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Quasi D-Limits Of Bounded Sequences And Regular Matrices 

 

Sakambari Mishra1, Balaji Padhy2* 

1 Department of Mathematics, College of Basic Science and Humanities, Odisha University of Agriculture and 

Technology, Bhubaneswar 751003, Odisha, India. Email ID: sakambari@ouat.ac.in (S.Mishra) 

*2 Department of Mathematics, Centurion University of Technology and Management, Paralakhemundi 761211, 

Odisha, India. E-mail: balaji.padhy@cutm.ac.in ∗Corresponding author 

Article History: 

Received: 14-11-2024 

Revised:26-12-2024 

Accepted:10-01-2025 

Abstract:  

In this paper, we explore the concept of quasi-D-limits for bounded sequences through 

the nonnegative regular matrix transformation D. Quasi D-limits extend spaces of D-

limits notions, providing a framework to analyse the convergence properties of 

sequences that exhibit specific bounded behaviour. We establish criteria for quasi-D-

limits, demonstrate their existence via regular matrix transformations, and apply these 

results to various mathematical and applied contexts. 

2010 Mathematics Subject Classification: Primary 40A05. Secondary 40C05. 

Key Words.  Linear functional, D-limit, D-Invariant, regular matrices and quasi-D-

convergence. 

 

1. Introduction:  

The study of limits in sequence convergence has been a foundational aspect of analysis. In 

particular, bounded sequences whose terms are confined within a specific range often require 

advanced techniques to elucidate their limiting behaviour. quasi-D-limits offer an innovative 

approach, enabling us to extend conventional limit concepts. The idea of Quasi almost 

convergence in normed space was featured by Hajdukovic[4]. Further the concept of quasi 

Banach limit  is introduced by Das and Mishra in [3] and the concept of quasi-invariant limit 

is recently  explained by Mishra[5]  in the space of real bounded sequences which yields quasi-

invariant convergent sequences. 

 This paper aims to investigate the quasi-D-limits of bounded sequences utilizing matrix 

transformations. We first introduce the necessary background on quasi-D-limits and matrix 

transformations, followed by an exploration of their interrelations. 

2. Preliminaries:  

        Let 𝑚 be the space of real bounded sequences 𝑥 = {𝑥𝑛}  normed by ‖𝑥‖ = sup
𝑛
|𝑥𝑛|. Let 

𝑚∗ denote the set of all continuous linear functional on m. Throughout this work, we have used 

an infinite matrix D whose scalar entries 𝑑𝑛𝑘 are in 𝑚. Consider a sequence 𝑥 in 𝑚. Let 𝐷𝑥 be 

the transformed sequence whose general term is written as 

                                   (𝐷𝑥)𝑛 = ∑ 𝑑𝑛𝑘𝑥𝑘
∞
𝑘=0  

 and it is convergent for each 𝑛 ≥ 0. 



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  Let us recall the definition of conservative matrix ( due to Stieglitz [8]). 

Definition 1: A matrix  𝐷 = (𝑑𝑛𝑘) is  said to be conservative matrix iff it satisfies the following 

properties: 

(a) ‖𝐷‖ < ∞  i.e., sup
𝑛
∑ |𝑑𝑛𝑘| < ∞.∞
𝑘=0  

(b) lim
𝑛
𝑑𝑛𝑘 = 𝑑𝑘  for every fixed k. 

(c) lim
𝑛
∑ 𝑑𝑛𝑘 = 𝑑  
∞
𝑘=0 .    

If 𝑑𝑘 = 0 and 𝑑 = 1,  then 𝐷 is called regular (see [9] p.64).  

Let 𝐷 = (𝑑𝑛𝑘) be the a fixed regular matrix with ‖𝐷‖ < 1( This is assumed throughout the 

paper). 

Definition 2:  A linear functional 𝜓 ∈ 𝑚∗ is said to be an 𝐷-mean or  𝐷-limit if and only if the 

following properties hold good: 

(i) For  𝑥 = {𝑥𝑛} , 𝜓(𝑥) ≥ 0 if 𝑥𝑛 ≥ 0 for all 𝑛. 

(ii) 𝜓(𝑒) = 1 for 𝑒 = (1,1,1, … . ). 

(iii) 𝜓(𝐷𝑥) = 𝜓(𝑥) for all 𝑥 𝜖 𝑚 . 

Let 𝐷 = 𝐵 be the translation matrix i.e.  (𝐵𝑥)𝑛 = 𝑥𝑛+1  then it is called a 𝐵 -limit and is often 

called as a Banach limit [1]. Some inequalities are shown in [6] between sublinear functionals 

emerging from Banach limits of some sequences and their matrix transformations using 

conservative and regular matrices.  

Definition 3: A matrix 𝐶 ∈ 𝑚 is called a 𝐷-Invariant matrix if 𝐶(𝐷 − 𝐼) behaves as a zero map 

and maps every bounded sequence  to null sequence. (I is the Identity matrix.) 

The concept of  𝐷-limits were first introduced by Bell[2], where he assumed that 𝐷 is a positive 

matrix such that  ‖𝐷‖ = 1. From definition of 𝐷-limit ,  it follows that  

                                                                           𝑥 ≤ 𝑦 ⇒  𝜓(𝑥) ≤ 𝜓(𝑦) .  

and from (i) and (ii) that ‖𝜓‖ = 1. Here the product of 𝐷 with itself 𝑝 times is denoted by 𝐷𝑝. 

 Let 𝑄𝐷 be the set of all bounded sequences having equal 𝐷-means. So, we can write 

                        𝑄𝐷 = {𝑥 ∈ 𝑚: lim
𝑝
 ℎ𝑝𝑛(𝑥) = 𝑙  uniformly in 𝑛, 𝑙 = 𝐷 − 𝑙𝑖𝑚𝑥}  

where for 𝑝 ≥ 0, 𝑛 > 0. 

 

                                               ℎ𝑝𝑛(𝑥) =
𝑥𝑛+(𝐷𝑥)𝑛+⋯+(𝐷

𝑝𝑥)𝑛

𝑝+1
 .                                          ( 2.1)  

We know that a sublinear functional P on 𝑚 generates 𝐷-limit if 𝜓 ∈ 𝑚∗and 𝜓 < 𝑃 ⇒ 𝜓 is a 

𝐷-limit.   Here  𝜓 < 𝑃  𝑚𝑒𝑎𝑛𝑠  𝜓(𝑥) ≤ 𝑃(𝑥)  for all 𝑥 ∈ 𝑚. P is said to dominate  𝐷-limit if 

every 𝐷-limit 



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𝜓 < 𝑃. i.e. 𝜓 ∈ 𝑄𝐷 ⇒  𝜓 < 𝑃, where 𝑄𝐷 is the set of all 𝐷-limits. It is provided that 𝑄𝐷 is a 

closed convex set. It is proved that 𝜓 ∈ 𝑚∗ is a 𝐷-limit if and only if 𝜓(𝑥) ≤ 𝑡(𝑥) ,where 𝑡(𝑥) 

is a sublinear functional on 𝑚 defined by 

𝑡(𝑥) = lim
𝑝
𝑠𝑢𝑝 sup

𝑛
ℎ𝑝𝑛(𝑥). 

i.e.                                             𝑡(𝑥) = lim
𝑝
𝑠𝑢𝑝 sup

𝑛
 
1

𝑝+1
∑ (𝐷𝑘𝑥)𝑛
𝑝
𝑘=0 .                                      (2.2) 

We now take the idea from above and purpose the following definition for a new family of 

functionals of the kind of 𝐷-mean called quasi 𝐷-limit. 

Definition 4: The linear functional 𝜓 ∈ 𝑚∗  is a quasi  𝐷-mean or quasi 𝐷-limit or quasi matrix 

invariant limit if    𝜓(𝑥) ≤ 𝑟(𝑥) , 𝑥 ∈ 𝑚, where 𝑟(𝑥)  is a sublinear functional on 𝑚 defined 

by  

             .                                   𝑟(𝑥) = lim
𝑝
𝑠𝑢𝑝 sup

𝑛
 
1

𝑝+1
∑ (𝐷𝑘𝑥)𝑛𝑝
𝑝
𝑘=0   .                                   (2.3) 

. Quasi 𝐷-limit need not exist for every regular matrix. 

Example 1: Consider 

                         𝑑𝑛𝑘 =

{
 
 

 
 
1               ,                   𝑖𝑓 𝑘 = 2𝑛
0                ,                  𝑖𝑓 𝑘 > 2𝑛
1

𝑛
       , 𝑖𝑓 𝑘 𝑖𝑠 𝑒𝑣𝑒𝑛 1 ≤ 𝑘 ≤ 2𝑛

−1

𝑛
        , 𝑖𝑓 𝑘 𝑖𝑠 𝑜𝑑𝑑, 1 ≤ 𝑘 ≤ 2𝑛

                                                                                  

Here 𝐷 is regular. So, for 𝑥 = {1,0,1,0, … . . },      𝐷𝑥 = {−1,−1,−1,−1,… . . }.  

If 𝜓 is a quasi 𝐷-limit then  0 ≤ 𝜓(𝑥) = 𝜓(𝐷𝑥) = −1  which is impossible. So quasi 𝐷-limit 

does not exist. 

        We now propose the following definitions for new kind of 𝐷-convergent sequences, called 

as quasi 𝐷-convergent sequence. For this we define it as follows 

Definition 5: A sequence  𝑥 ∈ 𝑚 is called a quasi 𝐷-convergent sequence if −𝑟(−𝑥) = 𝑟(𝑥).  

The following theorem is known. 

Theorem A [10]: For all 𝑥 ∈ 𝑚, the sublinear functional 𝑡(𝑥) both generates and dominates 

𝐷-limit 

𝜓(𝑥) if and only if 𝜓(𝑥) ≤ 𝑡(𝑥). 

Applying the above theorem, we will prove some new results for the sublinear functional 𝑟(𝑥), 

where these are based on the idea of quasi Banach Limit defined in [3] , quasi invariant limits 

and their convergence   suggested by  Mishra[5] and Nuray[7]. 

3. Main Results:  



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Now we are going to prove a theorem on existence o 𝐷-limit and then establish some 

inequalities on quasi 𝐷-limit . 

Theorem 1: (Existence Theorem) If there is a nonnegative regular and 𝐷-invariant matrix then  

𝐷-limits exist or quasi  𝐷-limits exist. 

Proof: Suppose 𝐶  is   such  a nonnegative regular  𝐷-invariant matrix. Define a  nonnegative 

homogeneous sublinear functional 𝑝 on 𝑚 by   𝑝(𝑥) = limsup𝐶𝑥 .If 𝑥 is a convergent 

sequence, then by regularity of 𝐶  we get  lim 𝑥 = 𝑝(𝑥) . Therefore, extending the above limit 

to 𝜓 we have 𝜓(𝑥) ≤ 𝑝(𝑥) for all 𝑥 ∈ 𝑚. 

Thus, −𝑝(−𝑥) ≤ − 𝜓(−𝑥) =  𝜓(𝑥) ≤ 𝑝(𝑥) for all 𝑥 ∈ 𝑚. Since   𝐶 is nonnegative ,we have  

0 ≤ liminf 𝐶𝑥 ≤  𝜓(𝑥).This implies 𝜓 is nonnegative. Here we have 𝐶(𝐷 − 𝐼)𝑥 =

0.Therefore 

−𝑝(−(𝐷 − 𝐼)𝑥) = 0 = 𝑝((𝐷 − 𝐼)𝑥) which implies 𝜓((𝐷 − 𝐼)𝑥) = 0. 

Hence, 𝜓 is a  𝐷-limit. Considering 𝑝(𝑥) as equal to the sublinear functional in (2.3),we can 

say that  

 𝜓 is a quasi  𝐷-limit . 

Example 2: Cesaro matrix is a nonnegative regular 𝐵 -Invariant matrix. So, its quasi Banach  

limit[3] exists. If we define  

                                    𝑑𝑛𝑘 = {
1

𝑛
      , 𝑖𝑓 1 ≤ 𝑘 ≤ 𝑛

0          , 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
                                                                      (3.1) 

                               

then the matrix 𝐷 = (𝑑𝑛𝑘) is a Cesaro matrix of order one which is a translation matrix and its 

quasi 𝐷-limit [3] or quasi Banach limit exists as it is a nonnegative, regular and 𝐷(= 𝐵)-

Invariant matrix. 

Theorem 2: For all 𝑥 ∈ 𝑚, the sublinear functional 𝑟(𝑥) generates   𝐷-limit .  

Proof: We first proceed to prove that the sublinear functional 𝜓 ∈ 𝑚∗ generates 𝐷-mean. 

Let 𝜓 ∈ 𝑚∗ and satisfies the inequality 

                                                    𝜓(𝑥) ≤ 𝑟(𝑥)    for all 𝑥 ∈ 𝑚                                                    (3.2) 

So by linearity of   𝜓(𝑥)and sub-linearity of 𝑟(𝑥) we have from (3.2), that is  

                                                 −𝑟(−𝑥) ≤  𝜓(𝑥) ≤ 𝑟(𝑥)                                                             (3.3) 

This indicates that for 𝑥 ≥ 0 implies 𝑟(𝑥) ≥ 0 and −𝑟(−𝑥) ≥ 0  and so from (3.3), we get  

                                                       𝜓(𝑥) ≥ 0, for all 𝑥 ≥ 0. 

Also −𝑟(−𝑒) = 𝑟(𝑒) = 1, where 𝑒 = (1,1,1,1,… . ,1). Hence from (3.3), we get  𝜓(𝑒) = 1. 



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Taking 𝐷 as a nonnegative regular matrix, we have 

limsup
𝑝

 sup
𝑛
 
1

𝑝 + 1
∑(𝐷𝑘+1−𝐷𝑘)𝑥𝑛𝑝

𝑝

𝑘=0

= limsup
𝑝

 sup
𝑛
 
(𝐷𝑝+1−𝐼)𝑥𝑛𝑝

𝑝 + 1
 

                                                                                            ≤ ‖𝑥‖ limsup
𝑝

  
1

𝑝+1
= 0. 

This shows 𝑟(𝐷𝑥 − 𝑥) = 0. 

Similarly, we also obtained 𝑟(𝑥 − 𝐷𝑥) = 0. 

Hence from (3.3) we have 𝜓(𝐷𝑥 − 𝑥) = 0. Since 𝜓 is linear, 𝜓(𝐷𝑥) = 𝜓(𝑥), 𝑥 ∈ 𝑚. 

This proves that  

𝜓(𝑥) ≤ 𝑟(𝑥) ⇒  𝜓 is a 𝐷-limit                                                                                                    (3.4) 

So 𝑟(𝑥) also generates 𝐷 –limit as required. 

  

Note:    From above we say that  𝑟(𝑥) also generates quasi 𝐷 -limit. 

Next consider 𝜓 is a quasi 𝐷-limit. Then 

𝜓(𝑥𝑛𝑝) = 𝜓(𝐷𝑥)𝑛𝑝 = 𝜓(𝐷2𝑥)𝑛𝑝 = ⋯… = 𝜓(𝐷𝑝𝑥)𝑛𝑝 

This implies 

                       𝜓(𝑥) = 𝜓 (
𝑥𝑛𝑝+(𝐷𝑥)𝑛𝑝+(𝐷

2𝑥)𝑛𝑝+⋯…+(𝐷
𝑝𝑥)𝑛𝑝

𝑝+1
) 

                                ≤ limsup
𝑝

sup
𝑛
 (
𝑥𝑛𝑝+(𝐷𝑥)𝑛𝑝+(𝐷

2𝑥)𝑛𝑝+⋯…+(𝐷
𝑝𝑥)𝑛𝑝

𝑝+1
) = 𝑟(𝑥) 

i.e., 𝑟(𝑥) dominates quasi 𝐷 –limit. 

Theorem 3: Let 𝑡(𝑥)  and 𝑟(𝑥)  be two sublinear functionals defined in (2.2) and (2.3) 

respectively.Then     

                                          𝑟(𝑥) ≤ 𝑡(𝑥)     for all 𝑥 ∈ 𝑚   .                              

 

Proof: Combining the given condition of (3.4) and the inequality of Theorem A, we will get  

that for all 𝜓 ∈ 𝑚∗. 

                                             𝜓(𝑥) ≤ 𝑟(𝑥) ⇒  𝜓(𝑥) ≤ 𝑡(𝑥)                                                         (3.5)  

claim :           

                                                𝑟(𝑥) ≤ 𝑡(𝑥)  for all 𝑥 ∈ 𝑚 .                                                          (3.6) 

Suppose to the contrary, that (3.6) is false. Then there exists a sequence 𝑦 ∈ 𝑚 such that  

                                                     𝑟(𝑦) > 𝑡(𝑦)  .                                                                              (3.7) 



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Since  𝑟 is sublinear, by Hahn-Banach Theorem, there exist a linear functional 𝑔 on 𝑚 such 

that 

                                                       𝑔(𝑦) = 𝑟(𝑦) .  

From (3.7), it follows that 

                                                       𝑔(𝑦) > 𝑡(𝑦) . 

However, from the given condition (3.5), we know that for all 𝜓 ∈ 𝑚∗      

                                                   𝜓(𝑥) ≤ 𝑟(𝑥) ⇒  𝜓(𝑥) ≤ 𝑡(𝑥)      

Substituting 𝑔(𝑦) into this condition, we have:  𝑔(𝑦) ≤ 𝑡(𝑦)   

which contradicts our earlier assertion      𝑔(𝑦) > 𝑡(𝑦)                                      

 Thus, our assumption that 𝑔(𝑦) > 𝑡(𝑦)  is untenable. Therefore, we conclude that: 

                                                      𝑟(𝑥) ≤ 𝑡(𝑥) for all 𝑥 ∈ 𝑚.                                                     (3.8) 

Hence proved the result. 

    

 Next, we are going to prove an important corollary of above theorem. From the definition and 

from above theorem, the set of 𝐷-convergent sequences    𝑄𝐷 also can be written  as: 

                                    𝑄𝐷 = {𝑥 ∈ 𝑚: −𝑡(−𝑥) = 𝑡(𝑥)} .                                                            (3.9) 

This  identifies the structure of sequences 𝑥 for which −𝑡(−𝑥) = 𝑡(𝑥) which are also referred 

to  define  quasi 𝐷-convergent sequences. We denote the set of quasi 𝐷-convergent sequences 

as 𝑄𝐷
∗ where  

                                     𝑄𝐷
∗ = {𝑥 ∈ 𝑚: −𝑟(−𝑥) = 𝑟(𝑥)} .                                                         (3.10) 

 Then we now prove the following corollary. 

Corollary: 𝑄𝐷 ⊆ 𝑄𝐷
∗ 

where 𝑄𝐷 is the set of 𝐷-convergent sequences, and 𝑄𝐷
∗is the set of quasi  𝐷-convergent 

sequences. 

Proof: Since  𝑡(𝑥) and 𝑟(𝑥) are sublinear, it follows from above theorem that  

                                  −𝑡(−𝑥) ≤ −𝑟(−𝑥) ≤ 𝑟(𝑥) ≤ 𝑡(𝑥) for all  𝑥 ∈ 𝑚. 

If 𝑥 ∈ 𝑄𝐷 then by definition,       

                                                           −𝑡(−𝑥) = 𝑡(𝑥)  

Substituting this condition into the chain of inequalities above, we have:  

−𝑡(−𝑥) = 𝑡(𝑥) ⇒ −𝑟(−𝑥) = 𝑟(𝑥).  

This implies that 𝑥 ∈ 𝑄𝐷
∗, where 𝑄𝐷

∗represents the set of quasi 𝐷-convergent sequences. 



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Thus, every 𝑥 ∈ 𝑄𝐷 ( 𝐷-convergent sequence) is also an element of 𝑄𝐷
∗(quasi 𝐷-convergent 

sequence). 

Therefore, we conclude that:  𝑄𝐷 ⊆ 𝑄𝐷
∗ 

This completes the proof. 

4. Conclusion: 

This study presented the quasi 𝐷-limit for bounded sequences and demonstrated their existence 

using matrix transformations. The findings indicate that matrix transformations are an effective 

tool for examining the limiting behaviour of various types of sequences within their respective 

spaces.  

Furthermore, the inclusions for quasi 𝐷-limit convergent and 𝐷-convergent sequences pave the 

way for more in-depth sequence analysis. These findings have major implications for a variety 

of mathematical topics, including functional analysis, topology, and numerical analysis, while 

also laying the groundwork for future study into the interactions between sequence spaces and 

their transformations. 

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[3] Das, Gokulananda , Mishra, Sakambari and Ray,Braja Kishore. “Quasi Banach limits”, 

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[4] Hajduković, Dimitrije. "Quasi-almost convergence in a normed space", Univ. Beograd. 

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[7]  Nuray, Fatih. “Quasi-Invariant convergence in a normed space “, Annals of the 

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[8] Stieglitz, Michael. "Eine verallgemeinerung des begriffs der fastkonvergenz",Math. 

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[10] Tripathy,Nandita.: Ph.D thesis , approved by Utkal Univ. BBSR(Orissa),2006.