕ define the continuous linear functionals 𝜑𝑚𝑛 (𝑥) on 𝑏𝑣 𝑢, 𝑝  by 𝜑𝑚𝑛  𝑥 =  𝑎 𝑛, 𝑘, 𝑚 𝑥𝑘𝑘  ; 𝑛, 𝑘, 𝑚 ∈ ℕ. Since 𝑏𝑣 𝑢, 𝑝 is complete and sup𝑚 ,𝑛   𝑎  𝑛, 𝑘, 𝑚 𝑥𝑘𝑘  < ∞, so by uniform bounded principle , there exists 𝑀 > 0 such that                   𝑠𝑢𝑝𝑚 ,𝑛  𝜑𝑚𝑛  𝑥  = 𝑠𝑢𝑝𝑚 ,𝑛   𝑎 𝑛, 𝑘, 𝑚 𝑥𝑘𝑘                                                = 𝑠𝑢𝑝𝑚 ,𝑛 |  𝑎  𝑛, 𝑘, 𝑚 𝑥𝑘 |𝑘 ≤ 𝑀 < ∞.  This implies that 𝑠𝑢𝑝𝑚 ,𝑛  |𝑎  𝑛, 𝑘, 𝑚 𝑥𝑘 |𝑝𝑘′𝑘 < ∞, which shows the necessity of the condition (2.2) and the proof of (i) is complete.  Theorem 2.4 : (a) Let 1 < 𝑝𝑘 ≤ 𝑀 < ∞ for every 𝑘 ∈ ℕ. Then 𝐴 ∈  𝑏𝑣 𝑢, 𝑝 : 𝑓∞  if and only if (i)  the condition (2.2)-(2.5) of  Theorem 2.3 holds  (ii)  there is a sequence  𝛽𝑘  of scalars such that  𝑙𝑖𝑚𝑚𝑎  𝑛, 𝑘, 𝑚 = 𝛽𝑘 ,  uniformly in 𝑛.                    (2.6)  Proof: Sufficiency: Suppose that the conditions (2.2)-(2.6) hold and 𝑥 ∈ 𝑏𝑣(𝑢, 𝑝). Then 𝐴𝑥 exists and we have by (2.6) that  𝑎  𝑛, 𝑘, 𝑚 𝐵−1 𝑝𝑘′→  𝛽𝑘𝐵−1 𝑝𝑘′  as 𝑚 → ∞ uniformly in 𝑛 for each 𝑘 ∈ ℕ, which leads us with (2.2) that                          𝛽𝑗𝐵−1 𝑝𝑘′𝑘𝑗 =0 =   𝑎  𝑛, 𝑗, 𝑚 𝐵−1 𝑝𝑘′𝑘𝑗 =0                                                  ≤ 𝑠𝑢𝑝𝑚 ,𝑛   𝑎  𝑛, 𝑗, 𝑚 𝐵−1 𝑝𝑘′< ∞𝑗 ,   holding for every 𝑘 ∈ ℕ. Consequently reasoning as in the proof of the sufficiency of Theorem 2.3, the series  𝑎 𝑛, 𝑘, 𝑚 𝑥𝑘𝑘  and  𝛽𝑘𝑥𝑘𝑘  converges for every 𝑛, 𝑚 and for every 𝑥 ∈𝑏𝑣(𝑢, 𝑝). Now, for given 𝜀 > 0 and 𝑥 ∈ 𝑏𝑣(𝑢, 𝑝), choose a fixed 𝑘0 ∈ ℕ  such that KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 8, No. II, DECEMBER, 2012, pp 89-92 92     𝑥𝑘  𝑝𝑘∞𝑘=𝑘0+1 1𝐻 < 𝜀, where 𝐻 = 𝑠𝑢𝑝𝑘𝑝𝑘 . Then, there is some 𝑚0 ∈ ℕ, by  condition (ii) such that     𝑎(𝑛, 𝑘, 𝑚) − 𝛽𝑘 𝑘0𝑘=1  < 𝜀, for every  𝑚 ≥ 𝑚0 and uniformly in 𝑛.    Now, since  𝑎 𝑛, 𝑘, 𝑚 𝑥𝑘𝑘  and  𝛽𝑘𝑥𝑘𝑘  converges (absolutely) uniformly in 𝑛, 𝑚  and for  𝑥 ∈ 𝑏𝑣(𝑢, 𝑝), we have that    𝑎(𝑛, 𝑘, 𝑚) − 𝛽𝑘 ∞𝑘0+1𝑥𝑘 <𝜀2 , converges uniformly in 𝑛, 𝑚 and 𝑥 ∈ 𝑏𝑣(𝑢, 𝑝). Hence by conditions (i) and (ii) we have    𝑎(𝑛, 𝑘, 𝑚) − 𝛽𝑘 ∞𝑘0+1<𝜀2 for all  𝑚 ≥ 𝑚0 , uniformly in 𝑛. Therefore,     𝑎(𝑛, 𝑘, 𝑚) − 𝛽𝑘 ∞𝑘0+1 → 0  𝑚 → ∞  uniformly in  𝑖. 𝑒.,    𝑙𝑖𝑚𝑚  𝑎 𝑛, 𝑘, 𝑚 𝑥𝑘𝑘 =  𝛽𝑘𝑥𝑘𝑘  uniformly in 𝑛.                         (2.7)   Hence, 𝐴𝑥 ∈ 𝑓, which proves sufficiency.  Necessity: Suppose that  𝐴 ∈   𝑏𝑣 𝑢, 𝑝 : 𝑓  . Then, since 𝑓 ⊂ 𝑓∞( by Lemma 2.1 ), the necessities of condition (i) is immediately obtained from Theorem 2.1 . To prove the necessity of (ii) 𝑖. 𝑒., (2.6), consider the sequence 𝑒𝑘 = ( 0,0, …  , 1𝑘𝑡ℎ−𝑝𝑙𝑎𝑐𝑒 , 0,0, …  ) ∈ 𝑏𝑣 𝑢, 𝑝 , condition (ii) follows immediately by (2.7) and the   proof is complete.  Collary 2.5: 𝐴 ∈   𝑏𝑣 𝑢, 𝑝 : 𝑓0  if and only if condition (i) and (ii) of above Theorem holds along with 𝛽𝑘 = 0 for each  𝑘 ∈ ℕ.  Proof: The proof follows from theorem 2.4 by taking  𝛽𝑘 = 0 for each  𝑘 ∈ ℕ.  REFERENCES [1] Banach, S,  Theòries des operations linėaries, Warszawa, 1932.  [2] Başar, F, Altay, B  & Mursaleen, M , Some generalizations of the space bvp  of            p-bounded variation sequences, Nonlinear Anal., 68 (2)(2008), 273-287.  [3] Lorentz, G G, A contribution to the theory of divergent series, Acta Math., (80) (1948), 167-190.  [4] Mursaleen, M, Infinite matrices and almost convergent sequences, Southeast Asian Bulletin of Math., 19(1) (1995), 45-48.  [5]  Mursaleen, M,  Jarrah, A. M.  &  Mohiuddine, S. A., Almost convergence through the generalized de la Vallee-Pousin mean, Iranian J. Sci. Tech. Trans. A,  33(A2) (2009), 169-177.  [6] Nanda, S, Matrix transformations and almost boundedness, Glasnik Mat., 14(34) (1979), 99-107.  [7] Yasida, K, Functional Analysis, Springer-Verlag, Berlin Heidelberg, New York, 1966. 

Some matrix transformations and almost convergence

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Some matrix transformations and almost convergence

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