Multipliers on Generalized Mixed Norm Sequence Spaces

Sequence Spaces ◽

Mixed Norm ◽

Mixed Norm Space ◽

Norm Space ◽

Given1≤p,q≤∞and sequences of integers(nk)kand(nk′)ksuch thatnk≤nk′≤nk+1, the generalized mixed norm spaceℓℐ(p,q)is defined as those sequences(aj)jsuch that((∑j∈Ik‍|aj|p)1/p)k∈ℓqwhereIk={j∈ℕ0 s.t. nk≤j<nk′},k∈ℕ0. The necessary and sufficient conditions for a sequenceλ=(λj)jto belong to the space of multipliers(ℓℐ(r,s),ℓ𝒥(u,v)), for different sequencesℐand𝒥of intervals inℕ0, are determined.

A note on Köthe-Toeplitz duals of certain sequence spaces and their matrix transformations

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171295000871 ◽

1995 ◽

Vol 18 (4) ◽

pp. 681-688 ◽

Cited By ~ 5

Author(s):

B. Choudhary ◽

S. K. Mishra

Keyword(s):

Sufficient Conditions ◽

Sequence Spaces ◽

Matrix Transformations ◽

In this paper we define the sequence spacesSℓ∞(p),Sc(p)andSc0(p)and determine the Köthe-Toeplitz duals ofSℓ∞(p). We also obtain necessary and sufficient conditions for a matrixAto mapSℓ∞(p)toℓ∞and investigate some related problems.

Mixed-Norm Amalgam Spaces and Their Predual

Symmetry ◽

10.3390/sym14010074 ◽

2022 ◽

Vol 14 (1) ◽

pp. 74

Author(s):

Houkun Zhang ◽

Jiang Zhou

Keyword(s):

Duality Theory ◽

Fractional Integral ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Mixed Norm ◽

Fractional Integral Operators ◽

Primal Space ◽

Amalgam Spaces

In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.

On the generalized Riesz B-difference sequence spaces

Filomat ◽

10.2298/fil1004035b ◽

2010 ◽

Vol 24 (4) ◽

pp. 35-52 ◽

Cited By ~ 13

Author(s):

Metin Başarir

Keyword(s):

Sufficient Conditions ◽

Sequence Spaces ◽

Matrix Transformations ◽

Difference Sequence ◽

Topological Properties ◽

Linear Spaces ◽

The Matrix ◽

Difference Sequence Spaces

In this paper, we define the new generalized Riesz B-difference sequence spaces rq? (p, B), rqc (p, B), rq0 (p, B) and rq (p, B) which consist of the sequences whose Rq B-transforms are in the linear spaces l?(p), c (p), c0(p) and l(p), respectively, introduced by I.J. Maddox[8],[9]. We give some topological properties and compute the ?-, ?- and ?-duals of these spaces. Also we determine the necessary and sufficient conditions on the matrix transformations from these spaces into l? and c.

On Certain Sequence Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1981-027-5 ◽

1981 ◽

Vol 24 (2) ◽

pp. 169-176 ◽

Cited By ~ 276

Author(s):

H. Kizmaz

Keyword(s):

Sufficient Conditions ◽

Sequence Spaces ◽

AbstractIn this paper define the spaces l∞(Δ), c(Δ), and c0(Δ), where for instance l∞(Δ) = {x=(xk):supk |xk -xk + l|< ∞}, and compute their duals (continuous dual, α-dual, β-dual and γ-dual). We also determine necessary and sufficient conditions for a matrix A to map l∞(Δ) or c(Δ) into l∞ or c, and investigate related questions.

Masked factorable matrices

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500001608 ◽

2002 ◽

Vol 132 (1) ◽

pp. 245-254

Author(s):

Gord Sinnamon

Keyword(s):

Sufficient Conditions ◽

Sequence Spaces ◽

Factorable Matrices

The class of masked factorable matrices is introduced and simple necessary and sufficient conditions are given for matrices in the class to represent bounded transformations between Lebesgue sequence spaces.

FRÉCHET FRAMES, GENERAL DEFINITION AND EXPANSIONS

Analysis and Applications ◽

10.1142/s0219530514500018 ◽

2014 ◽

Vol 12 (02) ◽

pp. 195-208 ◽

Cited By ~ 2

Author(s):

STEVAN PILIPOVIĆ ◽

DIANA T. STOEVA

Keyword(s):

Banach Spaces ◽

Sufficient Conditions ◽

Sequence Spaces ◽

General Definition ◽

Fréchet Space ◽

Frechet Space ◽

Bk Space ◽

Frame Expansions ◽

We define an (X1, Θ, X2)-frame with Banach spaces X2 ⊆ X1, ‖ ⋅ ‖1 ≤ ‖ ⋅ ‖2, and a BK-space [Formula: see text]. Then by the use of decreasing sequences of Banach spaces [Formula: see text] and of sequence spaces [Formula: see text], we define a General Fréchet frame on the Fréchet space [Formula: see text]. We obtain frame expansions of elements of XF and its dual [Formula: see text], as well of some of the generating spaces of XF with convergence in appropriate norms. Moreover, we determine necessary and sufficient conditions for a General pre-Fréchet frame to be a General Fréchet frame, as well as for the complementedness of the range of the analysis operator U : XF → ΘF. Several examples illustrate our investigations.

New matrix domain derived by the matrix product

Filomat ◽

10.2298/fil1605233k ◽

2016 ◽

Vol 30 (5) ◽

pp. 1233-1241

Author(s):

Vatan Karakaya ◽

Necip Imşek ◽

Kadri Doğan

Keyword(s):

Topological Structure ◽

Sufficient Conditions ◽

Sequence Spaces ◽

Matrix Transformation ◽

Matrix Product ◽

Difference Matrix ◽

Matrix Domain ◽

The Matrix ◽

In this work, we define new sequence spaces by using the matrix obtained by product of factorable matrix and generalized difference matrix of order m. Afterward, we investigate topological structure which are completeness, AK-property, AD-property. Also, we compute the ?-, ?- and ?- duals, and obtain bases for these sequence spaces. Finally we give necessary and sufficient conditions on matrix transformation between these new sequence spaces and c,??.

The application domain of difference type matrix D(r,0,s,0,t) on some sequence spaces

Yugoslav journal of operations research ◽

10.2298/yjor200618032p ◽

2020 ◽

pp. 32-32

Author(s):

Avinoy Paul ◽

Binod Tripathy

Keyword(s):

Sufficient Conditions ◽

Sequence Spaces ◽

Application Domain ◽

Topological Properties ◽

Difference Type ◽

The Matrix ◽

Matrix Mappings

In this paper we introduce new sequence spaces with the help of domain of matrix D(r,0,s,0,t), and study some of their topological properties. Further, we determine ? and ? duals of the new sequence spaces and finally, we establish the necessary and sufficient conditions for characterization of the matrix mappings.

Measures of noncompactness in (N¯qΔ)summable difference sequence spaces

Filomat ◽

10.2298/fil1815459m ◽

2018 ◽

Vol 32 (15) ◽

pp. 5459-5470

Author(s):

Ishfaq Malik ◽

Tanweer Jalal

Keyword(s):

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Linear Operators ◽

Sequence Spaces ◽

Difference Sequence ◽

Hausdorff Measure Of Noncompactness ◽

Measures Of Noncompactness ◽

Difference Sequence Spaces

In this paper we first introduce N?q?summable difference sequence spaces and prove some properties of these spaces. We then obtain the necessary and sufficient conditions for infinite matrices A to map these sequence spaces into the spaces c,c0, and l?. Finally, the Hausdorff measure of noncompactness is then used to obtain the necessary and sufficient conditions for the compactness of the linear operators defined on these spaces.

Compact operators on some generalized mixed norm spaces

Filomat ◽

10.2298/fil1405019a ◽

2014 ◽

Vol 28 (5) ◽

pp. 1019-1026

Author(s):

A. Alotaibi ◽

E. Malkowsky ◽

H. Nergiz

Keyword(s):

Hausdorff Measure ◽

Measure Of Noncompactness ◽

Sufficient Conditions ◽

Compact Operators ◽

Mixed Norm ◽

Hausdorff Measure Of Noncompactness ◽

Mixed Norm Spaces ◽

We establish identities or estimates for the Hausdorff measure of noncompactness of operators from some generalized mixed norm spaces into any of the spaces c0, c, ?1, and [?1, ??]<m(?)>. Furthermore we give necessary and sufficient conditions for the operators in these cases to be compact. Our results are complementary to those in [1, 3, 13].