Microsoft Word - AB Hamis Ganie _97-102_ Ab Hamis Ganie / BIBECHANA 9 (2013) 97-101 : BMHSS, p.97 (Online Publication: Nov., 2012) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Almost boundedness and matrix transformation Ab Hamid Ganie Department of Mathematics, National Institute of Technology, Srinagar (India)- 190006. Email: ashamidg@rediffmail.com Article history: Received 25 September, 2012; Accepted 12 November, 2012 Abstract The sequence space ��� have been defined and the various classes of infinite matrices have been characterized by Aydin and Başar, ( see, [1] ) , where 1 p≤ ≤ ∞ . In this paper we characterize the classes ( : )r c a f∞ , ( : )r c a f and 0 ( : )r c a f ,where ,f f∞ and 0 f denote respectively the spaces of almost bounded sequences , almost convergent sequences and almost convergent null sequences. Keywords: Sequence space of non-absolute type;, almost convergent sequences; β-duals and Matrix Transformations. 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 denote the space of all sequences ( real or complex ). Let and �be two non-empty subsets of . Let � = ���� , �, � ∈ ℕ�, be an infinite matrix of real or complex numbers. We write ���� = �� �� = ∑ ����� � . Then �� = ��� ��� is called the �-transform of �, whenever �� �� = ∑ ����� � < ∞ for all � ∈ ℕ. We write lim� �� = lim� �� ��. If � ∈ implies �� ∈ �, we say that �-defines a matrix transformations from into �, denoted by �: → �. By : ��, we mean the class of all matrices � such that �: → �. For a sequence space, the matrix domain of an infinite matrix � is defined as (1) = �� = ��� ∈ : �� ∈ � Let ℓ" and # be the Banach spaces of bounded and convergent sequences � = �� ����$%" with supremum norm ‖�‖ = '()�|� ��|. Let + denote the shift operator on , that is , +� = �� ����$," , +-� = �� ����$-" and so on. A Banach limit .is a non-negative linear functional Ab Hamis Ganie / BIBECHANA 9 (2013) 97-102: BMHSS, p.98 (Online Publication: Nov., 2012) on ℓ" such that . is invariant under the shift operator and . /� = 1, where / = 1,1, … � (see, [2]), that is, a functional .: ℓ" → ℝ is called a Banach limit if (i) . is linear, (ii) . �� ≥ 0if�� ≥ 0 for all �. (iii) . �� = . +��where+ is shift operator on . (iv) . /� = 1, where / = 1, 1 , … �. Since the Hahn-Banach norm preserving extension is not unique, there must be many Banach limits in the dual space of ℓ", and usually different Banach limits have different values at the same element in ℓ". However, there indeed exists sequences whose values of all Banach limits are same. If � = �����$%" ∈ #, where # is a Banach space of ℓ" consisting of convergent sequences, then . �� = lim� �� is a trivial example. Besides this there also exists non-convergent sequences satisfying this property. For example � = �1, 0, 1, 0, … �the value of . �� = , - is same for every Banach limit. Lorentz (see, [4]) called a sequence � = �����$," almost convergent if all Banach limits of �, . �� , are same, and this unique Banach limit is called 5-lim of �. In his paper Lorentz proved the following criterion for almost convergent sequences. A � = �����$%" ∈ ℓ" is almost convergent with 5-limit . �� if and only if lim6→" 78� ��= . ��, where, 78� �� = , 6 ∑ +9��6:, 9$% , +% = 0�, uniformly in � ≥ 0. The above limit can be rewritten in detail as ∀< > 0�, ∃)%� ∀) > )%� ∀�� ?�� + ⋯ + ��B6:, ) − .? < <. We denote the set of almost convergent sequences by D. D = �� ∈ E" ∶ lim8 78� �� /�G'7' , (�GDHIJEK G� ��. Nanda [6] has defined a new set of sequences f∞ as follows: D" = �� ∈ E" ∶ lim8|78� ��| < ∞�. We call D"the set of all almost bounded sequences. 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., ( see, [1, 5, 7] ). Following (see, [1], [7]), the sequence space r c a is defined as the set of all sequences whose rA -transform is in c , that is, 0 1 ( ) :lim ( 1 ) 1 n r k c k k n k a x x r x exists n ω =   = = ∈ +  +  ∑ Ab Hamis Ganie / BIBECHANA 9 (2013) 97-102 : BMHSS, p.99 (Online Publication: Nov., 2012) ; where , 1 , 0 , 1 0 , . k r nk r k n n a k n  + ≤ ≤ + =   >   With the notation of (1) that, ( ) r r c A a c= . 2. Main Results Define the sequence ( ( )) k y y r= which will be used, by the rA -transform of a sequence ( ) k x x= , that is, (2) 0 1 ( ) ; 1 jk k j j r y r x k= + = +∑ for .k∈ℕ For brevity in notation, we write 0 1 ( ) ( ) ( , , ) 1 m mn n i k j k t Ax A x a n k m x m + = = = + ∑ ∑ where , , 0 1 ( , , ) ; ( , , ) 1 m n j k j a n k m a n k m m + = = ∈ + ∑ ℕ Also , 1 ( , , ) ( , , ) ( , 1, ) ( , , ) ( 1) ( 1) 1 1 1k k k a n k m a n k m a n k m a n k m k k r r r + +   =∆ + = − +   + + +    ɶ We denote by X β , the β − deal of a sequence space X and mean the set of all the sequences ( ) k x x= such that ( ) k k xy x y cs= ∈ for all ( ) k y y X= ∈ . Now, we give the following lemmas which will be needed in proving the main Theorems. Lemma 2.1[1]: Define the sets 1 ( )D p and 2 ( )D p as follows 1 ( ) : ( 1) 1 r k k k k a D a a k r ω     = = ∈ ∆ + < ∞  +    ∑ 2 ( ) : 1 r k k k a D a a cs r ω    = = ∈ ∈  +   where , 1 k k a r   ∆  +  = 1 k k a r+ - 11 k k a r ++ Then , 1 2 r r r c a D D β   =  ∩ Lemma 2.2 [5]: f f∞⊂ Theorem 2.1: ( : )r c A a f∞∈ if and only if Ab Hamis Ganie / BIBECHANA 9 (2013) 97-102 : BMHSS, p.100 (Online Publication: Nov., 2012) (3) , sup ( , , ) n m k a n k m ∈ < ∞∑ ℕ ɶ and (4) 1 nk k k a cs r ∈   ∈  +  ℕ , for all n∈ℕ Proof: Sufficiency: Suppose the conditions (3) & (4) holds and .r c x a∈ Then { },n k k a ∈ ∈ ℕ r c a β    for every n∈ℕ ,the A -transform of x exists. Since r c x a∈ , by hypothesis, and r c a c≅ ( see,[1] ) , we have y c∈ . Thus , we can find 0K > such that sup k k y K< . ( ) ( , , ) mn k k t Ax a n k m x= ∑ ( , , ) k k a n k m y= ∑ ɶ ( , , ) k k a n k m y≤ ∑ ɶ ( , , ) k K a n k m≤ ∑ ɶ Taking , sup m n on both sides, we get Ax f∞∈ for every .r c x a∈ Necessity: Suppose that ( : )r c A a f∞∈ . Then Ax exists for every r c x a∈ and this implies that { },n k k a ∈ ∈ ℕ r c a β    for every n∈ℕ , the necessity of (4) is immediate. Now, ( , , ) k k a n k m x∑ exists for each m , n and r c x a∈ , the sequences { ( , , )} k a n k m ∈ℕ define the continuous linear functionals ( ) mn xψ on r c a by ( ) ( , , ) mn k k x a n k m xψ =∑ ; ( , , )n k m∈ℕ . Since r c a and c are norm isomorphic ( see [1] , ), it should follow with (2) that ( ) ( , , ) mn x a n k mψ = ɶ holds for every k∈ℕ . This implies that the functionals defined by mn ψ on r c a are point wise bounded, so by uniform bounded principle , there exists 0M > such that ( ) mn x Mψ ≤ for every ,m n∈ℕ . Thus we conclude that , , , sup ( ) sup ( , , ) sup ( , , ) mn k k m n m n m nk k x a n k m x a n k m y Mψ = = <∑ ∑ ɶ This implies that , sup ( , , ) n m k a n k m ∈ < ∞∑ ℕ ɶ , which shows the necessity of the condition (3) and the proof of (i) is complete.□ Theorem 2.2 : ( : )r c A a f∈ if and only if (3) ,(4) and (5) lim ( , , ) , k m a n k m β=ɶ uniformly in n , and for each k∈ℕ . (6) lim ( , , ) 0k m a n k m β− =∑ ɶ , uniformly in n . Ab Hamis Ganie / BIBECHANA 9 (2013) 97-102 : BMHSS, p.101 (Online Publication: Nov., 2012) Proof: Sufficiency: Suppose that the conditions (3), (4), (5) and (6) hold and r c x a∈ . Then Ax exists and at this stage , we observe with the help of (5 ) & (6) that ,0 sup ( , , ) k j m nj j a n j mβ = = < ∞∑ ∑ ɶ holds for every k .This gives that ( ) 1k lβ ∈ .Since r c x a∈ by hypothesis and r c a c≅ ( see,[1] ) , we have y c∈ .Therefore , we can easily see that ( ) 1k k y lβ ∈ for each y c∈ and also there exists 0K > such that sup k k y K< . Now for 0ε > , choose a fixed 0 k ∈ℕ ,there is some 0 m ∈ℕ such that 0 0 ( , , ) 2 k k k k a n k m y ε β = − <∑ ɶ for every 0 m m≥ and 0 k ∈ℕ . Also by (6) , there is some 1 m ∈ℕ , such that 0 1 ( , , ) 2 k k k a n k m ε β ∞ = + − <∑ ɶ for every 1 m m≥ uniformly in n .Thus , we have ( ) 0 1 ( , , ) 1 m k k k kn j j k k Ax y a n k m y m β β + = − = − + ∑ ∑ ∑ ɶ 0 00 1 ( , , ) ( , , ) k k k k k k k k a n k m y a n k m yβ β ∞ = = + ≤ − + −∑ ∑ɶ ɶ 0 1 ( , , ) 2 k k k k a n k m y ε β ∞ = + < + −∑ ɶ 2 2 K K ε ε ε< + = for all sufficiently large m ,uniformly in n . Hence, Ax f∈ , which proves sufficiency. Necessity: Suppose that ( : )r c A a f∈ . Then , since f f∞⊂ ( by Lemma 2.2 ), the necessities of (3) and (4) are immediately obtained from Theorem 2.1 . To prove the necessity of (5), consider the sequence ( )( ) ( )( ) ( )k k n b r b r= for every k∈ℕ , where ( ) 1 ( 1) , 1 ( ) 1 0 , 0 1 n k k k n k k n k b r r n k or n k − + − ≤ ≤ + = +  ≤ < > + Since Ax exists and is in f for each r c x a∈ , one can easily see that Ab Hamis Ganie / BIBECHANA 9 (2013) 97-102 : BMHSS, p.102 (Online Publication: Nov., 2012) ( ) ( ) ( 1) 1 k nk k n a Ab r k f r ∈    = ∆ + ∈  +   ℕ for all k∈ℕ ,which proves the necessity of (6). Similarly taking r c x e a= ∈ , we shall get ( 1) 1 nk k k n a Ax k f r ∈    = ∆ + ∈  +   ∑ ℕ ,which proves the necessity of (5).This concludes the proof. Note that if we replace f by 0 f , then Theorem 2.2 is reduced to the following corollary: Corollary: 0 ( : )r c A a f∈ if and only if (3) ,(4), (5) and (6) holds with 0 k β = for each k∈ℕ . References [1] C. Aydinand F. Başar, On the new sequence space of which include the spaces 0 c and c , Hokkaido Math. J., 33(2) (2004) 83-398. [2] S. Banach, Theỏries des operations linẻaries, Warszawa, (1932). [3] C. G. Lascarides and I. J. Maddox, Matrix transformations between some classes of sequences, Proc. Camb. Phil. Soc., 68 (1970) 99-104. [4] G. G. Lorentz, A contribution to the theory of divergent series, Acta Math., 80(1948)167-190. [5] Mursaleen, Infinite matrices and almost convergent sequences, Southeast Asian Bulletin of Math. 19(1995) 45-48. [6] S. Nanda, Matrix transformations and almost boundedness, Glasnik Mat., 14(1979) 99-107. [7] N. A. Sheikh, and A. H. Ganie,On the λ-convergent sequence and almost convergence ( to be appeared in Thai J. of Math, vol.3 (2012).