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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 612671, 10 pages
http://dx.doi.org/10.1155/2012/612671
Research Article

Some Matrix Transformations of Convex and Paranormed Sequence Spaces into the Spaces of Invariant Means

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 9 February 2012; Accepted 13 April 2012

Academic Editor: AlbertoΒ Fiorenza

Copyright Β© 2012 M. Mursaleen and S. A. Mohiuddine. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We determine the necessary and sufficient conditions to characterize the matrices which transform convex sequences and Maddox sequences into π‘‰πœŽ(πœƒ) and π‘‰βˆžπœŽ(πœƒ).

1. Introduction and Preliminaries

By 𝑀, we denote the space of all real-valued sequences π‘₯=(π‘₯π‘˜)βˆžπ‘˜=1. Any vector subspace of 𝑀 is called a sequence space. We write that β„“βˆž, 𝑐, and 𝑐0 denote the sets of all bounded, convergent, and null sequences, respectively, and note that π‘βŠ‚β„“βˆž; also 𝑐𝑠 and ℓ𝑝 are the set of all convergent and 𝑝-absolutely convergent series, respectively, where β„“π‘βˆ‘βˆΆ={π‘₯βˆˆπ‘€βˆΆβˆžπ‘˜=0|π‘₯π‘˜|𝑝<∞} for 1≀𝑝<∞. In the theory of sequence spaces, a beautiful application of the well-known Hahn-Banach extension theorem gave rise to the concept of the Banach limit. That is, the lim functional defined on 𝑐 can be extended to the whole β„“βˆž, and this extended functional is known as the Banach limit [1]. In 1948, Lorentz [2] used this notion of a weak limit to define a new type of convergence, known as the almost convergence. Later on, Raimi [3] gave a slight generalization of almost convergence and named it the 𝜎-convergence.

A sequence space 𝑋 with a linear topology is called a 𝐾-π‘ π‘π‘Žπ‘π‘’ if each of the maps π‘π‘–βˆΆπ‘‹β†’β„‚ defined by 𝑝𝑖(π‘₯)=π‘₯𝑖 is continuous for all π‘–βˆˆβ„•. A 𝐾-space 𝑋 is called an 𝐹𝐾-π‘ π‘π‘Žπ‘π‘’ if 𝑋 is a complete linear metric space. An 𝐹𝐾-space whose topology is normable is called a 𝐡𝐾-space. An 𝐹𝐾-space 𝑋 is said to have 𝐴𝐾 property if π‘‹βŠƒπœ™ and (𝑒(π‘˜)) is a basis for 𝑋, where (𝑒(π‘˜)) is a sequence whose only nonzero term is a 1 in π‘˜th place for each π‘˜βˆˆβ„• and πœ™=span{𝑒(π‘˜)}, the set of all finitely nonzero sequences. If πœ™ is dense in 𝑋, then 𝑋 is called an 𝐴𝐷-π‘ π‘π‘Žπ‘π‘’, thus 𝐴𝐾 implies 𝐴𝐷. For example, the spaces 𝑐0, 𝑐𝑠, and ℓ𝑝 are 𝐴𝐾-spaces.

Let 𝑋 and π‘Œ be two sequence spaces, and let 𝐴=(π‘Žπ‘›π‘˜)βˆžπ‘›;π‘˜=1 be an infinite matrix of real or complex numbers. We write 𝐴π‘₯=(𝐴𝑛(π‘₯)), π΄π‘›βˆ‘(π‘₯)=π‘˜π‘Žπ‘›π‘˜π‘₯π‘˜ provided that the series on the right converges for each 𝑛. If π‘₯=(π‘₯π‘˜)βˆˆπ‘‹ implies that 𝐴π‘₯βˆˆπ‘Œ, then we say that 𝐴 defines a matrix transformation from 𝑋 into π‘Œ, and by (𝑋,π‘Œ), we denote the class of such matrices.

Let 𝜎 be a one-to-one mapping from the set of natural numbers into itself. A continuous linear functional πœ‘ on the space β„“βˆž is said to be an π‘–π‘›π‘£π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘‘π‘šπ‘’π‘Žπ‘› or a 𝜎-π‘šπ‘’π‘Žπ‘› if and only if (i) πœ‘(π‘₯)β‰₯0 if π‘₯β‰₯0 (i.e., π‘₯π‘˜β‰₯0 for all π‘˜), (ii) πœ‘(𝑒)=1, where 𝑒=(1,1,1,…), and (iii) πœ‘(π‘₯)=πœ‘((π‘₯𝜎(π‘˜))) for all π‘₯βˆˆβ„“βˆž.

Throughout this paper, we consider the mapping 𝜎 which has no finite orbits, that is, πœŽπ‘(π‘˜)β‰ π‘˜ for all integer π‘˜β‰₯0 and 𝑝β‰₯1, where πœŽπ‘(π‘˜) denotes the 𝑝th iterate of 𝜎 at π‘˜. Note that a 𝜎-mean extends the limit functional on the space 𝑐 in the sense that πœ‘(π‘₯)=lim

π‘₯ for all π‘₯βˆˆπ‘, (cf. [4]). Consequently, π‘βŠ‚π‘‰πœŽ, the set of bounded sequences all of whose 𝜎-means are equal. We say that a sequence π‘₯=(π‘₯π‘˜) is 𝜎-π‘π‘œπ‘›π‘£π‘’π‘Ÿπ‘”π‘’π‘›π‘‘ if and only if π‘₯βˆˆπ‘‰πœŽ. Using this concept, Schaefer [5] defined and characterized 𝜎-conservative, 𝜎-regular, and 𝜎-coercive matrices. If 𝜎 is translation, then π‘‰πœŽ is reduced to the set 𝑓 of almost convergent sequences [2].

The notion of 𝜎-convergence for double sequences has been introduced in [6] and further studied in [7–9].

Recently, the sequence spaces π‘‰πœŽ(πœƒ) and π‘‰βˆžπœŽ(πœƒ) have been introduced in [10] which are related to the concept of 𝜎-mean and the lacunary sequence πœƒ=(π‘˜π‘Ÿ).

In this section, we establish the necessary and sufficient conditions on the matrix 𝐴=(π‘Žπ‘›π‘˜)βˆžπ‘›,π‘˜=1 which transforms π‘Ÿ-convex sequences in to the spaces π‘‰πœŽ(πœƒ) and π‘‰βˆžπœŽ(πœƒ).

By a lacunary sequence, we mean an increasing integer sequence πœƒ=(π‘˜π‘Ÿ) such that π‘˜0=0 and β„Žπ‘ŸβˆΆ=π‘˜π‘Ÿβˆ’π‘˜π‘Ÿβˆ’1β†’βˆž as π‘Ÿβ†’βˆž. Throughout this paper, the intervals determined by πœƒ will be denoted by πΌπ‘ŸβˆΆ=(π‘˜π‘Ÿβˆ’1,π‘˜π‘Ÿ], and the ratio π‘˜π‘Ÿ/π‘˜π‘Ÿβˆ’1 will be abbreviated by π‘žπ‘Ÿ.

A bounded sequence π‘₯=(π‘₯π‘˜) of real numbers is said to be 𝜎-lacunary convergent to a number 𝐿 if and only if limπ‘Ÿβ†’βˆž(1/β„Žπ‘Ÿ)βˆ‘π‘—βˆˆπΌπ‘Ÿπ‘₯πœŽπ‘—(𝑛)=𝐿, uniformly in 𝑛, and let π‘‰πœŽ(πœƒ) denote the set of all such sequences, that is,π‘‰πœŽξƒ―(πœƒ)=π‘₯βˆˆβ„“βˆžβˆΆlimπ‘šβ†’βˆž1β„Žπ‘Ÿξ“π‘—βˆˆπΌπ‘Ÿπ‘₯πœŽπ‘—(𝑛)ξƒ°=𝐿uniformlyin𝑛.(1.1) In this case, 𝐿 is called the (𝜎,πœƒ)-limit of π‘₯. We remark that(i) if 𝜎(𝑛)=𝑛+1, then π‘‰πœŽ(πœƒ) is reduced to the space 𝑓(πœƒ) (cf. [11]),(ii)π‘βŠ‚π‘‰πœŽ(πœƒ)βŠ‚β„“βˆž.

A bounded sequence π‘₯=(π‘₯π‘˜) of real numbers is said to be 𝜎-lacunary bounded if and only if supπ‘Ÿ,𝑛|(1/β„Žπ‘Ÿ)βˆ‘π‘—βˆˆπΌπ‘Ÿπ‘₯πœŽπ‘—(𝑛)|<∞, and let π‘‰βˆžπœŽ(πœƒ) denote the set of all such sequences, that is,π‘‰βˆžπœŽβŽ§βŽͺ⎨βŽͺ⎩(πœƒ)=π‘₯βˆˆβ„“βˆžβˆΆsupπ‘Ÿ,𝑛|||||1β„Žπ‘Ÿξ“π‘—βˆˆπΌπ‘Ÿπ‘₯πœŽπ‘—(𝑛)|||||⎫βŽͺ⎬βŽͺ⎭<∞.(1.2) We remark that π‘βŠ‚π‘‰πœŽ(πœƒ)βŠ‚π‘‰βˆžπœŽ(πœƒ)βŠ‚β„“βˆž and the spaces π‘‰βˆžπœŽ(πœƒ) and π‘‰βˆžπœŽ(πœƒ) are BK spaces with the normβ€–π‘₯β€–=supπ‘Ÿ,𝑛|||||1β„Žπ‘Ÿξ“π‘—βˆˆπΌπ‘Ÿπ‘₯πœŽπ‘—(𝑛)|||||.(1.3)

2. Convex Sequence Spaces

Pati and Sinha [12] defined π‘Ÿ-convex sequences as follows: a real sequence π‘₯=(π‘₯π‘˜)βˆžπ‘˜=0 is said to be π‘Ÿ-π‘π‘œπ‘›π‘£π‘’π‘₯,π‘Ÿβˆˆβ„•, if Ξ”π‘Ÿπ‘₯π‘˜β‰₯0 for all π‘˜βˆˆβ„•, where Ξ”π‘Ÿπ‘₯π‘˜ is defined byΞ”0π‘₯π‘˜=π‘₯π‘˜,Ξ”1π‘₯π‘˜=π‘₯π‘˜βˆ’π‘₯π‘˜+1,Ξ”π‘Ÿπ‘₯π‘˜ξ€·Ξ”=Ξ”π‘Ÿβˆ’1π‘₯π‘˜ξ€Έ,π‘Ÿβˆˆβ„•.(2.1) The space of all bounded π‘Ÿ-convex sequences with π‘Ÿβ‰₯2 is denoted by SCπ‘Ÿ, that is,SCπ‘Ÿξ€½βˆΆ=π‘₯βˆˆπ‘™βˆžβˆΆΞ”π‘Ÿπ‘₯π‘˜ξ€Ύ,β‰₯0βˆ€π‘˜βˆˆβ„•SC1ξ€½βˆΆ=π‘₯βˆˆπ‘™βˆžβˆΆπ‘₯π‘˜βˆ’π‘₯π‘˜+1ξ€Ύ.β‰₯0(2.2) It is clear that SC1βŠ†π‘.

It is well known that (Zygmund [13]) a bounded convex sequence (π‘₯π‘˜) is nonincreasing. It is easy to prove the identity Ξ”(π‘Ÿ+𝑠)π‘₯π‘˜=Ξ”π‘Ÿ(Δ𝑠π‘₯π‘˜),π‘Ÿ,𝑠β‰₯0,which shows that SCπ‘ŸβŠ‚SCπ‘Ÿβˆ’1, when π‘Ÿβ‰₯2. Properties of bounded π‘Ÿ-convex sequences have been investigated by Rath [14]. Note that SCπ‘ŸβŠ‚π‘£βŠ‚π‘βŠ‚β„“βˆž. Recently, using the generalized difference operator Ξ”π‘Ÿ, Γ‡olak and Et [15], and Et and Γ‡olak [16] defined and studied the sequence spaces 𝑐0(Ξ”π‘Ÿ),𝑐(Ξ”π‘Ÿ), and β„“βˆž(Ξ”π‘Ÿ). In this section, we establish the necessary and sufficient conditions on the matrix 𝐴=(π‘Žπ‘›π‘˜)βˆžπ‘›,π‘˜=1 which transforms π‘Ÿ-convex sequences into the spaces π‘‰πœŽ(πœƒ) and π‘‰βˆžπœŽ(πœƒ).

Write1𝑑(𝑛,π‘˜,π‘š)=β„Žπ‘šξ“π‘—βˆˆπΌπ‘šπ‘ŽπœŽπ‘—(𝑛),π‘˜,𝑔(π‘Ÿ)βˆ‘(𝑛,π‘˜,π‘š)=π‘˜π‘—=1ξ€·π‘Ÿβˆ’1π‘˜βˆ’π‘—ξ€Έπ‘‘(𝑛,𝑗,π‘š)π‘˜π‘Ÿβˆ’1,πœ†π‘šπ‘›(π‘₯)=βˆžξ“π‘˜=1𝑔(𝑛,π‘˜,π‘š)π‘₯π‘˜,(2.3) where for our convenience, we use 𝑔(𝑛,π‘˜,π‘š) instead of 𝑔(π‘Ÿ)(𝑛,π‘˜,π‘š) for π‘Ÿβ‰₯2 throughout the paper.

Theorem 2.1. 𝐴∈(π‘†πΆπ‘Ÿ,π‘‰πœŽ(πœƒ)) if and only if(i)sup𝑖,𝑝|βˆ‘βˆžπ‘˜=π‘π‘Žπ‘–π‘˜|<∞, (ii) there exists a constant 𝑀 such that for 𝑠,𝑛=1,2,…,supπ‘šβˆžξ“π‘˜=𝑠||||𝑔(𝑛,π‘˜,π‘š)≀𝑀,(2.4)(iii)limπ‘šπ‘”(𝑛,π‘˜,π‘š)=π›Όπ‘˜, uniformly in 𝑛, for each π‘˜βˆˆβ„•,(iv)limπ‘šβˆ‘π‘˜π‘”(𝑛,π‘˜,π‘š)=𝛼, uniformly in 𝑛.

Proof. In [17], a characterization of 𝐴∈(SCπ‘Ÿ,𝐹ℬ) was given, where 𝐹ℬ, in the sense of [18], is the bounded domain of a sequence ℬ=(𝐡(𝑖)) of matrices 𝐡(𝑖)=(𝑏(𝑖)π‘Ÿπ‘˜). Now, by the setting 𝑏(𝑖)π‘Ÿπ‘˜=ξƒ―1β„Žπ‘Ÿ0ifπ‘˜=πœŽπ‘—(𝑖),π‘—βˆˆπΌπ‘Ÿ,otherwise.(π‘Ÿ,π‘˜,π‘–βˆˆβ„•),(2.5) then π‘‰πœŽ(πœƒ)=𝐹ℬ, and the proof follows from Theorem 2.1 of [17].

Theorem 2.2. 𝐴∈(π‘†πΆπ‘Ÿ,π‘‰βˆžπœŽ(πœƒ)) if and only if the condition (i) of Theorem 2.1 holds and sup𝑛,π‘šξ“π‘˜||||𝑑(𝑛,π‘˜,π‘š)<∞.(2.6)

Proof. Sufficiency
Suppose that the conditions (i) and (2.6) hold and π‘₯=(π‘₯π‘˜)∈SCπ‘ŸβŠ‚β„“βˆž. Therefore, 𝐴π‘₯ is bounded, and we have ||πœ†π‘šπ‘›||≀(π‘₯)π‘˜||𝑔(𝑛,π‘˜,π‘š)π‘₯π‘˜||β‰€ξƒ©ξ“π‘˜||||ξƒͺ𝑔(𝑛,π‘˜,π‘š)supπ‘˜||π‘₯π‘˜||ξ‚Ά.(2.7) Taking the supremum over 𝑛,π‘š on both sides and using (2.6), we get 𝐴π‘₯βˆˆπ‘‰βˆžπœŽ(πœƒ) for π‘₯∈SCπ‘Ÿ.

Necessity
Let 𝐴∈(SCπ‘Ÿ,π‘‰βˆžπœŽ(πœƒ)). Condition (i) follows as in the proof of Theorem 2.1. Write π‘žπ‘›(π‘₯)=supπ‘š|πœ†π‘šπ‘›(𝐴π‘₯)|. It is easy to see that π‘žπ‘› is a continuous seminorm on SCπ‘Ÿ, since for π‘₯∈SCπ‘ŸβŠ‚β„“βˆž, ||π‘žπ‘›||(π‘₯)≀𝑀‖π‘₯β€–,𝑀>0.(2.8) Suppose that (2.6) is not true, then there exists π‘₯∈SCπ‘Ÿ with supπ‘›π‘žπ‘›(π‘₯)=∞. By the principle of condensation of singularities (cf. [19]), the set ξ‚»π‘₯∈SCπ‘ŸβˆΆsupπ‘›π‘žπ‘›ξ‚Ό(π‘₯)=∞(2.9) is of second category in SCπ‘Ÿ and hence nonempty, that is, there is π‘₯∈SCπ‘Ÿ with supπ‘›π‘žπ‘›(π‘₯)=∞. But this contradicts the fact that π‘žπ‘› is pointwise bounded on SCπ‘Ÿ. Now, by the Banach-Steinhaus theorem, there is a constant 𝑀 such that π‘žπ‘›(π‘₯)≀𝑀‖π‘₯β€–.(2.10) Now, we define a sequence π‘₯=(π‘₯π‘˜) by π‘₯π‘˜=ξƒ―sgn𝑔(𝑛,π‘˜,π‘š)π‘˜ξ€·foreachπ‘š,𝑛1β‰€π‘˜β‰€π‘˜0ξ€Έ,0forπ‘˜>π‘˜0.(2.11) Then, π‘₯∈SCπ‘Ÿ. Applying this sequence to (2.10), we get (2.6).
This completes the proof of the theorem.

3. Maddox Sequence Spaces

A linear topological space 𝑋 over the real field ℝ is said to be a paranormed space if there is a subadditive function π‘”βˆΆπ‘‹β†’β„ such that 𝑔(πœƒ)=0,𝑔(π‘₯)=𝑔(βˆ’π‘₯), and scalar multiplication is continuous, that is, |π›Όπ‘›βˆ’π›Ό|β†’0 and 𝑔(π‘₯π‘›βˆ’π‘₯)β†’0 imply 𝑔(𝛼𝑛π‘₯π‘›βˆ’π›Όπ‘₯)β†’0 for all π‘₯,π‘₯𝑛 in 𝑋 and 𝛼,𝛼𝑛 in ℝ, where πœƒ is the zero vector in the linear space 𝑋. Assume here and after that π‘₯=(π‘₯π‘˜) is a sequence such that π‘₯π‘˜β‰ 0 for all π‘˜βˆˆβ„•. Let 𝑝=(π‘π‘˜)βˆžπ‘˜=0 be a bounded sequence of positive real numbers with supπ‘˜π‘π‘˜=𝐻 and 𝑀=max{1,𝐻}. The sequence spaces𝑐0ξ‚»(𝑝)∢=π‘₯βˆˆπœ”βˆΆlimπ‘˜||π‘₯π‘˜||π‘π‘˜ξ‚Ό,ξ€½=0𝑐(𝑝)∢=π‘₯βˆˆπœ”βˆΆπ‘₯βˆ’π‘™π‘’βˆˆπ‘0ξ€Ύ,𝑙(𝑝)forsomeπ‘™βˆˆβ„‚βˆžξ‚»(𝑝)∢=π‘₯βˆˆπœ”βˆΆsupπ‘˜||π‘₯π‘˜||π‘π‘˜ξ‚Ό,ξƒ―<βˆžπ‘™(𝑝)∢=π‘₯βˆˆπœ”βˆΆβˆžξ“π‘˜=0||π‘₯π‘˜||π‘π‘˜ξƒ°,<∞(3.1) were defined and studied by Et and Γ‡olak [16] and Pati and Sinha [12]. If π‘π‘˜=𝑝(π‘˜=0,1,…) for some constant 𝑝>0, then these sets reduce to 𝑐0,𝑐,π‘™βˆž, and 𝑙𝑝, respectively. Note that 𝑐0(𝑝) is a linear metric space paranormed by 𝑔(π‘₯)=supπ‘˜||π‘₯π‘˜||π‘π‘˜/𝑀(3.2) where 𝑀=max(1,supπ‘π‘˜). π‘™βˆž(𝑝) and 𝑐(𝑝) fail to be linear metric spaces because the continuity of scalar multiplication does not hold for them, but these two turn out to be linear metric spaces if and only if infπ‘˜π‘π‘˜>0. 𝑙(𝑝) is linear metric space paranormed by β„Ž1βˆ‘(π‘₯)=(π‘˜|π‘₯π‘˜|π‘π‘˜)1/𝑀. All these spaces are complete in their respective topologies; however, these are not normed spaces in general (cf. [20]).

In this section, we characterize the matrix classes (𝑙(𝑝),π‘‰πœŽ(πœƒ)) and (π‘™βˆž(𝑝),π‘‰πœŽ(πœƒ)).

Let 𝐴π‘₯ be defined, then, for all π‘Ÿ,𝑛, we writeπœπ‘Ÿ(𝐴π‘₯)=βˆžξ“π‘˜=1𝑑(𝑛,π‘˜,π‘Ÿ)π‘₯π‘˜,(3.3) where1𝑑(𝑛,π‘˜,π‘Ÿ)=β„Žπ‘Ÿξ“π‘—βˆˆπΌπ‘Ÿπ‘Žξ€·πœŽπ‘—ξ€Έ(𝑛),π‘˜,(3.4) and π‘Ž(𝑛,π‘˜) denotes the element π‘Žπ‘›π‘˜ of the matrix 𝐴.

Theorem 3.1. 𝐴∈(β„“(𝑝),π‘‰πœŽ(πœƒ)) if and only if there exists 𝐡>1 such that for every 𝑛,(i)supπ‘Ÿξ“π‘˜||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜π΅βˆ’π‘žπ‘˜ξ€·<∞,1<π‘π‘˜ξ€Έ,𝑝<βˆžπ‘˜βˆ’1+π‘žπ‘˜βˆ’1ξ€Έ;=1supπ‘Ÿ,π‘˜||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘π‘˜ξ€·<∞,0<π‘π‘˜ξ€Έ,≀1(3.5)(ii)π‘Ž(π‘˜)={π‘Žπ‘›π‘˜}βˆžπ‘›=1βˆˆπ‘‰πœŽ(πœƒ) for each π‘˜, that is, limπ‘Ÿπ‘‘(𝑛,π‘˜,π‘Ÿ)=π‘’π‘˜ uniformly in 𝑛.In this case, the (𝜎,πœƒ)-limit of 𝐴π‘₯ is βˆ‘π‘˜π‘’π‘˜π‘₯π‘˜.

Proof. Necessity
We consider the case 1<π‘π‘˜<∞. Let 𝐴∈(β„“(𝑝),π‘‰πœŽ(πœƒ)). Since π‘’π‘˜βˆˆβ„“(𝑝), the condition (ii) holds. Put π‘“π‘Ÿπ‘›(π‘₯)=πœπ‘Ÿπ‘›(𝐴π‘₯), since πœπ‘Ÿπ‘›(𝐴π‘₯) exists for each π‘Ÿ and π‘₯βˆˆπ‘™(𝑝), therefore {π‘“π‘Ÿπ‘›(π‘₯)}π‘Ÿ is a sequence of continuous real functionals on 𝑙(𝑝) and further supπ‘Ÿ|π‘“π‘Ÿπ‘›(π‘₯)|<∞ on 𝑙(𝑝). Now condition (i) follows by arguing with uniform boundedness principle. The case 0<π‘π‘˜β‰€1 can be proved similarly.

Sufficiency
Suppose that the conditions (i) and (ii) hold and π‘₯βˆˆβ„“(𝑝). Now for every π‘šβ‰₯1, we have π‘šξ“π‘˜=1||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜π΅βˆ’π‘žπ‘˜β‰€supπ‘Ÿξ“π‘˜||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜π΅βˆ’π‘žπ‘˜.(3.6) Therefore, ξ“π‘˜||π‘’π‘˜||π‘žπ‘˜π΅βˆ’π‘žπ‘˜=limπ‘šlimπ‘Ÿπ‘šξ“π‘˜=1||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜π΅βˆ’π‘žπ‘˜β‰€supπ‘Ÿξ“π‘˜||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜π΅βˆ’π‘žπ‘˜<∞.(3.7) Thus, the series βˆ‘π‘˜π‘‘(𝑛,π‘˜,π‘Ÿ)π‘₯π‘˜ and βˆ‘π‘˜π‘’π‘˜π‘₯π‘˜ converge for each π‘Ÿ and π‘₯βˆˆβ„“(𝑝). For a given πœ€>0 and π‘₯βˆˆβ„“(𝑝), choose π‘˜0 such that βŽ›βŽœβŽœβŽβˆžξ“π‘˜=π‘˜0+1||π‘₯π‘˜||π‘π‘˜βŽžβŽŸβŽŸβŽ 1/𝐻<πœ€,(3.8) where 𝐻=supπ‘˜π‘π‘˜. Since (ii) holds, therefore there exists π‘Ÿ0 such that |||||π‘˜0ξ“π‘˜=1𝑑(𝑛,π‘˜,π‘Ÿ)βˆ’π‘’π‘˜ξ€Έ|||||<πœ€,(3.9) for every π‘Ÿ>π‘Ÿ0. Hence, by the condition (ii), it follows that |||||βˆžξ“π‘˜=π‘˜0+1𝑑(𝑛,π‘˜,π‘Ÿ)βˆ’π‘’π‘˜ξ€Έ|||||(3.10) is arbitrary small, and we have limπ‘Ÿξ“π‘˜π‘‘(𝑛,π‘˜,π‘Ÿ)π‘₯π‘˜=ξ“π‘˜π‘’π‘˜π‘₯π‘˜,(3.11) uniformly in 𝑛.
This completes the proof of the theorem.

Theorem 3.2. 𝐴∈(β„“βˆž(𝑝),π‘‰πœŽ(πœƒ)) if and only if there exists 𝑁>1 such that(i)𝑀𝑛=supπ‘Ÿβˆ‘π‘˜|𝑑(𝑛,π‘˜,π‘Ÿ)|𝑁1/π‘π‘˜<∞ for every 𝑛,(ii)π‘Ž(π‘˜)={π‘Žπ‘›π‘˜}βˆžπ‘›=1βˆˆπ‘‰πœŽ(πœƒ) for each π‘˜, that is, limπ‘Ÿπ‘‘(𝑛,π‘˜,π‘Ÿ)=π‘’π‘˜ uniformly in 𝑛,(iii)limπ‘Ÿβˆ‘π‘˜|𝑑(𝑛,π‘˜,π‘Ÿ)βˆ’π‘’π‘˜|=0 uniformly in 𝑛.In this case, the (𝜎,πœƒ)-limit of 𝐴π‘₯ is βˆ‘π‘˜π‘’π‘˜π‘₯π‘˜.

Proof. Necessity
Let 𝐴∈(β„“βˆž(𝑝),π‘‰πœŽ(πœƒ)), then 𝐴∈(β„“βˆž,π‘‰πœŽ(πœƒ)), and the conditions (ii) and (iii) follow from Theorem 3 of Schaefer [5]. Now on the contrary, suppose that (i) does not hold, then there exists 𝑁>1 such that 𝑀𝑛=∞. Therefore, by Theorem 3 of Schaefer [5], the matrix 𝑏𝐡=π‘›π‘˜ξ€Έ=ξ€·π‘Žπ‘›π‘˜π‘1/π‘π‘˜ξ€Έβˆ‰ξ€·β„“βˆž,π‘‰πœŽξ€Έ(πœƒ),(3.12) that is, there exists π‘₯βˆˆβ„“βˆž such that 𝐡π‘₯βˆ‰π‘‰πœŽ(πœƒ). Now, let 𝑦=(𝑁1/π‘π‘˜π‘₯π‘˜), then π‘¦βˆˆβ„“βˆž(𝑝) and 𝐡π‘₯=π΄π‘¦βˆ‰π‘‰πœŽ(πœƒ), which contradicts that 𝐴∈(β„“βˆž(𝑝),π‘‰πœŽ(πœƒ)). Therefore, (i) must hold.

Sufficiency
Suppose that the conditions hold and π‘₯βˆˆβ„“βˆž(𝑝), then for every 𝑛, |||||ξ“π‘˜π‘‘(𝑛,π‘˜,π‘Ÿ)π‘₯π‘˜|||||≀supπ‘˜||π‘₯π‘˜||π‘π‘˜ξ‚Άξƒ©supπ‘Ÿξ“π‘˜||||𝑁𝑑(𝑛,π‘˜,π‘Ÿ)1/π‘π‘˜ξƒͺ<∞.(3.13) Therefore 𝐴π‘₯ is defined. Now arguing as in Theorem 3.1, we get 𝐴π‘₯βˆˆπ‘‰πœŽ(πœƒ), and the series βˆ‘π‘˜π‘‘(𝑛,π‘˜,π‘Ÿ)π‘₯π‘˜ and βˆ‘π‘˜π‘’π‘˜π‘₯π‘˜ converge for π‘₯βˆˆβ„“βˆž(𝑝). Hence, by the condition (iii), we get limπ‘Ÿξ“π‘˜π‘‘(𝑛,π‘˜,π‘Ÿ)π‘₯π‘˜=ξ“π‘˜π‘’π‘˜π‘₯π‘˜,(3.14) uniformly in 𝑛.
This completes the proof of the theorem.

Theorem 3.3. Let 1<π‘π‘˜<supπ‘˜π‘π‘˜=𝐻<∞ for every π‘˜, then 𝐴∈(β„“(𝑝),π‘‰βˆžπœŽ(πœƒ)) if and only if there exists an integer 𝑁>1 such thatsupπ‘Ÿ,π‘›ξ“π‘˜||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜π‘βˆ’π‘žπ‘˜<∞.(3.15)

Proof. Sufficiency
Let (3.15) hold and that π‘₯βˆˆβ„“(𝑝) using the following inequality (see [21]): ||||ξ€·π‘Žπ‘β‰€πΆ|π‘Ž|π‘žπΆβˆ’π‘ž+||𝑏||𝑝,(3.16) for 𝐢>0 and π‘Ž,𝑏, are two complex numbers (π‘žβˆ’1+π‘βˆ’1=1), we have ||πœπ‘Ÿ(||=𝐴π‘₯)π‘˜||𝑑(𝑛,π‘˜,π‘Ÿ)π‘₯π‘˜||β‰€ξ“π‘˜π‘ξ€Ί||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜π‘βˆ’π‘žπ‘˜+||π‘₯π‘˜||π‘π‘˜ξ€»,(3.17) where π‘žπ‘˜βˆ’1+π‘π‘˜βˆ’1=1. Taking the supremum over π‘Ÿ,𝑛 on both sides and using (3.15), we get 𝐴π‘₯βˆˆπ‘‰βˆžπœŽ(πœƒ)) for π‘₯βˆˆβ„“(𝑝), that is, 𝐴∈(β„“(𝑝),π‘‰βˆžπœŽ(πœƒ)).

Necessity
Let 𝐴∈(β„“(𝑝),π‘‰βˆžπœŽ(πœƒ)). Write π‘žπ‘›(π‘₯)=supπ‘Ÿ|πœπ‘Ÿ(𝐴π‘₯)|. It is easy to see that for 𝑛β‰₯0, π‘žπ‘› is a continuous seminorm on β„“(𝑝), and (π‘žπ‘›) is pointwise bounded on β„“(𝑝). Suppose that (3.15) is not true, then there exists π‘₯βˆˆβ„“(𝑝) with supπ‘›π‘žπ‘›(π‘₯)=∞. By the principle of condensation of singularities [19], the set ξ‚»π‘₯βˆˆβ„“(𝑝)∢supπ‘›π‘žπ‘›ξ‚Ό(π‘₯)=∞(3.18) is of second category in β„“(𝑝) and hence nonempty, that is, there is π‘₯βˆˆβ„“(𝑝) with supπ‘›π‘žπ‘›(π‘₯)=∞. But this contradicts the fact that (π‘žπ‘›) is pointwise bounded on β„“(𝑝). Now, by the Banach-Steinhaus theorem, there is constant 𝑀 such that π‘žπ‘›(π‘₯)≀𝑀𝑔(π‘₯).(3.19) Now, we define a sequence π‘₯=(π‘₯π‘˜) by π‘₯π‘˜=𝛿𝑀/π‘π‘˜||||(sgn𝑑(𝑛,π‘˜,π‘Ÿ))𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜βˆ’1π‘†βˆ’1π‘βˆ’π‘žπ‘˜/π‘π‘˜,1β‰€π‘˜β‰€π‘˜0,0,forπ‘˜>π‘˜0,(3.20) where 0<𝛿<1 and 𝑆=π‘˜0ξ“π‘˜=1||||𝑑(𝑛,π‘˜,π‘Ÿ)π‘žπ‘˜π‘βˆ’π‘žπ‘˜.(3.21) Then it is easy to see that π‘₯βˆˆβ„“(𝑝) and 𝑔(π‘₯)≀𝛿. Applying this sequence to (3.19), we get the condition (3.15).
This completes the proof of the theorem.

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