On matrix transformations and sequence spaces

In this paper are given results on the spacesw _τ (μ) andc _τ (μ, μ′) the second one generalizing the well-known spacec _∞ (μ) of sequences that are strongly bounded. Then we deal with matrix transformations into these spaces. These results generalize those given in [7].     

Some paranormed Euler sequence spaces of difference sequences of order m

The main purpose of this work is to extend the sequence spaces which are defined in [KARAKAYA, V.—POLAT, H.: Some new paranormed sequence spaces defined by Euler and difference operators, Acta Sci. Math. (Szeged) 76 (2010), 87–100] and [POLAT, H.—BASAR, F.: Some Euler spaces of difference sequences of order m, Acta Math. Sci. Ser. B Engl. Ed. 27 (2007), 254–266] by using difference operator of order m, and to give their alpha, beta and gamma duals. Furthermore, we characterize some classes of the related matrix transformations.     

Sum of sequence spaces and matrix transformations

Summary We are interested in the study of the sum <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>E+F$ and the product $E*F$, when $E$ and $F$ are of the form $s_{\xi}$, or $s_{\xi}^{\circ}$, or $s_{\xi}^{(c)}$. Then we deal with the identities $(E+F) (\Delta^{q}) \eg E$ and $(E+F) (\Delta^{q}) \eg F$. Finally we consider matrix transformations in the previous sets and study the identities $\big((E^{p_{1}}+F^{p_{2}}) (\Delta^{q}),s_{\mu}\big) \eg S_{\alpha^{p_{1}}\pl \beta^{p_{2}},\mu}$ and $\big(E+F(\Delta^{q}),s_{\gamma}\big) \eg S_{\beta,\gamma}$.     

On some new paranormed euler sequence spaces and Euler Core

In this paper, the sequence spaces e0^τ（u, p） and ec^τ（u, p） of non-absolute type which are the generalization of the Maddox sequence spaces have been introduced and it is proved that the spaces e0^τ（u,p） and ec^τ（u,p） are linearly isomorphic to spaces co（p） and c（p）, respectively. Furthermore, the α-, β- and γ-duals of the spaces 0^τ（u,p） and ec^τ（u,p） have been computed and their bases have been constructed and some topological properties of these spaces have been investigated. Besides this, the class of matrices （e0^τ）（u, p） ： μ） has been characterized, where μ is one of the sequence spaces l∞, c and co and derives the other characterizations for the special cases of μ. In the last section, Euler Core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.     

Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness

The spaces and introduced by Ayd?n and Ba?ar [C. Ayd?n, F. Ba?ar, Some new difference sequence spaces, Appl. Math. Comput. 157 (3) (2004) 677-693] can be considered as the matrix domains of a triangle in the sets of all sequences that are summable to zero, summable, and bounded by the Cesàro method of order 1. Here we define the sets of sequences which are the matrix domains of that triangle in the sets of all sequences that are summable, summable to zero, or bounded by the strong Cesàro method of order 1 with index p?1. We determine the β-duals of the new spaces and characterize matrix transformations on them into the sets of bounded, convergent and null sequences.     

On matrix mappings into some strong Cesàro sequence spaces

We determine the classes (X, Y_T) of matrix transformations from X into Y_T where X is one of the classical sequence spaces c₀, c, ?_∞ and ?₁ of all null, convergent and bounded complex sequences and all absolutely convergent complex series, T is a triangle, Y_T is the matrix domain of T in Y and Y is any of the sets of all sequences that are summable, summable to zero or bounded by the strong Cesàro method of order 1, with index 1 ? p < ∞. Furthermore, we determine the representations of the general bounded linear operators from c into Y. We also establish estimates for the norms of the operators in each case.     

A note on hyperbolic transformations

We shall discuss the so‐called hyperbolic Householder and Givens transformations applied to complex matrices, including the case of zero hyperbolic energy of a transformed vector. For each case a numerically stable algorithm is available. Copyright © 2001 John Wiley & Sons, Ltd.     

On some generalized difference paranormed sequence spaces associated with multiplier sequence defined by modulus function

In this article we introduce the paranormed sequence spaces(f,Λ,△m,p),c0(f,Λ,△m,p) and ■∞(f,Λ,△m,p),associated with the multiplier sequence Λ =(λk),defined by a modulus function f.We study their different properties like solidness,symmetricity,completeness etc.and prove some inclusion results.     

Sum and product of certain BK spaces and matrix transformations between these spaces

We deal with the sum and the product of particular BK spaces and give necessary and sufficient conditions to have w _α(λ) + w _β(μ) = w _{α +β} (μ). Some results on matrix transformations mapping the space w _α(λ) + w _β(μ) into a given BK space are also given. This study generalizes some results obtained in [7] and [8]. This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalized difference sequence spaces and their dual spaces

The definition of the pα-, pβ- and pγ-duals of a sequence space was defined by Et [Internat. J. Math. Math. Sci. 24 (2000) 785-791]. In this paper we compute pα- and N-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀, and compute β- and γ-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀.     

Some new sequence spaces derived by The composition of binomial matrix and double band matrix

In this paper, we construct three new sequence spaces $b^{{r,s}}_{0}(G)$, $b^{{r,s}}_{c}(G)$ and $b^{{r,s}}_{\infty}(G)$ and mention some inclusion relations, where $G$ is generalized difference matrix. Moreover, we give Schauder basis of the spaces $b^{{r,s}}_{0}(G)$ and $b^{{r,s}}_{c}(G)$. Afterward, we determine $\alpha-$, $\beta-$ and $\gamma-$duals of those spaces. Finally, we characterize some matrix classes related to the space $b^{{r,s}}_{c}(G)$.     

A note on compact operators on matrix domains

We establish some identities or estimates for the operator norms and Hausdorff measures of noncompactness of linear operators given by infinite matrices that map the matrix domains of triangles in arbitrary BK spaces with AK, or in the spaces of all convergent or bounded sequences, into the spaces of all null, convergent or bounded sequences, or of all absolutely convergent series. Furthermore, we apply these results to the characterizations of compact operators on the matrix domains of triangles in the classical sequence spaces, and on the sequence spaces studied in [I. Djolovi?, Compact operators on the spaces and , J. Math. Anal. Appl. 318 (2) (2006) 658-666; I. Djolovi?, On the space of bounded Euler difference sequences and some classes of compact operators, Appl. Math. Comput. 182 (2) (2006) 1803-1811].     

A note on Kamenev type theorems for second order matrix differential systems

Some oscillation criteria are given for the second order matrix differential system , where and are real continuous matrix functions with symmetric, . These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation , where is the identity matrix, is a given function on and .

     

Measurable linear transformations on abstract Wiener spaces

Measurable linear transformations from an abstract Wiener space to a Hilbert space are characterized. It is shown that the measure on any infinite dimensional abstract Wiener space can be transformed to that on any other by a measurable linear transformation.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.