Convergent and bounded Ces�ro sections inFK-spaces

Buntinas, Martin

doi:10.1007/bf01111591

Cited by 18 publications

(10 citation statements)

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“…It can be topologized with the seminorms Pi(x) = |xi|, (i = 1,2,...), and any vector subspace of w is called a sequence space. A sequence space X, with a vector space topology T, is a K-space provided that the inclusion mapping < oo}, bv 0 := bvD c 0 (see [2], [4] and [6]). Troughout the paper e denotes the sequence of ones, (1,1,..., 1,...); (j = 1,2,...), the sequence (0,0,... ,0,1,0,...) with the one in the j-th position.…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

“…It follows from a result of Buntinas [4] that q C bv C c, thus q C T(h). We now prove the reverse inclusion.…”

Section: " N=lmentioning

confidence: 96%

“…The sequence spaces The subspace of a locally convex sequence space X consisting of the sequences with the property AK (respectively oK) is denoted by XAK (respectively X a x). Every AK-space is a oK-space, [4]. For example Finally, s = {s n }^Li always denotes a strictly increasing sequence of non-negative integers with si = 0.…”

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Cesáro Conull Fk-Spaces

İnce¹

2000

Demonstratio Mathematica

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Abstract. The purpose of this paper is to study the (strongly) Cesaro conull FKspaces and to give some characterizations. IntroductionThe classification of conservative matrices as conull or coregular, due to Wilansky [14], has been extended by Yurimyae [17] and Snyder [13] to all FK-spaces. Bennett [2] continued work on conull FK-spaces; and improved some results of Sember [10]- [12].Motivating by Bennett's paper [2] and his talks at the Ankara University during the summer of 1996 we study the (strongly) Cesaro conull FK-spaces.In Section 2 we introduce the notation and terminology while in Section 3 we study the Cesaro conull FK-spaces and provide some examples to illustrate the differences between the conull and Cesaro conull FK-spaces. Section 4 deals with the strongly Cesaro conull FK-spaces; and gives a relationship between the (Cesaro wedge) weak Cesaro wedge and (strongly) Cesaro conull FK-spaces. In Section 5 we obtain some results for a summability domain EA to be (strongly) Cesaro conull. Section 6 presents some applications to summability domains. Notation and preliminary resultsLet w denote the space of all real or complex-valued sequences. It can be topologized with the seminorms Pi(x) = |xi|, (i = 1,2,...), and any vector subspace of w is called a sequence space. A sequence space X, with a vector space topology T, is a K-space provided that the inclusion mapping By m,c, Co we denote the spaces of all bounded sequences, convergent sequences and null sequences, respectively. These are FK-spaces under ||a:|| = sup fc By £ p , (1 < p < oo), and cs we shall denote the space of all absolutely p-summable sequences, and convergent series, respectively. As usual i 1 is replaced by I. The sequence spaces The subspace of a locally convex sequence space X consisting of the sequences with the property AK (respectively oK) is denoted by XAK (respectively X a x). Every AK-space is a oK-space, [4]. For example Finally, s = {s n }^Li always denotes a strictly increasing sequence of non-negative integers with si = 0. We shall also be interested in spaces of the form SN+L c|s| := |a; G w : limij = 0 and sup ^ j|Aa;j|

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Section: Notation and Preliminary Resultsmentioning

confidence: 99%

“…It follows from a result of Buntinas [4] that q C bv C c, thus q C T(h). We now prove the reverse inclusion.…”

Section: " N=lmentioning

confidence: 96%

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Cesáro Conull Fk-Spaces

İnce¹

2000

Demonstratio Mathematica

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“…(see [3], [6], [8] and [9]). Throughout the paper e denotes the sequences of ones, (1,1,..., 1,...); <5 J , (j = 1,2,...), the sequence (0,0,..., 0,1,0,...) with the one in the jth position; (j > the linear span of the <5 J 's.…”

Section: Tlmentioning

confidence: 99%

Cλ-Conull FK-SPACES

Dagadur¹

2002

Demonstratio Mathematica

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Abstract. Recall that a C\ method is obtained by deleting a set of rows from the Cesaro matrix C\. In this paper we study the (strongly) C\-conull FK-spaces and give some characterizations. We also apply these results to summability domains. Introduction and notationThe classification of conservative matrices as conull or coregular was defined by Wilansky in [16] In section 2, for an FK-space X, the concepts of CA-conullity have been defined. Their relationship to ordinary conullity and Cesaro conullity, weak CA-wedgeness and CVwedgeness have also been examined.In section 3 we study the subspaces C\W and C\S of an FK-space X.In section 4 we obtain some results for a summability domain Ya to be (strongly) CA-conull.Let F be an infinite subset of N and F as the range of a strictly increasing sequence of positive integers, say F = {A(n)}^L 1 . The Cesaro submethod C\ is defined as where {x*} is a sequence of a real or complex numbers. Therefore, the C A -method yields a subsequence of the Cesaro method C\, and hence it 1991 Mathematics Subject Classification: Primary 46A45; Secondary 47B37, 40H05.

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“…For an FF-space F, various forms of sectional boundedness and sectional convergence have been shown to be equivalent to invariances of the form F = D-E with respect to coordinatewise multiplication by some space D. Such statements show the equivalence of topological properties of F with algebraic properties of F. In 1968 Garling [9] showed that an FF-space F has the property of sectional boundedness AB if and only if F is invariant with respect to the space bv of sequences of bounded variation, and that F has the property of sectional convergence AK if and only if F = bvQ • F. In 1970 Buntinas [4] showed that, for an FF-space F, Cesàro sectional boundedness oB is equivalent to invariance with respect to the space q of bounded quasiconvex sequences and that Cesàro sectional convergence oK is equivalent to invariance with respect to the space q0 = q n c0 of quasiconvex null sequences. In 1973 results were obtained for more general Toeplitz sections [5].…”

Section: Introductionmentioning

confidence: 99%

Absolute boundedness and absolute convergence in sequence spaces

Buntinas¹,

Tanović-Miller²

1991

Proc. Amer. Math. Soc.

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Abstract.Let ft* be the set of all sequences h = (hk)k*Ax of Os and Is. A sequence x in a topological sequence space E has the property of absolute boundedness \AB\ if ft* • x = {y\yk = hkxk , h € ft*} is a bounded subset of E . The subspace E,AB, of all sequences with absolute boundedness in E has a natural topology stronger than that induced by E. A sequence x has the property of absolute sectional convergence \AK\ if, under this stronger topology, the net {h • x} converges to x , where h ranges over all sequences in ft* with a finite number of Is ordered coordinatewise (h1 < h" iff V/c, hk < hk ). Absolute boundedness and absolute convergence are investigated. It is shown that, for an F.K-space E, we have E = E,AB, if and only if E = l°° • E, and every element of E has the property \AK\ if and only if E = c0 • E . Solid hulls and largest solid subspaces of sequence spaces are also considered. The results are applied to standard sequence spaces, convergence fields of matrix methods, classical Banach spaces of Fourier series and to more recently introduced spaces of absolutely and strongly convergent Fourier series.

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Convergent and bounded Ces�ro sections inFK-spaces

Cited by 18 publications

References 11 publications

Cesáro Conull Fk-Spaces

Cesáro Conull Fk-Spaces

Cλ-Conull FK-SPACES

Absolute boundedness and absolute convergence in sequence spaces

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