Transpose of Nörlund matrices on the domain of summability matrices

Abstract

Let $E=(E_{n,k})_{n,k\geq 0}$ be an invertible summability matrix with bounded absolute row sums and column sums, and let $E_{p}$ denote the domain of E in the sequence space $\ell _{p}$ $(1\le p<\infty )$. In this paper, we consider the transpose of Nörlund matrix associated with a nonnegative and nonincreasing sequence as an operator mapping $\ell _{p}$ into the sequence space $E_{p}$ and establish a general upper estimate for its operator norm, which depends on the $\ell _{1}$-norm of the rows and columns of the matrix E. In particular, we apply our result to domains of some summability matrices such as Fibonacci, Karamata, Euler, and Taylor matrices. Our result is an extension of those given by G. Talebi and M.A. Dehghan (Linear Multilinear Algebra 64(2):196–207, 2016). It also provides some analogues of the results by G. Talebi (Indag. Math. 28(3):629–636, 2017).

1 Introduction

Let ω be the space of all real- or complex-valued sequences. We denote by $\ell _{1}$ and $\ell _{p}$ the spaces of all absolutely and p-absolutely convergent series, respectively, where $1< p<\infty$.

Let $E=(E_{n,k})_{n,k\geq 0}$ be an arbitrary invertible summability matrix with bounded absolute row and column sums, and let $E_{p}$ denote the domain of E in the sequence space $\ell _{p}$, that is,

$$\begin{aligned} E_{p}:=(\ell _{p})_{E}= \bigl\{ {x= ( {{x_{n}}} )\in \omega , Ax \in \ell _{p}} \bigr\} . \end{aligned}$$

We summarize some important properties of the sequence space $E_{p}$ and refer the reader to [13] for more details. The set $E_{p}$ is a linear space with the coordinatewise addition and scalar multiplication, which is a normed space with the norm ${ \Vert x \Vert _{{E_{p}}}}: = { \Vert {{E}x} \Vert _{ {\ell _{p}}}}$; in particular, $E_{p}$ is a BK space if the matrix E is lower triangular. Moreover, the inclusion $\ell _{p}\subseteq E _{p}$ holds for $1\le p<\infty$ whenever the absolute row sums and column sums of the matrix E are bounded. In addition, the inclusion $E_{q}\subseteq E_{p}$ holds for $1\le q\le p$. Further, if the map $E: E_{p}\to \ell _{p}$ is onto, then the space $E_{p}$ is linearly isomorphic to $\ell _{p}$, and in such a case the columns of the matrix $E^{-1}$ form a Schauder basis for $E_{p}$. We also refer the reader to [14], where the α-, β-, and γ-duals of the space $E_{p}$ for $1\le p\le \infty$ are established.

Let $W=(w_{n})_{n=0}^{\infty }$ be a sequence of nonnegative real numbers with $w_{0}>0$. Set $W_{n}:=\sum_{k=0}^{n}w_{k}$, $n\geq 0$, and define the Nörlund matrix $A_{W}^{\mathrm{NM}}:=A(w_{n})=(a_{n,k})_{n,k \ge 0}$, associated with the sequence W by

$$\begin{aligned} a_{n,k}= \textstyle\begin{cases} \frac{w_{n-k}}{W_{n}} & ( 0\leq k\leq n), \\ 0& \mathrm{otherwise.} \end{cases}\displaystyle \end{aligned}$$

Since $A(w_{n})=A(cw_{n})$ for any $c>0$, we may as well assume that $w_{0}=1$.

Recently, in [13], Theorem 4.2, the author considered the Nörlund matrices as operators from $\ell _{p}$ into $E_{p}$ and established a general upper estimate for their operator norms. In this paper, we consider the same problem for the transpose of Nörlund matrices associated with the nonnegative and nonincreasing sequences as operators from $\ell _{p}$ into $E_{p}$. We obtain again a general upper estimate for their operator norms, which depend on the $\ell _{1}$-norm of the columns and the rows of the summability matrix E. In particular, we apply our results to domains of some summability matrices such as Fibonacci, Karamata, Euler, and Taylor matrices. Our result is an extension of Theorem 3.8 in [15] and provides some analogue of those given in [13].

Throughout the paper we suppose that $1< p<\infty$ and $E=(e_{n,k})_{n,k \ge 0}$ is a summability matrix whose absolute row sums and column sums are bounded.

Theorem 1.1

Suppose that $W=(w_{n})_{n=0}^{\infty }$ is a nonnegative and nonincreasing sequence of real numbers with $w_{0}=1$. Then the transpose of the associated Nörlund matrix maps $\ell _{p}$ into $E_{p}$, and we have

$$\bigl\Vert {{{ \bigl( {A_{W}^{\mathrm{NM}}} \bigr)}^{t}}} \bigr\Vert _{{\ell _{p}}, {E_{p}}} \le p \Bigl( \sup_{k \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{n,k}}} \}} _{n \in \mathbb{N}^{0}}}} \bigr\Vert }_{{\ell _{1}}}} \Bigr)^{ \frac{1}{p}}{ \Bigl( \sup_{n \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{n,k}}} \}}_{k \in \mathbb{N}^{0}}}} \bigr\Vert }_{{\ell _{1}}}} \Bigr)^{ \frac{p-1}{p} }}. $$

Proof

Let us take any $x\in \ell _{p}$. Applying Hölder’s inequality, we have

$$\begin{aligned} \bigl\Vert {{{ \bigl( {A_{W}^{\mathrm{NM}}} \bigr)}^{t}}x} \bigr\Vert _{E_{p}} ^{p} &= \sum_{k = 0}^{\infty }{{{ \Biggl\vert {\sum_{n = 0}^{\infty }{{e_{k,n}} \sum_{j = n}^{\infty }{\frac{{{w_{j - n}}}}{{{W_{j}}}} {x_{j}}} } } \Biggr\vert }^{p}}} \\ & \le \sum_{k = 0}^{\infty }{\sum _{n = 0}^{\infty }{ \vert {{e_{k,n}}} \vert {{ \Biggl\vert {\sum_{j = n}^{\infty }{ \frac{ {{w_{j - n}}}}{{{W_{j}}}}{x_{j}}} } \Biggr\vert }^{p}}} {{ \Biggl\vert {\sum_{n = 0}^{\infty }{{e_{k,n}}} } \Biggr\vert }^{p - 1}}} \\ & \le \sum_{k = 0}^{\infty }{\sum _{n = 0}^{\infty }{ \vert {{e_{k,n}}} \vert } {{ \Biggl\vert {\sum_{j = n}^{\infty }{ \frac{ {{w_{j - n}}}}{{{W_{j}}}}{x_{j}}} } \Biggr\vert }^{p}}} { \Biggl( {\sum_{n = 0}^{\infty }{ \vert {{e_{k,n}}} \vert } } \Biggr) ^{p - 1}} \\ & \le \sum_{k = 0}^{\infty }{\sum _{n = 0}^{\infty }{ \vert {{e_{k,n}}} \vert } {{ \Biggl\vert {\sum_{j = n}^{\infty }{ \frac{ {{w_{j - n}}}}{{{W_{j}}}}{x_{j}}} } \Biggr\vert }^{p}}} { \Bigl( \sup_{k \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{k,n}}} \}} _{n \in \mathbb{N}^{0}}}} \bigr\Vert }_{{\ell _{1}}}} \Bigr)^{p - 1}} \\ & = { \Bigl( {\sup_{k \in {\mathbb{N}^{0}}} {{ \bigl\Vert {{{ \{ {{e_{k,n}}} \}}_{n \in \mathbb{N}^{0}}}} \bigr\Vert } _{{\ell _{1}}}}} \Bigr)^{p - 1}}\sum_{n = 0}^{\infty }{{{ \Biggl\vert {\sum_{j = n}^{\infty }{ \frac{{{w_{j - n}}}}{{{W_{j}}}}{x_{j}}} } \Biggr\vert }^{p}}\sum _{k = 0}^{\infty }{ \vert {{e_{k,n}}} \vert } } \\ & \le { \Bigl( {\sup_{k \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{k,n}}} \}}_{n \in \mathbb{N}^{0}}}} \bigr\Vert } _{{\ell _{1}}}}} \Bigr)^{p - 1}}\sum_{n = 0}^{\infty }{{{ \Biggl\vert {\sum_{j = n}^{\infty }{ \frac{{{w_{j - n}}}}{{{W_{j}}}}{x_{j}}} } \Biggr\vert }^{p}} \Bigl( { \sup_{n \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{k,n}}} \}}_{k \in \mathbb{N}^{0}}}} \bigr\Vert }_{{\ell _{1}}}}} \Bigr)} \\ &\le \Bigl( {\sup_{n \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{k,n}}} \}}_{k \in \mathbb{N}^{0}}}} \bigr\Vert } _{{\ell _{1}}}}} \Bigr){ \Bigl( {\sup_{k \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{k,n}}} \}} _{n \in \mathbb{N}^{0}}}} \bigr\Vert }_{{\ell _{1}}}}} \Bigr)^{p - 1}} \sum_{n = 0}^{\infty }{{{ \Biggl\vert {\sum_{j = n}^{\infty }{ \frac{{{w_{j - n}}}}{{{W_{j}}}}{x_{j}}} } \Biggr\vert }^{p}}} \\ & = \Bigl( {\sup_{n \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{n,k}}} \}}_{k \in \mathbb{N}^{0}}}} \bigr\Vert } _{{\ell _{1}}}}} \Bigr){ \Bigl( {\sup_{k \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{k,n}}} \}} _{n \in \mathbb{N}^{0}}}} \bigr\Vert }_{{\ell _{1}}}}} \Bigr)^{p - 1}} \bigl\Vert {{{ \bigl( {A_{W}^{\mathrm{NM}}} \bigr)}^{t}}x} \bigr\Vert _{\ell _{p}} ^{p} \\ & \le \Bigl( {\sup_{n \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{n,k}}} \}}_{k \in \mathbb{N}^{0}}}} \bigr\Vert } _{{\ell _{1}}}}} \Bigr){ \Bigl( {\sup_{k \in \mathbb{N}^{0}} {{ \bigl\Vert {{{ \{ {{e_{k,n}}} \}} _{n \in \mathbb{N}^{0}}}} \bigr\Vert }_{{\ell _{1}}}}} \Bigr)^{p-1}} {p^{p}}\sum_{j = 0}^{\infty }{{{ \vert {{x_{j}}} \vert } ^{p}}}, \end{aligned}$$

where the last inequality is based on [5], Theorem 1. This leads us to the desired inequality and completes the proof. □

Theorem 1.1 extends [15], Theorem 3.8, from the Fibonacci sequence space to the domains of invertible summability matrices in $\ell _{p}$. To see this, consider the Fibonacci sequence space $F_{p}$ defined in [7]:

$${F_{p}} = \Biggl\{ { ( {{x_{n}}} )\in \omega : \sum _{n = 0}^{\infty }{{{ \Biggl\vert { \frac{1}{{{f_{n}}{f_{n + 1}}}} \sum_{k = 0}^{n} {f_{k}^{2}{x_{k}}} } \Biggr\vert }^{p}}} < \infty } \Biggr\} , $$

which is the domain of the Fibonacci matrix $F=(F_{n,k})_{n,k\ge 0}$ with the entries

$${F_{n,k}} = \textstyle\begin{cases} \frac{{f_{k}^{2}}}{{{f_{n}}{f_{n + 1}}}}, & 0 \le k \le n, \\ 0 & \text{otherwise}. \end{cases} $$

Here the Fibonacci numbers are the sequence of numbers $\{f_{n}\}_{n=0} ^{\infty }$ defined by the linear recurrence equations

$${f_{n}} = {f_{n - 1}} + {f_{n - 2}},\quad n \ge 1, $$

where $f_{-1}=0$ and $f_{0}=1$. For this matrix, the sums of all rows are 1, and by Lemma 2.4 of [7] its column sums are bounded. Hence, applying Theorem 1.1 to the Fibonacci sequence space $F_{p}$, we have the following result, which was previously obtained in Theorem 3.8 of [15].

Corollary 1.2

Let $W=(w_{n})_{n=0}^{\infty }$ be a nonnegative and nonincreasing sequence of real numbers with $w_{0}=1$. Then the transpose of the associated Nörlund matrix maps $\ell _{p}$ into the Fibonacci sequence space $F_{p}$, and we have

$${ \bigl\Vert {{{ \bigl( {A_{W}^{\mathrm{NM}}} \bigr)}^{t}}} \bigr\Vert _{{\ell _{p}},{ F_{p}}}} \le p{ \Biggl( {\sup_{k \in {\mathbb{N}^{0}}} \sum _{n = k}^{\infty }{\frac{ {f_{k}^{2}}}{{{f_{n}}{f_{n + 1}}}}} } \Biggr)^{\frac{1}{p}}}. $$

For further details on the normed spaces derived by the Fibonacci matrix, we refer the readers to the recent papers [4, 6, 8,9,10].

In the following, we present some additional particular cases of Theorem 1.1. First, consider the Karamata sequence space $\mathcal{K} _{p}^{{\alpha ,\beta }}$ defined by [2]

$$\mathcal{K}_{p}^{\alpha ,\beta }= \Biggl\{ (x_{n} ) \in \omega :\sum_{n = 0}^{\infty } {\Biggl\vert {\sum _{k = 0}^{\infty }{\sum_{v = 0}^{k} \begin{pmatrix} n\\ v \end{pmatrix} {{ ( {1 - \alpha - \beta } )}^{v}} {\alpha ^{n - v}} \begin{pmatrix} n + k - v - 1\\ k - v \end{pmatrix} {\beta ^{k - v}}} {x_{k}}} \Biggr\vert }^{p}< \infty \Biggr\} , $$

which is the domain of the Karamata matrix $K[\alpha ,\beta ]= (a _{n,k} )_{n,k\ge 0}$ in the sequence space $\ell _{p}$ with entries

$${a_{n,k}} = \sum_{v = 0}^{k} { \begin{pmatrix} n \\ v \end{pmatrix}{{ ( {1 - \alpha - \beta } )}^{v}} {\alpha ^{n - v}} \begin{pmatrix} n + k - v - 1 \\ k - v \end{pmatrix} {\beta ^{k - v}}}, $$

where $\alpha ,\beta \in (0,1)$. This matrix is a particular case of Sonnenschein matrices [3].

For the Karamata matrix, it is proved in [2], Theorem 1.2, that the sum of the first column is $\frac{1}{1-\alpha }$, the sums of all other columns are $\frac{1-\beta }{1-\alpha }$, and the sums of all rows are 1. Hence, applying Theorem 1.1 to the Karamata sequence space $\mathcal{K}_{p}^{{\alpha ,\beta }}$, we have the following result.

Corollary 1.3

Let $W=(w_{n})_{n=0}^{\infty }$ be a nonnegative and nonincreasing sequence of real numbers with $w_{0}=1$. Then the transpose of the associated Nörlund matrix maps $\ell _{p}$ into the Karamata sequence space $\mathcal{K}_{p}^{{\alpha ,\beta }}$, and we have

$${ \bigl\Vert {{{ \bigl( {A_{W}^{\mathrm{NM}}} \bigr)}^{t}}} \bigr\Vert _{{\ell _{p}},{\kern1pt} \mathcal{K}_{p}^{{\alpha ,\beta }}}} \le p \biggl(\frac{1}{1- \alpha } \biggr)^{\frac{1}{p}}. $$

Next, consider the Euler sequence space $e^{\alpha }_{p}$ defined by [1]

$$e_{p}^{\alpha }= \left \{ {(x_{n} )\in \omega : \sum_{n = 0} ^{\infty }{ \left \vert {\sum _{k = 0}^{n} {\begin{pmatrix} n \\ k \end{pmatrix}(1 - \alpha )^{n - k} \alpha ^{k} x_{k} } } \right \vert ^{p}< \infty } } \right \}, $$

where $\alpha \in (0,1)$ (see also [11]). Clearly, $e^{\alpha }_{p}=\mathcal{K}_{p}^{{1-\alpha ,0 }}$. Therefore we have the following estimation for the operator norm of the transpose of Nörlund matrix as operator mapping $\ell _{p}$ into $e^{\alpha } _{p}$.

Corollary 1.4

Let $W=(w_{n})_{n=0}^{\infty }$ be a nonnegative and nonincreasing sequence of real numbers with $w_{0}=1$. Then the transpose of the associated Nörlund matrix maps $\ell _{p}$ into the Euler sequence space $e^{\alpha }_{p}$, and we have

$${ \bigl\Vert {{{ \bigl( {A_{W}^{\mathrm{NM}}} \bigr)}^{t}}} \bigr\Vert _{{\ell _{p}},{\kern1pt} e^{\alpha }_{p}}} \le p \biggl(\frac{1}{\alpha } \biggr) ^{\frac{1}{p}}. $$

As the third particular case, we consider the Taylor sequence space [13]

$$t_{p}^{\theta }= \left \{ { ( {{x_{n}}} )\in \omega : \sum_{n = 0}^{\infty }{{{ \left \vert {\sum _{k = n}^{\infty }{\begin{pmatrix} n \\ k \end{pmatrix}{{ ( {1 - \theta } )}^{n + 1}} {\theta ^{k - n}} {x_{k}}} } \right \vert }^{p}} < \infty } } \right \},\quad \theta \in (0,1), $$

which is the domain of the Taylor matrix $T^{\theta }=(t_{n,k}^{ \theta })_{n,k\ge 0}$ in $\ell _{p}$ with entries [12]

$$t_{n,k}^{\theta }=\{\begin{array}{cc}0,\hfill & 0\le k<n,\hfill \\ \left(\begin{array}{c}n\\ k\end{array}\right){(1-\theta )}^{n+1}{\theta }^{k-n},\hfill & k\ge n.\hfill \end{array}$$

For this matrix, we have $\sup_{n} { \Vert {{{ \{ {t_{n,k}^{\theta }} \}}_{k \in {\mathbb{N}^{0}}}}} \Vert _{{\ell _{1}}}} = 1$ and $\sup_{k} { \Vert {{{ \{ {t_{n,k}^{\theta }} \}}_{n \in {\mathbb{N}^{0}}}}} \Vert _{{\ell _{1}}}} = ( {1 - \theta } )$. Hence Theorem 1.1 enables us to obtain the following estimation for the operator norm of the transpose of Nörlund matrix as an operator mapping $\ell _{p}$ into $t_{p}^{\theta }$.

Corollary 1.5

Let $W=(w_{n})_{n=0}^{\infty }$ be a nonnegative and nonincreasing sequence of real numbers with $w_{0}=1$. Then the transpose of the associated Nörlund matrix maps $\ell _{p}$ into the Taylor sequence space $t_{p}^{\theta }$, and we have

$${ \bigl\Vert {{{ \bigl( {A_{W}^{\mathrm{NM}}} \bigr)}^{t}}} \bigr\Vert _{{\ell _{p}}, t_{p}^{\theta }}} \le p{ ( {1 - \theta } )^{ \frac{1}{p}}}. $$

2 Conclusions

In this paper, we consider the transpose of Nörlund matrix associated with a nonnegative and nonincreasing sequence as an operator mapping $\ell _{p}$ into the sequence space $E_{p}$ and establish a general upper estimate for its operator norm, which depends on the $\ell _{1}$-norm of the rows and the columns of the matrix E, where $E=(E_{n,k})_{n,k\geq 0}$ is an invertible summability matrix with bounded absolute row sums and column sums, and $E_{p}$ denotes the domain of E in $\ell _{p}$.