A new series space 𝑵𝑵
𝒑𝒑𝜽𝜽𝝁𝝁 and matrix operators with applications
Fadime Gökçe∗, Mehmet A. Sarıgöl
Dept. of Mathematics, University of Pamukkale, TR-20070 Denizli, Turkey
*Corresponding author: fgokce@pau.edu.tr
Abstract The space 𝑁𝑁!!
!of all series summable by the absolute weighted mean method has recently been introduced and studied in several publications. In the present paper, we define a new notion of generalized absolute summability, which includes several well-known summability methods, and construct a series space 𝑁𝑁!! 𝜇𝜇 corresponding to it. Further, we obtain several properties of the new space and characterize certain matrix transformations on that space. We also deduce some important results as special cases.
Keywords: Absolute weighted summability; 𝐵𝐵𝐵𝐵 − 𝐴𝐴𝐴𝐴 spaces; bounded linear operators; matrix transformations;
sequence spaces.
Mathematics Subject Classification: 40C05, 40D25, 40F05, 46A45, 46B03, 46B28.
1. Introduction
Let 𝑋𝑋, 𝑌𝑌 be any two subsets of 𝜔𝜔, the set of all sequences of complex numbers, and 𝐴𝐴 = (𝑎𝑎!") be an infinite matrix of complex numbers for 𝑛𝑛, 𝜈𝜈 ≥ 0. By 𝐴𝐴(𝑥𝑥) = (𝐴𝐴!(𝑥𝑥)), we denote the 𝐴𝐴-transform of a sequence 𝑥𝑥 = (𝑥𝑥!), i.e., 𝐴𝐴! 𝑥𝑥 = 𝑎𝑎!"𝑥𝑥!
!
!!!
1
provided that the series are convergent for 𝑛𝑛 ≥ 0. If 𝐴𝐴(𝑥𝑥) ∈ 𝑌𝑌, whenever 𝑥𝑥 ∈ 𝑋𝑋, then we say that 𝐴𝐴 defines a matrix mapping from 𝑋𝑋 into 𝑌𝑌 and denote it by 𝐴𝐴 ∶ 𝑋𝑋 → 𝑌𝑌. By (𝑋𝑋, 𝑌𝑌), we mean the class of all infinite matrices 𝐴𝐴 such that 𝐴𝐴 ∶ 𝑋𝑋 → 𝑌𝑌, and also the matrix domain 𝑋𝑋! of an infinite matrix 𝐴𝐴 in a sequence space 𝑋𝑋 is defined by
𝑋𝑋! = 𝑥𝑥 = 𝑥𝑥! ∈ 𝜔𝜔: 𝐴𝐴(𝑥𝑥) ∈ 𝑋𝑋 . (2) A subspace 𝑋𝑋 is called an 𝐹𝐹𝐹𝐹 space if it is a Frechet space, that is, a complete locally convex linear metric space, with continuous coordinates 𝑅𝑅!∶ 𝑋𝑋 → ℂ (𝑛𝑛 = 0,1, 2, … ), where 𝑅𝑅! 𝑥𝑥 = 𝑥𝑥! for all 𝑥𝑥 ∈ 𝑋𝑋; an 𝐹𝐹𝐹𝐹 space whose metric is given by a norm is said to be a 𝐵𝐵𝐵𝐵 space. An 𝐹𝐹𝐹𝐹 space 𝑋𝑋 ⊃ 𝜙𝜙, the set of all finite sequences, is said to have the 𝐴𝐴𝐴𝐴 property if
!→!lim 𝑥𝑥[!]= lim!→! 𝑥𝑥!𝑒𝑒(!)
!
!!!
= 𝑥𝑥,
for every sequence 𝑥𝑥 ∈ 𝑋𝑋, where 𝑒𝑒(!) is a sequence whose only non-zero term is one in 𝑛𝑛-th place for 𝑛𝑛 ≥ 0.
For example, it is well known that Maddox's space 𝑙𝑙 𝜇𝜇 = 𝑥𝑥 = 𝑥𝑥! ∶ 𝑥𝑥!!!
!
!!!
< ∞
is an 𝐹𝐹𝐹𝐹 space with 𝐴𝐴𝐴𝐴 with respect to its natural paranorm
𝑔𝑔 𝑥𝑥 = 𝑥𝑥! !!
!
!!!
!\!
where 𝑀𝑀 = max {1; sup!𝜇𝜇!}; it is even a 𝐵𝐵𝐵𝐵 space if 𝜇𝜇!≥ 1 for all 𝑛𝑛 ∈ ℕ with respect to the norm
𝑥𝑥 = inf 𝛿𝛿 > 0: 𝑥𝑥! 𝛿𝛿 !!≤ 1
!
!!!
, (Maddox 1969; 1968; 1967; Nakano 1951).
Research on absolute summability factors and the comparison of summability methods plays an important role in Fourier Analysis and Approximation Theory and has been pursued by many authors (see, for example, Altay & Basar 2006; Bor et al. 2015; Bor 1985; Borwein
& Cass 1968; Bosanquet & Chow 1957; Bosanquet 1950;
Bosanquet 1945; Hazar & Sarıgöl 2018; Das 1970; Flett 1957; Kalaivani & Youvaraj 2013; Mazhar 1971;
McFadden 1942; Mehdi 1960; Orhan & Sarıgöl 1993;
Sarıgöl 2016a; Sarıgöl 2016b; Sarıgöl 2015; Sarıgöl 2011;
Sarıgöl 2010; Sarıgöl 1993; Sarıgöl 1991a; Sarıgöl 1991b;
Sulaiman 1992; Tanaka 1978).
Here, we note that these problems correspond to the special matrix transformations such as identity matrix and diagonal matrix. Concerning these topics, some sequence spaces have been generated and examined by several authors (see Altay & Basar 2006; Choudhary & Mishra 1993; Grosse-Erdmann 1993; Maddox 1968; Maddox 1969; Malkowsky & Rakocevic 2007; Mohapatra &
Sarıgöl 2018; Mursaleen & Noman 2011; Mursaleen &
Noman 2010; Nakano 1951).
The space 𝑁𝑁!! ! has recently been derived by Sarıgöl (2011) using Sulaiman’s (1992) summability method 𝑁𝑁, 𝑝𝑝!, 𝜃𝜃! !, and studied by Mohapatra & Sarıgöl (2018), Sarıgöl (2016b) and Ozarslan & Ozgen (2015). The purpose of the present paper is to generalize this space to a new space 𝑁𝑁!! 𝜇𝜇 , show that it is a 𝐵𝐵𝐵𝐵-space with 𝐴𝐴𝐴𝐴 and characterize certain matrix transformations on that space. In doing so, we also deduce some important results of Bosanquet (1950), Mohapatra & Sarıgöl (2018), Sarıgöl (2011), Orhan & Sarıgöl (1993) and Sunouchi (1949) as special cases.
First, to define the space 𝑁𝑁!! 𝜇𝜇 , we need a new notion of the generalized absolute summability method that also includes some well-known summability methods. Let 𝑎𝑎! be a given infinite series with 𝑠𝑠! as its
𝑛𝑛-th partial sum, 𝐴𝐴 is an infinite matrix of complex numbers and 𝜃𝜃! is any positive sequence. Let 𝜇𝜇! be any bounded sequence of positive real numbers. Then we say that the series 𝑎𝑎! is summable 𝐴𝐴, 𝜃𝜃 𝜇𝜇 , if
𝜃𝜃!!!!! 𝐴𝐴! 𝑠𝑠 − 𝐴𝐴!!! 𝑠𝑠 !! <
!
!!!
∞. (3)
Then it should be noted that the summability 𝐴𝐴, 𝜃𝜃 𝜇𝜇 includes the following well-known summability methods for special cases of 𝜇𝜇, 𝜃𝜃 and 𝑘𝑘 ≥ 1:
(a) If 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝐴𝐴, 𝜃𝜃! (Sarıgöl, 2010).
(b) If 𝜇𝜇!= 𝑘𝑘 and 𝜃𝜃!= 𝑛𝑛 or 𝜃𝜃!= 1 𝑎𝑎!! for all 𝑛𝑛, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝐴𝐴!
(Sarıgöl, 1991b).
(c) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑛𝑛 for all 𝑛𝑛 and 𝐴𝐴 = 𝐶𝐶, 𝛼𝛼 , Cesaro means of order 𝛼𝛼 > −1, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝐶𝐶, 𝛼𝛼! (Flett, 1957).
(d) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝛼𝛼!! (!!!) for all 𝑛𝑛 and A = (C, 𝛼𝛼), then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability
𝐶𝐶, 𝛼𝛼, 𝛼𝛼! ! (Bor et al., 2015).
(e) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑛𝑛 for all 𝑛𝑛, and 𝐴𝐴 = (𝐶𝐶, 𝛼𝛼, 𝛽𝛽), Cesaro means of order 𝛼𝛼, 𝛽𝛽 , 𝛼𝛼 + 𝛽𝛽 ≠ −1, −2, …, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability
𝐶𝐶, 𝛼𝛼, 𝛽𝛽 ! (Das, 1970).
(f) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑛𝑛 for all 𝑛𝑛, and 𝐴𝐴 = (𝑅𝑅, 𝑝𝑝!), Riesz means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability
𝑅𝑅, 𝑝𝑝! ! (Sarıgöl, 1993).
(g) If 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛, and 𝐴𝐴 = 𝑁𝑁, 𝑝𝑝! , the weighted means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability
𝑁𝑁, 𝑝𝑝!, 𝜃𝜃! ! (Sulaiman, 1992).
(h) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑃𝑃! 𝑝𝑝! for all 𝑛𝑛, and 𝐴𝐴 = 𝑁𝑁, 𝑝𝑝! , the weighted means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝑁𝑁, 𝑝𝑝! ! (Bor, 1985).
(i) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑛𝑛 for all 𝑛𝑛 and 𝐴𝐴 = 𝑁𝑁, 𝑝𝑝! , Nörlund means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝑁𝑁, 𝑝𝑝! ! (Borwein & Cass, 1968).
(j) If 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛 and 𝐴𝐴 is the generalized Nörlund means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝑁𝑁, 𝑝𝑝!, 𝑞𝑞! !. In particular, for 𝑘𝑘 = 1, it is reduced the summability 𝑁𝑁, 𝑝𝑝!, 𝑞𝑞! (Tanaka, 1978 ).
(k) If 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛 and 𝜃𝜃!= 𝛾𝛾 𝑛𝑛 𝑛𝑛1 𝑘𝑘∗, where 𝛾𝛾: [1, ∞) → [1, ∞) is a nondecreasing function, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝐴𝐴, 𝛾𝛾!
(Kalaivani & Youvaraj, 2013).
Definition. Let (𝑝𝑝!) be a positive sequence with 𝑃𝑃!= 𝑝𝑝!+ 𝑝𝑝!+ ⋯ + 𝑝𝑝!→ ∞ as 𝑛𝑛 → ∞, (𝑃𝑃!! = 𝑝𝑝!! = 0). We define a space 𝑁𝑁!! 𝜇𝜇 as the set of all series summable by the absolute summability 𝐴𝐴, 𝜃𝜃 𝜇𝜇 , where 𝐴𝐴 is the weighted mean matrix:
𝑎𝑎!"= 𝑝𝑝! 𝑃𝑃!, 0 ≤ 𝜈𝜈 ≤ 𝑛𝑛 0, 𝜈𝜈 > 𝑛𝑛.
Then, it can be written from (1) that
𝐴𝐴! 𝑠𝑠 − 𝐴𝐴!!! 𝑠𝑠 = 𝑝𝑝!
𝑃𝑃!𝑃𝑃!!! 𝑃𝑃!!!𝑎𝑎! .
!
!!!
which implies 𝐴𝐴! 𝑠𝑠 = 𝑎𝑎! , and for 𝑛𝑛 ≥ 1, 𝐴𝐴! 𝑠𝑠 − 𝐴𝐴!!! 𝑠𝑠 = 𝑝𝑝!
𝑃𝑃!𝑃𝑃!!! 𝑃𝑃!!!𝑎𝑎! .
!
!!!
To understand the space 𝑁𝑁!! 𝜇𝜇 better, it is useful to state it in terms of the series 𝑎𝑎!. In fact, it is clear by (3) that the space 𝑁𝑁!! 𝜇𝜇 can be written as
𝑁𝑁!! 𝜇𝜇 = 𝑎𝑎 ∶ 𝜃𝜃!!!!!
!
!!!
𝜒𝜒!(𝑎𝑎)!!< ∞ where
𝜒𝜒! 𝑎𝑎 = 𝑝𝑝!
𝑃𝑃!𝑃𝑃!!! 𝑃𝑃!!!𝑎𝑎! .
!
!!!
It is also trivial that, in the special case 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛 ≥ 0, the series space 𝑁𝑁!! 𝜇𝜇 is reduced to the space 𝑁𝑁!! !(Sarıgöl, 2011) and the space 𝑅𝑅! ! with 𝜃𝜃!= 𝑛𝑛 (Orhan & Sarıgöl, 1993). Further, with the notation (2), it can be redefined by 𝑁𝑁!! 𝜇𝜇 = 𝑙𝑙 𝜇𝜇 ! !,!,!, where the matrix 𝑇𝑇 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 is given by
𝑡𝑡!" 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 =
1, 𝑛𝑛 = 0, 𝜈𝜈 = 0 𝜃𝜃!! !!∗!!!!!!!
!!!!!, 1 ≤ 𝜈𝜈 ≤ 𝑛𝑛 0, 𝜈𝜈 > 𝑛𝑛, to which the inverse is 𝑆𝑆 𝜃𝜃, 𝜇𝜇, 𝑝𝑝
𝑠𝑠!! 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 = 1,
𝑠𝑠!" 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 =
−𝜃𝜃!!!!! !!!!∗ !!!!!
!!!, 𝜈𝜈 = 𝑛𝑛 − 1 𝜃𝜃!!! !!∗ !!!
!, 𝜈𝜈 = 𝑛𝑛 0, 𝜈𝜈 ≠ 𝑛𝑛 − 1, 𝑛𝑛
where 𝜇𝜇!∗ is the conjugate of 𝜇𝜇!, i.e. 1 𝜇𝜇!+ 1 𝜇𝜇!∗ = 1, 𝜇𝜇!> 1, and 1 𝜇𝜇!∗ = 0 for 𝜇𝜇!= 1.
In addition, for simplicity of presentation we take for all 𝑛𝑛, 𝜈𝜈 ≥ 0,
𝑎𝑎!"= 𝑃𝑃!
𝜃𝜃!! !!∗𝑝𝑝! 𝑎𝑎!"−𝑃𝑃!!!
𝑃𝑃! 𝑎𝑎!,!!! .
With these notations, we establish the following theorems.
Theorem 1.1. Let (𝜃𝜃!) be a sequence of positive numbers and (𝜇𝜇!) be a bounded sequence of positive numbers.
Then the set 𝑁𝑁!! 𝜇𝜇 becomes a linear space with the coordinate-wise addition and scalar multiplication. It is also an 𝐹𝐹𝐹𝐹-space with 𝐴𝐴𝐴𝐴 in respect to the paranorm ℎ(𝑥𝑥) = 𝑔𝑔(𝑇𝑇(𝑥𝑥)) with
𝑔𝑔(𝑇𝑇(𝑥𝑥)) = 𝜃𝜃!!!!! 𝑇𝑇!(𝑥𝑥)!!
!
!!!
! !
where 𝜃𝜃!= 1 and 𝑀𝑀 = max {1, sup!𝜇𝜇!}.
Theorem 1.2. Let 𝐴𝐴 = (𝑎𝑎!") be an infinite matrix of complex numbers and (𝜃𝜃!) be a sequence of positive numbers. If (𝜇𝜇!) is an arbitrary bounded sequence of positive numbers such that 𝜇𝜇!> 1 for all 𝑛𝑛, then (4)
(5) 𝑛𝑛-th partial sum, 𝐴𝐴 is an infinite matrix of complex
numbers and 𝜃𝜃! is any positive sequence. Let 𝜇𝜇! be any bounded sequence of positive real numbers. Then we say that the series 𝑎𝑎! is summable 𝐴𝐴, 𝜃𝜃 𝜇𝜇 , if
𝜃𝜃!!!!! 𝐴𝐴! 𝑠𝑠 − 𝐴𝐴!!! 𝑠𝑠 !! <
!
!!!
∞. (3)
Then it should be noted that the summability 𝐴𝐴, 𝜃𝜃 𝜇𝜇 includes the following well-known summability methods for special cases of 𝜇𝜇, 𝜃𝜃 and 𝑘𝑘 ≥ 1:
(a) If 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝐴𝐴, 𝜃𝜃! (Sarıgöl, 2010).
(b) If 𝜇𝜇!= 𝑘𝑘 and 𝜃𝜃!= 𝑛𝑛 or 𝜃𝜃!= 1 𝑎𝑎!! for all 𝑛𝑛, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝐴𝐴!
(Sarıgöl, 1991b).
(c) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑛𝑛 for all 𝑛𝑛 and 𝐴𝐴 = 𝐶𝐶, 𝛼𝛼 , Cesaro means of order 𝛼𝛼 > −1, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝐶𝐶, 𝛼𝛼! (Flett, 1957).
(d) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝛼𝛼!! (!!!) for all 𝑛𝑛 and A = (C, 𝛼𝛼), then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability
𝐶𝐶, 𝛼𝛼, 𝛼𝛼! ! (Bor et al., 2015).
(e) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑛𝑛 for all 𝑛𝑛, and 𝐴𝐴 = (𝐶𝐶, 𝛼𝛼, 𝛽𝛽), Cesaro means of order 𝛼𝛼, 𝛽𝛽 , 𝛼𝛼 + 𝛽𝛽 ≠ −1, −2, …, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability
𝐶𝐶, 𝛼𝛼, 𝛽𝛽 ! (Das, 1970).
(f) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑛𝑛 for all 𝑛𝑛, and 𝐴𝐴 = (𝑅𝑅, 𝑝𝑝!), Riesz means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability
𝑅𝑅, 𝑝𝑝! ! (Sarıgöl, 1993).
(g) If 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛, and 𝐴𝐴 = 𝑁𝑁, 𝑝𝑝! , the weighted means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability
𝑁𝑁, 𝑝𝑝!, 𝜃𝜃! ! (Sulaiman, 1992).
(h) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑃𝑃! 𝑝𝑝! for all 𝑛𝑛, and 𝐴𝐴 = 𝑁𝑁, 𝑝𝑝! , the weighted means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝑁𝑁, 𝑝𝑝! ! (Bor, 1985).
(i) If 𝜇𝜇!= 𝑘𝑘, 𝜃𝜃!= 𝑛𝑛 for all 𝑛𝑛 and 𝐴𝐴 = 𝑁𝑁, 𝑝𝑝! , Nörlund means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝑁𝑁, 𝑝𝑝! ! (Borwein & Cass, 1968).
(j) If 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛 and 𝐴𝐴 is the generalized Nörlund means, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝑁𝑁, 𝑝𝑝!, 𝑞𝑞! !. In particular, for 𝑘𝑘 = 1, it is reduced the summability 𝑁𝑁, 𝑝𝑝!, 𝑞𝑞! (Tanaka, 1978 ).
(k) If 𝜇𝜇! = 𝑘𝑘 for all 𝑛𝑛 and 𝜃𝜃!= 𝛾𝛾 𝑛𝑛 𝑛𝑛1 𝑘𝑘∗, where 𝛾𝛾: [1, ∞) → [1, ∞) is a nondecreasing function, then 𝐴𝐴, 𝜃𝜃 𝜇𝜇 is reduced to the summability 𝐴𝐴, 𝛾𝛾 !
(Kalaivani & Youvaraj, 2013).
Definition. Let (𝑝𝑝!) be a positive sequence with 𝑃𝑃!= 𝑝𝑝!+ 𝑝𝑝!+ ⋯ + 𝑝𝑝!→ ∞ as 𝑛𝑛 → ∞, (𝑃𝑃!! = 𝑝𝑝!! = 0). We define a space 𝑁𝑁!! 𝜇𝜇 as the set of all series summable by the absolute summability 𝐴𝐴, 𝜃𝜃 𝜇𝜇 , where 𝐴𝐴 is the weighted mean matrix:
𝑎𝑎!"= 𝑝𝑝! 𝑃𝑃!, 0 ≤ 𝜈𝜈 ≤ 𝑛𝑛 0, 𝜈𝜈 > 𝑛𝑛.
Then, it can be written from (1) that
𝐴𝐴! 𝑠𝑠 − 𝐴𝐴!!! 𝑠𝑠 = 𝑝𝑝!
𝑃𝑃!𝑃𝑃!!! 𝑃𝑃!!!𝑎𝑎! .
!
!!!
which implies 𝐴𝐴! 𝑠𝑠 = 𝑎𝑎! , and for 𝑛𝑛 ≥ 1, 𝐴𝐴! 𝑠𝑠 − 𝐴𝐴!!! 𝑠𝑠 = 𝑝𝑝!
𝑃𝑃!𝑃𝑃!!! 𝑃𝑃!!!𝑎𝑎! .
!
!!!
To understand the space 𝑁𝑁!! 𝜇𝜇 better, it is useful to state it in terms of the series 𝑎𝑎!. In fact, it is clear by (3) that the space 𝑁𝑁!! 𝜇𝜇 can be written as
𝑁𝑁!! 𝜇𝜇 = 𝑎𝑎 ∶ 𝜃𝜃!!!!!
!
!!!
𝜒𝜒!(𝑎𝑎)!!< ∞ where
𝜒𝜒! 𝑎𝑎 = 𝑝𝑝!
𝑃𝑃!𝑃𝑃!!! 𝑃𝑃!!!𝑎𝑎! .
!
!!!
It is also trivial that, in the special case 𝜇𝜇!= 𝑘𝑘 for all 𝑛𝑛 ≥ 0, the series space 𝑁𝑁!! 𝜇𝜇 is reduced to the space 𝑁𝑁!!!(Sarıgöl, 2011) and the space 𝑅𝑅! ! with 𝜃𝜃!= 𝑛𝑛 (Orhan & Sarıgöl, 1993). Further, with the notation (2), it can be redefined by 𝑁𝑁!! 𝜇𝜇 = 𝑙𝑙 𝜇𝜇 ! !,!,! , where the matrix 𝑇𝑇 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 is given by
𝑡𝑡!" 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 =
1, 𝑛𝑛 = 0, 𝜈𝜈 = 0 𝜃𝜃!! !!∗ !!!!!!!
!!!!!, 1 ≤ 𝜈𝜈 ≤ 𝑛𝑛 0, 𝜈𝜈 > 𝑛𝑛, to which the inverse is 𝑆𝑆 𝜃𝜃, 𝜇𝜇, 𝑝𝑝
𝑠𝑠!! 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 = 1,
𝑠𝑠!" 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 =
−𝜃𝜃!!!!! !!!!∗ !!!!!!!!, 𝜈𝜈 = 𝑛𝑛 − 1 𝜃𝜃!!! !!∗ !!!
!, 𝜈𝜈 = 𝑛𝑛 0, 𝜈𝜈 ≠ 𝑛𝑛 − 1, 𝑛𝑛
where 𝜇𝜇!∗ is the conjugate of 𝜇𝜇!, i.e. 1 𝜇𝜇!+ 1 𝜇𝜇!∗ = 1, 𝜇𝜇!> 1, and 1 𝜇𝜇!∗ = 0 for 𝜇𝜇!= 1.
In addition, for simplicity of presentation we take for all 𝑛𝑛, 𝜈𝜈 ≥ 0,
𝑎𝑎!"= 𝑃𝑃!
𝜃𝜃!! !!∗𝑝𝑝! 𝑎𝑎!"−𝑃𝑃!!!
𝑃𝑃! 𝑎𝑎!,!!! .
With these notations, we establish the following theorems.
Theorem 1.1. Let (𝜃𝜃!) be a sequence of positive numbers and (𝜇𝜇!) be a bounded sequence of positive numbers.
Then the set 𝑁𝑁!! 𝜇𝜇 becomes a linear space with the coordinate-wise addition and scalar multiplication. It is also an 𝐹𝐹𝐹𝐹-space with 𝐴𝐴𝐴𝐴 in respect to the paranorm ℎ(𝑥𝑥) = 𝑔𝑔(𝑇𝑇(𝑥𝑥)) with
𝑔𝑔(𝑇𝑇(𝑥𝑥)) = 𝜃𝜃!!!!! 𝑇𝑇!(𝑥𝑥)!!
!
!!!
! !
where 𝜃𝜃!= 1 and 𝑀𝑀 = max {1, sup!𝜇𝜇!}.
Theorem 1.2. Let 𝐴𝐴 = (𝑎𝑎!") be an infinite matrix of complex numbers and (𝜃𝜃!) be a sequence of positive numbers. If (𝜇𝜇!) is an arbitrary bounded sequence of positive numbers such that 𝜇𝜇!> 1 for all 𝑛𝑛, then (4)
(5)
𝐴𝐴 ∈ 𝑁𝑁!! 𝜇𝜇 , 𝑁𝑁! if and only if there exists an integer 𝑀𝑀 > 1 such that, for 𝑛𝑛 = 0,1, …,
sup!
𝑀𝑀!!𝑃𝑃!𝑎𝑎!"
𝜃𝜃!! !!∗ 𝑝𝑝!
!!∗
<∞, (6)
𝑀𝑀!!𝑎𝑎!" !!∗
!
!!!
<∞, (7)
𝑀𝑀!!𝑞𝑞!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!
!
!!!
𝑎𝑎!"
!
!!!
!!∗
<∞
!
!!!
(8) where (𝑞𝑞!) is a positive sequence with 𝑄𝑄!= 𝑞𝑞!+ 𝑞𝑞!+
⋯ + 𝑞𝑞!→ ∞ as 𝑛𝑛 → ∞, (𝑄𝑄!!= 𝑞𝑞!! = 0).
Theorem 1.3. Let 𝐴𝐴 = (𝑎𝑎!") be an infinite matrix of complex numbers, (𝜃𝜃!) and (𝜓𝜓!) be sequences of positive numbers. If (𝜇𝜇!) and (𝜆𝜆!) are arbitrary bounded sequences of positive numbers and 𝜇𝜇!≤ 1 and 𝜆𝜆!≥ 1 for all 𝑛𝑛, then, 𝐴𝐴 ∈ 𝑁𝑁!! 𝜇𝜇 , 𝑁𝑁!! 𝜆𝜆 if and only if there exists an integer 𝑀𝑀 > 1 such that, for 𝑛𝑛 = 0, 1, …,
sup
! 𝑎𝑎!" !!<∞, (9) sup
!
𝑃𝑃!𝑎𝑎!"
𝜃𝜃!! !!∗ 𝑝𝑝! <∞, (10) and
sup
!
𝜓𝜓!! !!∗𝑞𝑞!𝑀𝑀!! !!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!
!
!!!
𝑎𝑎!"
!!
!
!!!
<∞. (11)
<∞. (11)
2. Needed Lemmas
We require the following lemmas for the proof of our theorems.
Lemma 2.1. (Stieglitz & Tietz, 1977) 𝐴𝐴 ∈ (𝑙𝑙, 𝑐𝑐) if and only if
𝑖𝑖 lim! 𝑎𝑎!"exists for each 𝜈𝜈, 𝑖𝑖𝑖𝑖 sup
!,! 𝑎𝑎!" <∞.
Lemma 2.2. (Grosse-Erdmann, 1993) Let (𝜇𝜇!) and (𝜆𝜆!) be any two bounded sequences of strictly positive numbers.
(i) If 𝜇𝜇!≤ 1, then, 𝐴𝐴 ∈ 𝑙𝑙 𝜇𝜇 , 𝑐𝑐 if and only if 𝑖𝑖 ! lim! 𝑎𝑎!"exists for each 𝜈𝜈, 𝑖𝑖𝑖𝑖 ! sup
! sup
! 𝑎𝑎!"!!<∞. (13) (ii) If 𝜇𝜇!> 1 for all 𝜈𝜈, then (𝑙𝑙(𝜇𝜇), 𝑐𝑐) iff
𝑖𝑖 ! (13) 𝑖𝑖 ! holds 𝑖𝑖𝑖𝑖 ! There exists an integer 𝑀𝑀 > 1 such that
sup! 𝑎𝑎!"𝑀𝑀!! !!∗
∞
!!!
<∞.
(𝑖𝑖𝑖𝑖𝑖𝑖) If 𝜇𝜇!> 1 for all 𝜈𝜈, then 𝐴𝐴 ∈ (𝑙𝑙(𝜇𝜇), 𝑙𝑙) if and only if there exists an integer
𝑀𝑀 > 1 such that
sup 𝑎𝑎!"𝑀𝑀!!
!∈!
!!∗
: 𝑁𝑁 ⊂ ℕ finite
!
!!!
<∞. (14)
<∞. (14)
(iv) If 𝜇𝜇!≤ 1 and 𝜆𝜆!≥ 1 for all 𝜈𝜈 ∈ ℕ 𝐴𝐴 ∈ 𝑙𝑙 𝜇𝜇 , 𝑙𝑙 𝜆𝜆 if and only if there exists some 𝑀𝑀 such that
sup! 𝑎𝑎!"𝑀𝑀!! !!!!
∞
!!!
<∞.
It may be noticed that the condition (14) exposes a rather difficult condition in applications. Thus, the following lemma, which derives a condition to be equivalent to (14), is more useful in many cases and also provides great convenience in computations.
Lemma 2.3. (Sarıgöl, 2013) Let 𝐴𝐴 = (𝑎𝑎!") be an infinite matrix with complex numbers, (𝜇𝜇!) be a bounded sequence of positive numbers,
𝑈𝑈! [𝐴𝐴] = 𝑎𝑎!"
!!!
!!
!
!!!
and 𝐿𝐿! 𝐴𝐴 = sup 𝑎𝑎!"
!∈!
!!
: 𝑁𝑁 ⊂ ℕ finite
!
!!!
.
If 𝑈𝑈! 𝐴𝐴 < ∞ or 𝐿𝐿! 𝐴𝐴 < ∞, then
(2𝐶𝐶)!!𝑈𝑈! 𝐴𝐴 ≤ 𝐿𝐿! 𝐴𝐴 ≤ 𝑈𝑈! 𝐴𝐴 , where 𝐶𝐶 = max 1, 2!!! , 𝐻𝐻 = sup!𝜇𝜇!.
Lemma 2.4. (Malkowsky & Rakocevic, 2007) Let 𝑋𝑋 be an 𝐹𝐹𝐹𝐹 space with 𝐴𝐴𝐴𝐴, 𝑇𝑇 be a triangle matrix, 𝑆𝑆 be its inverse and 𝑌𝑌 be an arbitrary subset of 𝜔𝜔. Then, we have 𝐴𝐴 ∈ (𝑋𝑋!, 𝑌𝑌) if and only if 𝐴𝐴 ∈ (𝑋𝑋, 𝑌𝑌) and 𝑉𝑉(!) ∈ (𝑋𝑋, 𝑐𝑐) for all 𝑛𝑛, where
𝑎𝑎!"= 𝑎𝑎!"𝑠𝑠!"; 𝑛𝑛, 𝜈𝜈 = 0,1, …, (15)
!
!!!
and
𝑣𝑣!"(!)= 𝑎𝑎!"𝑠𝑠!", 0 ≤ 𝜈𝜈 ≤ 𝑚𝑚
!
!!!
(16) 0, 𝜈𝜈 > 𝑚𝑚.
3. Proofs of Theorems
In this section, we only give the proofs of our theorems, making use of lemmas.
Proof of Theorem 1.1. The first part is a routine verification, so it is omitted. Let us consider the matrix 𝑇𝑇 defined by (4). Then 𝑇𝑇 defines a matrix map from 𝜔𝜔 into 𝜔𝜔 since it is a triangle matrix. Furthermore, since 𝜔𝜔 and 𝑙𝑙(𝜇𝜇) are FK spaces and 𝑁𝑁!! 𝜇𝜇 = 𝑙𝑙 𝜇𝜇 !, then 𝑇𝑇 is a continuous linear map. Thus, 𝑁𝑁!! 𝜇𝜇 is an 𝐹𝐹𝐹𝐹-space by Corollary 7.3.7 and Theorem 7.3.14 of Boos & Cass (2000). Finally, to show that 𝑁𝑁!! 𝜇𝜇 is a space with 𝐴𝐴𝐴𝐴, let us consider the base 𝑒𝑒! of 𝑙𝑙 𝜇𝜇 where 𝑒𝑒 ! is a (12)
sequence whose only non-zero term is one in 𝑛𝑛-th place for 𝑛𝑛 ≥ 1. Let 𝑟𝑟!= 𝑇𝑇!! 𝑒𝑒 ! , 𝑥𝑥 ∈ 𝑁𝑁!! 𝜇𝜇 and 𝑦𝑦 = 𝑇𝑇 𝑥𝑥 . Then, since 𝑦𝑦 ∈ 𝑙𝑙 𝜇𝜇 , there exists only a unique sequence of scalars 𝜆𝜆! such that 𝑔𝑔 𝑦𝑦 −
𝜆𝜆!𝑒𝑒!
!!!! → 0. Thus, it is clear that ℎ 𝑥𝑥 − 𝜆𝜆!𝑟𝑟!
!
!!!
= 𝑔𝑔 𝑦𝑦 − 𝜆𝜆!𝑒𝑒!
!
!!!
,
which gives the desired conclusion.
Proof of Theorem 1.2. Note that taking 𝜃𝜃!= 1 does not disrupt generality. Let 𝜇𝜇!> 1 for all 𝑛𝑛, 𝑇𝑇 = 𝑇𝑇 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 and 𝑇𝑇(!)= 𝑇𝑇 1,1, 𝑞𝑞 defined by (4). We can denote the inverse of the matrix 𝑇𝑇 by 𝑆𝑆 defined by (5). Then, it is clear that 𝑁𝑁!! 𝜇𝜇 = 𝑙𝑙 𝜇𝜇 ! and 𝑁𝑁! = 𝑙𝑙 !(!). So, by Lemma 2.4, we have 𝐴𝐴 ∈ 𝑁𝑁!! 𝜇𝜇 , 𝑁𝑁! if and only if 𝐴𝐴 ∈ 𝑙𝑙 𝜇𝜇 , 𝑁𝑁! and 𝑉𝑉(!)∈ 𝑙𝑙 𝜇𝜇 , 𝑐𝑐 , where 𝐴𝐴 and 𝑉𝑉(!) are given by (15) and (16), respectively. Besides, if 𝐵𝐵 = 𝑇𝑇(!)𝐴𝐴 then, it is easily seen that 𝐴𝐴 ∈ 𝑙𝑙 𝜇𝜇 , 𝑁𝑁! iff 𝐵𝐵 ∈ 𝑙𝑙 𝜇𝜇 , 𝑙𝑙 because, if 𝐴𝐴 𝑥𝑥 ∈ 𝑁𝑁! for all 𝑥𝑥 ∈ 𝑙𝑙 𝜇𝜇 , then 𝑇𝑇! 𝐴𝐴 𝑥𝑥 ∈ 𝑙𝑙, i.e. 𝐵𝐵 𝑥𝑥 ∈ 𝑙𝑙. Further, a few calculations reveal that for all 𝑛𝑛, 𝜈𝜈 ≥ 0,
𝑎𝑎!"= 𝑃𝑃!
𝜃𝜃!! !!∗𝑝𝑝!
𝑎𝑎!"−𝑃𝑃!!!
𝑃𝑃! 𝑎𝑎!,!!!
and
𝑣𝑣!"(!)=
𝑎𝑎!", 0 ≤ 𝜈𝜈 ≤ 𝑚𝑚 − 1 𝑃𝑃!𝑎𝑎!"
𝜃𝜃!! !!∗ 𝑝𝑝!
, 𝜈𝜈 = 𝑚𝑚, 𝑚𝑚 ≥ 1 (18) 0, 𝜈𝜈 > 𝑚𝑚.
Also, since the matrix 𝐵𝐵 is defined by 𝑏𝑏!"= 𝑡𝑡!"(!)𝑎𝑎!"
!
!!!
, we have for all 𝜈𝜈 ≥ 0,
𝑏𝑏!"=
𝑎𝑎!!, 𝑛𝑛 = 0 𝑞𝑞!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!
!
!!!
𝑎𝑎!", 𝑛𝑛 ≥ 1.
Now, applying Lemma 2.2 (ii) with the matrix 𝑉𝑉(!), since (13) 𝑖𝑖 ! holds, it follows that 𝑉𝑉(!)∈ (𝑙𝑙(𝜇𝜇), 𝑐𝑐) iff there exists an integer 𝑀𝑀 > 1 such that
sup! 𝑣𝑣!"(!)𝑀𝑀!! !!∗+ 𝑣𝑣!!(!)𝑀𝑀!! !!∗
!!!
!!!
< ∞,
which is satisfied iff the conditions (6) and (7) hold.
Again, if we apply Lemma 2.2 (iii) with the matrix 𝐵𝐵, then we have 𝐵𝐵 ∈ (𝑙𝑙(𝜇𝜇), 𝑙𝑙) iff there exists an integer 𝑀𝑀 > 1 such that (14) holds, equivalently, by Lemma 2.3,
𝑀𝑀!!𝑏𝑏!"
!
!!!
!!∗
< ∞. (19)
!
!!!
On the other hand, it is easily seen that (19) is satisfied iff (8) and the condition, which is satisfied by (7),
𝑀𝑀!!𝑎𝑎!! !!∗ < ∞
!
hold. Thus the proof is completed. !!!
Proof of Theorem 1.3. Let 𝜇𝜇!≤ 1 and 𝜆𝜆!≥ 1 for all 𝑣𝑣, 𝑇𝑇 = 𝑇𝑇 𝜃𝜃, 𝜇𝜇, 𝑝𝑝 and 𝑇𝑇(!)= 𝑇𝑇 𝜓𝜓, 𝜆𝜆, 𝑞𝑞 . Then, 𝑁𝑁!! 𝜇𝜇 =
𝑙𝑙 𝜇𝜇 ! !,!,! and 𝑁𝑁!! 𝜆𝜆 = 𝑙𝑙 𝜆𝜆 ! !,!,!. So, as in the above Theorem, 𝐴𝐴 ∈ 𝑁𝑁!! 𝜇𝜇 , 𝑁𝑁!! 𝜆𝜆 if and only if 𝐵𝐵 = 𝑇𝑇(!)𝐴𝐴 ∈ 𝑙𝑙 𝜇𝜇 , 𝑙𝑙 𝜆𝜆 and 𝑉𝑉(!)∈ 𝑙𝑙 𝜇𝜇 , 𝑐𝑐 , where the matrices 𝐴𝐴 and 𝑉𝑉(!) are defined by (17) and (18), respectively. Now considering that
𝑏𝑏!"= 𝑡𝑡!"(!)𝑎𝑎!"
!
!!!
, we get the matrix 𝐵𝐵 as for all 𝜈𝜈 ≥ 0,
𝑏𝑏!"=
𝑎𝑎!!, 𝑛𝑛 = 0 𝜓𝜓!! !!∗𝑞𝑞!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!
!
!!!
𝑎𝑎!", 𝑛𝑛 ≥ 1.
Now, applying Lemma 2.2 (i) and (iv) with the matrices 𝑉𝑉(!) and 𝐵𝐵, it follows that 𝑉𝑉(!)∈ 𝑙𝑙 𝜇𝜇 , 𝑐𝑐 iff, for 𝑛𝑛 = 0, 1, …, the conditions (9) and (10) hold, and that 𝐵𝐵 ∈ 𝑙𝑙 𝜇𝜇 , 𝑙𝑙 𝜆𝜆 iff there exists an integer 𝑀𝑀 such that
sup
! 𝑏𝑏!"𝑀𝑀!! !!!!
∞
!!!
<∞, (20) which is satisfied if and only if the condition (11) and the following condition hold:
sup
!!! 𝑎𝑎!!𝑀𝑀!! !! <∞. (21)
Note that condition (9) includes condition (21). In fact, if (9) holds, then there exists a number 𝐻𝐻 such that
𝜉𝜉! ≤ 𝐻𝐻! !! for all 𝑣𝑣, which implies 𝑀𝑀!! !!𝜉𝜉! ≤ 𝐻𝐻
𝑀𝑀
! !!
, where 𝜉𝜉! = 𝑎𝑎!!. This completes the proof.
4. Applications
Our theorems have several consequences depending on sequences 𝜆𝜆, 𝜇𝜇, 𝜃𝜃, 𝜓𝜓 and a matrix 𝐴𝐴 as parameters. For example, if 𝐴𝐴 is chosen as a diagonal matrix 𝑊𝑊 such as 𝑤𝑤!"= 𝜀𝜀! for 𝜈𝜈 = 𝑛𝑛, and zero otherwise, then 𝑊𝑊 ∈
𝑁𝑁!! 𝜇𝜇 , 𝑁𝑁!! 𝜆𝜆 leads to the conclusion that 𝜀𝜀!𝑥𝑥!
is summable 𝑁𝑁, 𝑞𝑞!, 𝜓𝜓! 𝜆𝜆 when 𝑥𝑥! is summable 𝑁𝑁, 𝑝𝑝!, 𝜃𝜃! 𝜇𝜇 . Hence, if 𝐼𝐼 ∈ 𝑁𝑁!! 𝜇𝜇 , 𝑁𝑁!! 𝜆𝜆 , where 𝐼𝐼 is the identity matrix, leads to the comparisons of these methods, i.e., 𝑁𝑁!! 𝜇𝜇 ⊂ 𝑁𝑁!! 𝜆𝜆 . Now one can easily obtain the following results.
Corollary 4.1. Let (𝜃𝜃!) be a sequence of positive numbers. If (𝜇𝜇!) is any bounded sequence of positive numbers such that 𝜇𝜇!> 1 for all 𝑛𝑛, then 𝑁𝑁!! 𝜇𝜇 ⊂ 𝑁𝑁! if and only if there exists an integer 𝑀𝑀 > 1 such that
𝑀𝑀!!!∗ 𝜃𝜃!
𝑞𝑞!𝑃𝑃!
𝑄𝑄!𝑝𝑝!+ 1 − 𝑞𝑞!𝑃𝑃! 𝑄𝑄!𝑝𝑝!
!!∗
< ∞. (22)
!
!!!
(17)
Proof. Take 𝐴𝐴 = 𝐼𝐼 in Theorem 1.2. Then (6) and (7) are directly satisfied, and (8) is reduced to (22). In fact, since for 𝜈𝜈 ≥ 0,
𝑄𝑄!!!𝑎𝑎!"
!
!!!
= 𝜃𝜃!
!!!!∗ 𝑄𝑄!!!𝑃𝑃! 𝑝𝑝!, 𝑛𝑛 = 𝜈𝜈
𝜃𝜃!
!!!!∗ 𝑄𝑄!−𝑞𝑞!𝑃𝑃!
𝑝𝑝! , 𝑛𝑛 > 𝜈𝜈 and
𝑞𝑞! 𝑄𝑄!𝑄𝑄!!!
!
!!!!!
= 1
𝑄𝑄!, we get
𝑞𝑞!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!𝑎𝑎!"
!
!!!
!
!!!
= 𝑞𝑞!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!𝑎𝑎!!
+ 𝑞𝑞!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!𝑎𝑎!"
!
!!!
!
!!!!!
= 𝜃𝜃!!! !!∗ 𝑞𝑞!𝑃𝑃!
𝑄𝑄!𝑝𝑝!+ 𝑄𝑄!−𝑃𝑃!𝑞𝑞!
𝑝𝑝!
1 𝑄𝑄! , and so (8) is the same as
𝑀𝑀!!!∗ 𝜃𝜃!
𝑞𝑞!𝑃𝑃!
𝑄𝑄!𝑝𝑝!+ 1 − 𝑞𝑞!𝑃𝑃!
𝑄𝑄!𝑝𝑝!
!!∗
< ∞.
!
!!!
This completes the proof.
Furthermore, taking 𝜃𝜃!= 𝑃𝑃! 𝑝𝑝! and 𝜇𝜇! = 𝑘𝑘 > 1 for all 𝜈𝜈 in Corollary 4.1, (22) is reduced to
𝑝𝑝!
𝑃𝑃!
𝑞𝑞!𝑃𝑃!
𝑄𝑄!𝑝𝑝!+ 1 − 𝑞𝑞!𝑃𝑃!
𝑄𝑄!𝑝𝑝!
!∗
< ∞.
!
!!!
But this is impossible, since 𝑝𝑝!
𝑃𝑃!
𝑞𝑞!𝑃𝑃!
𝑄𝑄!𝑝𝑝!+ 1 −𝑞𝑞!𝑃𝑃! 𝑄𝑄!𝑝𝑝!
!∗
≥𝑝𝑝!
𝑃𝑃!
for all 𝑣𝑣 and !!!
! is divergent by Abel-Dini Theorem. So we have the following result.
Corollary 4.2. If 𝜃𝜃!= 𝑃𝑃! 𝑝𝑝! for all 𝜈𝜈 ≥ 0 then 𝑁𝑁!! 𝜇𝜇 ⊈ 𝑁𝑁! for all sequences 𝑝𝑝! and 𝑞𝑞! , i.e.
there is a series 𝑎𝑎! summable by 𝑁𝑁, 𝑝𝑝!, 𝜃𝜃! ! but not summable by 𝑁𝑁, 𝑞𝑞!.
Also, choosing 𝜇𝜇!= 𝑘𝑘 > 1 for all 𝜈𝜈 ≥ 0 and 𝐴𝐴 is a triangle matrix, then Theorem 1.2 is reduced to the following main result given by Sarıgöl (2011).
Corollary 4.3. Let 𝐴𝐴 = (𝑎𝑎!") be an infinite triangle matrix of complex numbers and 𝜃𝜃! be a sequence of positive numbers. Then 𝐴𝐴 ∈ 𝑁𝑁!!
!, 𝑁𝑁! if and only if 𝑞𝑞!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!𝑎𝑎!"
!
!!!
!
!!!
!!∗
<∞.
!
!!!
Corollary 4.4. Let (𝜃𝜃!) and (𝜓𝜓!) be sequences of positive numbers. If (𝜇𝜇!) and (𝜆𝜆!) are arbitrary bounded sequences of positive numbers such that inf 𝜇𝜇!> 0, 𝜇𝜇!≤ 1 and 𝜆𝜆!≥ 1 for all 𝑛𝑛, then 𝑁𝑁!! 𝜇𝜇 ⊂ 𝑁𝑁!! 𝜆𝜆 if and only if there exists an integer 𝑀𝑀 > 1 such that
sup!
𝑀𝑀!! !!𝜓𝜓!! !!∗𝑞𝑞!𝑃𝑃!
𝜃𝜃!! !!∗𝑄𝑄!𝑝𝑝!
!!
<∞ and
sup!
𝑀𝑀!!!!𝜓𝜓!
!!!∗𝑞𝑞!
𝑄𝑄!𝑄𝑄!!!𝜃𝜃!
!!!∗
𝑄𝑄!−𝑞𝑞!𝑃𝑃! 𝑝𝑝!
!!
!
!!!!!
<∞.
To obtain this result, it is sufficient to take 𝐴𝐴 = 𝐼𝐼 in Theorem 1.3.
We remark that for the case 𝜇𝜇!= 𝜆𝜆!= 1 and 𝐴𝐴 = 𝐼𝐼, Corollary 4.4 gives the well known result of Bosanquet (1950) and Sunouchi (1949), as follows.
Corollary 4.5. 𝑁𝑁! ⊂ 𝑁𝑁! if and only if the following condition is satisfied:
sup!
𝑞𝑞!𝑃𝑃!
𝑄𝑄!𝑝𝑝!< ∞.
Corollary 4.6. Let 𝐴𝐴 be a triangle matrix and (𝜃𝜃!) be a sequence of positive numbers. Then, 𝐴𝐴 ∈ 𝑁𝑁! , 𝑁𝑁!!
! if and only if the following conditions are satisfied:
sup!
𝜃𝜃!! !∗𝑞𝑞! 𝑃𝑃!
𝑄𝑄!𝑝𝑝! 𝑎𝑎!! <∞, (23) sup!
𝑃𝑃!
𝑝𝑝!
!
𝜎𝜎!!− 𝜎𝜎!,!!!!
!
!!!!!
<∞ (24)
sup
! 𝜎𝜎!,!!! !
!
!!!!!
<∞ 25
where
𝜎𝜎!!=𝜃𝜃!! !∗𝑞𝑞!
𝑄𝑄!𝑄𝑄!!! 𝑄𝑄!!!𝑎𝑎!".
!
!!!
Proof. If we take 𝜇𝜇!= 1, 𝜆𝜆!= 𝑘𝑘 ≥ 1 for all 𝑛𝑛 ≥ 0, 𝜓𝜓 = 𝜃𝜃 and 𝐴𝐴 is a triangle matrix in Theorem 1.3, then the conditions (9) and (10) directly hold, and (11) is also reduced to
sup
!
!!
!!
! 𝜎𝜎!!−!!!!!
! 𝜎𝜎!,!!!
!<∞
!!!!
Note that the condition (26) is equivalent to the conditions (23), (24) and (25). In fact, we can write (26) as
sup!
𝜃𝜃!! !∗𝑞𝑞!𝑃𝑃! 𝑄𝑄!𝑝𝑝! 𝑎𝑎!!
!
+ Γ! <∞, (27) (26)
Bosanquet, L.S. (1950). Review on G. Sunouchi’s paper
“Notes on Fourier analysis, XVIII. Absolute summability of series with constant terms.” Tohoku Mathematical Journal, 1:
57-65, Mathematical Reviews, 11: 654.
Bosanquet, L.S. (1945). Note on convergence and summability factors. Journal of the London Mathematical Society, 20: 39-48.
Choudhary, B. & Mishra, S.K. (1993). On Köthe- Toeplitz duals of certain sequence spaces and their matrix transformations. Indian Journal of Pure and Applied Mathematics, 24: 291-301.
Das, G. (1970). A Tauberian theorem for absolute summability. Proceedings of the Cambridge Philosophical Society, 67: 321-326.
Flett, T.M. (1957). On an extension of absolute summability and some theorems of Littlewood and Paley. Proceedings of the London Mathematical Society, 7: 113-141.
Grosse-Erdmann, K.G. (1993). Matrix transformations between the sequence spaces of Maddox. Journal of Mathematical Analysis and Applications, 180: 223-238.
Hazar, G.C. & Sarıgöl, M.A. (2018). Compact and matrix operators on the space . Journal of Computational Analysis and Applications, 25: 1014-1024.
Kalaivani, K. & Youvaraj, G.P. (2013). Generalized absolute Hausdorff summability of orthogonal series. Acta Mathematica Hungarica, 140: 169-186.
Maddox, I.J. (1969). Some properties of paranormed sequence spaces. Journal of the London Mathematical Society, 1: 316-322.
Maddox, I.J. (1968). Paranormed sequence spaces generated by infinite matrices. Mathematical Proceedings of the Cambridge Philosophical Society, 64: 335-340.
Maddox, I.J. (1967). Spaces of strongly summable sequences. The Quarterly Journal of Mathematics 18: 345- 355.
Malkowsky, E. & Rakocevic, V. (2007). On matrix domains of triangles. Applied Mathematics and Computation, 189(2):
1146-1163.
Mazhar, S.M. (1971). On the absolute summability factors of infinite series. Tohoku Mathematical Journal, 23: 433-451.
McFadden, L. (1942). Absolute Nörlund summability. Duke Mathematical Journal, 9: 168-207.
Mehdi, M.R. (1960). Summability factors for generalized absolute summability I. Proceedings of the London Mathematical Society, 10(3): 180-199.
Mohapatra, R.N. & Sarıgöl, M.A. On matrix operators on the series space
. Ukrainian Mathematical Journal (69(11):1772-1783.).
Mursaleen, M. & Noman, A. K. (2011). On generalized means and some related sequence spaces. Computers Mathematics with Applications, 61: 988-999.
where
Γ!= 𝑃𝑃!
𝑝𝑝!
! 𝜎𝜎!!−𝑃𝑃!!!
𝑃𝑃! 𝜎𝜎!,!!! !
!
!!!!!
.
So it is easily seen from (27) that (23), (24) and (25) imply (26).
Conversely, if (26) is satisfied, then 𝐴𝐴: 𝑁𝑁! !→ 𝑁𝑁! is continuous linear mapping, so there exists a number 𝑀𝑀 such that
𝐴𝐴(𝑥𝑥) ≤ 𝑀𝑀 𝑥𝑥 for all 𝑥𝑥 ∈ 𝑁𝑁! !. (28) Taking any 𝜈𝜈 ≥ 0, we apply (28) with 𝑥𝑥!!!= 1, 𝑥𝑥!= 0, 𝑚𝑚 ≠ 𝜈𝜈 + 1. Hence, it can be obtained that for 𝜈𝜈 = 0,1, …,
! 𝜎𝜎!,!!! !
!!!!!
≤ 𝑀𝑀!. (29)
Therefore, it follows from (29) that (26) implies (23), (24) and (25). This result was given by Sarıgöl (2011).
Furthermore, by taking 𝜃𝜃!= 𝜓𝜓!= 𝑛𝑛, 𝜇𝜇!= 1, 𝜆𝜆!= 𝑘𝑘 > 1 and 𝐴𝐴 = 𝐼𝐼 in Theorem 1.3, we can deduce the following result according to Orhan & Sarıgöl (1993).
Corollary 4.7. Let 𝑘𝑘 ≥ 1. Then, 𝑅𝑅! ⊂ 𝑅𝑅! ! if and only if the following conditions are satisfied:
𝑖𝑖 sup! 𝑣𝑣!!∗!!!!!
!!! < ∞, 𝑖𝑖𝑖𝑖 sup
!
Ρ!𝑞𝑞!
𝑝𝑝! 𝑊𝑊!< ∞, 𝑖𝑖𝑖𝑖𝑖𝑖 sup
! 𝑄𝑄!𝑊𝑊!< ∞, where
𝑊𝑊!= 𝑛𝑛!!! 𝑞𝑞!
Q!Q!!!
∞ !
!!!!!
!!
.
ACKNOWLEDGMENTS
The authors would like to sincerely thank the referees for their valuable time and effort to improve the manuscript.
References
Altay, B. & Basar, F. (2006). On the paranormed Riesz sequence spaces of non-absolute type derived by weighted mean. Journal of Mathematical Analysis and Applications, 319: 494-508.
Boos, J. & Cass, P. (2000). Classical and modern methods in summability. Oxford University Press, New York.
Bor, H., Yu, D. & Zhou, P. (2015). Some new factor theorems for generalized absolute Cesáro summability.
Positivity 19: 111-120.
Bor, H. (1985). On 𝑁𝑁, 𝑝𝑝! ! summability factors of infinite series. Tamkang J. Math. 16(1): 13-20.
Borwein, D & Cass, F.P. (1968). Strong Nörlund summability. Mathematische Zeitschrift, 103: 94-111.
Bosanquet, L.S. & Chow, H.C. (1957). Some remarks on convergence and summability factors. Journal of the London Mathematical Society, 32: 73-82.
Bosanquet, L.S. (1950). Review on G. Sunouchi’s paper
“Notes on Fourier analysis, XVIII. Absolute summability of series with constant terms.” Tohoku Mathematical Journal, 1: 57-65, Mathematical Reviews, 11: 654.
Bosanquet, L.S. (1945). Note on convergence and summability factors. Journal of the London Mathematical Society, 20: 39-48.
Choudhary, B. & Mishra, S.K. (1993). On Köthe- Toeplitz duals of certain sequence spaces and their matrix transformations. Indian Journal of Pure and Applied Mathematics, 24: 291-301.
Das, G. (1970). A Tauberian theorem for absolute summability. Proceedings of the Cambridge Philosophical Society, 67: 321-326.
Flett, T.M. (1957). On an extension of absolute summability and some theorems of Littlewood and Paley.
Proceedings of the London Mathematical Society, 7: 113- 141.
Grosse-Erdmann, K.G. (1993). Matrix transformations between the sequence spaces of Maddox. Journal of Mathematical Analysis and Applications, 180: 223-238.
Hazar, G.C. & Sarıgöl, M.A. (2018). Compact and matrix operators on the space . Journal of Computational Analysis and Applications, 25: 1014-1024.
Kalaivani, K. & Youvaraj, G.P. (2013). Generalized absolute Hausdorff summability of orthogonal series.
Acta Mathematica Hungarica, 140: 169-186.
Maddox, I.J. (1969). Some properties of paranormed sequence spaces. Journal of the London Mathematical Society, 1: 316-322.
Maddox, I.J. (1968). Paranormed sequence spaces generated by infinite matrices. Mathematical Proceedings of the Cambridge Philosophical Society, 64: 335-340.
Maddox, I.J. (1967). Spaces of strongly summable sequences. The Quarterly Journal of Mathematics 18:
345-355.
Malkowsky, E. & Rakocevic, V. (2007). On matrix domains of triangles. Applied Mathematics and Computation, 189(2): 1146-1163.
Mazhar, S.M. (1971). On the absolute summability factors of infinite series. Tohoku Mathematical Journal, 23: 433-451.
McFadden, L. (1942). Absolute Nörlund summability.
Duke Mathematical Journal, 9: 168-207.
Mehdi, M.R. (1960). Summability factors for generalized absolute summability I. Proceedings of the London Mathematical Society, 10(3): 180-199.
Mohapatra, R.N. & Sarıgöl, M.A. On matrix operators on the series space . Ukrainian Mathematical Journal (69(11):1524-1533.).
Mursaleen, M. & Noman, A. K. (2011). On generalized means and some related sequence spaces. Computers Mathematics with Applications, 61: 988-999.
Mursaleen, M. & Noman, A. K. (2010). On some new difference sequence spaces of non-absolute type I.
Mathematical and Computer Modelling, 52: 603- 617.
Nakano, H. (1951). Modulared sequence space.
Proceedings of the Japan Academy, Ser. A Mathematical Sciences, 27: 508-512.
Orhan, C. & Sarıgöl, M.A. (1993). On absolute weighted mean summability. Rocky Mountain Journal of Mathematics, 23(3): 1091-1097.
Mursaleen, M. & Noman, A. K. (2010). On some new difference sequence spaces of non-absolute type I.
Mathematical and Computer Modelling, 52: 603- 617.
Nakano, H. (1951). Modulared sequence space. Proceedings of the Japan Academy, Ser. A Mathematical Sciences, 27:
508-512.
Orhan, C. & Sarıgöl, M.A. (1993). On absolute weighted mean summability. Rocky Mountain Journal of Mathematics, 23(3): 1091-1097.
Ozarslan, H.S. Ozgen, H.N. (2015). Necessary conditions for absolute matrix summability methods. Bollettino dell’
Unione Matematica Italiana, 8(3): 223-228.
Sarıgöl, M.A. (2016a). Spaces of series summable by absolute Cesáro and matrix operators. Communications in Mathematics and Applications, 7: 11-22.
Sarıgöl, M.A. (2016b). Norms and compactness of operators on abolute weighted mean summable series, Kuwait Journal of Science, 43(4): 68-74.
Sarıgöl, M.A. (2015). Extension of Mazhar’s theorem on summability factors, Kuwait Journal of Science, 42: 1-8.
Sarıgöl, M.A. (2013). An inequality for matrix operators and its applications. Journal of Classical Analysis, 2: 145- 150.
Sarıgöl, M.A. (2011). Matrix transformatins on fields of absolute weighted mean summability. Studia Scientiarum Mathematicarum Hungarica, 48(3): 331 341.
Sarıgöl, M.A. (2010). On local properties of factored
Fourier series. Applied Mathematics and Computation, 216:
3386-3390.
Sarıgöl, M.A. (1993). On two absolute summability Riesz summability of infinite series. Proceedings of the American Mathematical Society, 118: 485-488.
Sarıgöl, M.A. (1991a). Necessary and sufficient conditions for the equivalence of the summability methods
a n d . Indian Journal of Pure and Applied Mathematics, 22(6): 483-489.
Sarıgöl, M.A. (1991b). On absolute summability factors.
Commentationes Mathematicae, 31:157-163.
Sulaiman, W.T. (1992). On summability factors of infinite series. Proceedings of the American Mathematical Society, 115: 313-317.
Sunouchi, G. (1949). Notes on Fourier Analysis, 18, absolute summability of a series with constant terms.
Tohoku Mathematical Journal, 1:57-65.
Stieglitz, M. & Titez, H. (1977). Matrixtransformationen von Folgenraumen Eine Ergebnisüberischt. Mathematische Zeitschrift, 154: 1-16.
Tanaka, M. (1978). On generalized Nörlund methods of summability. Bulletin of the Australian Mathematical Society, 19:381-402.
Submission : 18/07/2017 Revision : 29/10/2017 Acceptance : 04/12/2017
