Criteria of boundedness and compactness of a class of matrix operators

Abstract

The aim of this article is to obtain criteria of boundedness and compactness for a wide class of matrix operators from one weighted l _p,v space of sequences to another weighted l _q,u space, in the case 1 < p ≤ q < ∞. We introduce a general class of matrices. Then we establish necessary and sufficient conditions for the boundedness and compactness of the operators ${\left(A^{+}f\right)}_i:=\sum _{j=1}^i{a}_{i,j}{f}_j$, i ≥ 1 and ${\left(A^{-}f\right)}_j:=\sum _{i=j}^{\infty }{a}_{i,j}{f}_i$, j ≥ 1 corresponding to matrices in such classes by using the method of localization. Our classes are more general than those for which corresponding Hardy inequalities are known in the literature.

2010 Mathematics Subject Classification: 26D15; 47B37.

1 Introduction

We consider the problem of boundedness from the weighted l _p,v space into the weighted l _q,u space of the matrix operators

$${\left(A^{+}f\right)}_i:=\sum _{j=1}^i{a}_{i,j}{f}_j,i\ge 1,$$

(1)

$${\left(A^{-}f\right)}_j:=\sum _{i=j}^{\infty }{a}_{i,j}{f}_i,j\ge 1,$$

(2)

which is equivalent to the validity of the following Hardy-type inequality

$${∥A^{\pm }f∥}_{q,u}\le C{∥f∥}_{p,v},\forall f\in l_{p,v},$$

(3)

where C is a positive finite constant independent of f and (a _i,j ) is a non-negative triangular matrix with entries a _i,j ≥ 0 for i ≥ j ≥ 1 and a _i,j = 0 for i < j.

Here and further 1 < p, q < ∞, $\frac{1}{p}+\frac{1}{p^{\prime }}=1$ and $u={\left\{u_i\right\}}_{i=1}^{\infty },v={\left\{v_i\right\}}_{i=1}^{\infty }$ are positive sequences of real numbers. l _p,v is the space of sequences $f={\left\{f_i\right\}}_{i=1}^{\infty }$ of real numbers such that

$$\left|\right|f|{|}_{p,v}:={\left(\sum _{i=1}^{\infty }|v_i{f}_i{|}^p\right)}^{\frac{1}{p}}<\infty ,1<p<\infty .$$

For a _i,j = 1, i ≥ j ≥ 1, the operators (1), (2) coincide with the discrete Hardy operators of the forms ${\left(A_0^{+}f\right)}_i:=\sum _{j=1}^i{f}_j,{\left(A_0^{-}f\right)}_j:=\sum _{i=j}^{\infty }{f}_i$, respectively. References about generalizations of the original forms of the discrete and continuous Hardy inequalities can be found in different books, see e.g., [1–3].

In [4, 5], necessary and sufficient conditions for the validity of (3) have been obtained for 1 < p, q < ∞ under the assumption that there exists d ≥ 1 such that the inequalities

$$\frac{1}{d}\left(a_{i,k}+a_{k,j}\right)\le a_{i,j}\le d\left(a_{i,k}+a_{k,j}\right),i\ge k\ge j\ge 1$$

(4)

hold.

A sequence ${\left\{a_i\right\}}_{i=1}^{\infty }$ is called almost non-decreasing (non-increasing), if there exists c > 0 such that ca _i ≥ a _k (a _k ≤ ca _j ) for all i ≥ k ≥ j ≥ 1.

In [6], estimate (3) has been studied under the assumption that there exist d ≥ 1 and a sequence of positive numbers ${\left\{{\omega }_k\right\}}_{k=1}^{\infty }$, and a non-negative matrix (b _i,j ), where b _i,j is almost non-decreasing in i and almost non-increasing in j, such that the inequalities

$$\frac{1}{d}\left(b_{i,k}{\omega }_j+a_{k,j}\right)\le a_{i,j}\le d\left(b_{i,k}{\omega }_j+a_{k,j}\right)$$

(5)

hold for all i ≥ k ≥ j ≥ 1.

In [7, 8], inequality (3) has been considered under the assumption that there exist d ≥ 1, a sequence of positive numbers ${\left\{{\omega }_k\right\}}_{k=1}^{\infty }$, and a non-negative matrix (b _i,j ), whose entries b _i,j are almost non-decreasing in i and almost non-increasing in j such that the inequalities

$$\frac{1}{d}\left(a_{i,k}+b_{k,j}{\omega }_i\right)\le a_{i,j}\le d\left(a_{i,k}+b_{k,j}{\omega }_i\right)$$

(6)

hold for all i ≥ k ≥ j ≥ 1.

Conditions (5) and (6) include conditions (4), and complement each other.

In this article, we introduce a general class of matrices. We establish necessary and sufficient conditions for the boundedness and compactness of the operators (1) and (2), where the corresponding matrices belong to such classes. Such classes of matrices are wider than those which have been previously studied in the theory of discrete Hardy-type inequalities.

The content of the article is as follows. In Section 2, we introduce our classes of matrices and their properties. Moreover, in this section we give some auxiliary statements. Section 3 contains the main results. In Section 4, we prove the theorems, which give criteria of boundedness of the operators defined by (1) and (2). In Section 5, we obtain compactness criteria for the operators defined by (1) and (2). Then based on these statements, we prove our main theorems in Section 6. Moreover, in this section we show that one can imply our main results in order to obtain necessary and sufficient conditions for boundedness and compactness of the composition of operators.

Notation: If M and K are real valued functionals of sequences, then we understand that the symbol M ≪ K means that there exists c > 0 such that M ≤ cK, where c is a constant which may depend only on parameters such as p, q, r _n and h _n . If M ≪ K ≪ M, then we write M ≈ K.

2 Preliminaries and notation

For n ≥ 1, we introduce the classes ${\mathcal{O}}_n^{+}$ and ${\mathcal{O}}_n^{-}$ of matrices (a _i,j ). We assume that $a_{i,j}\equiv a_{i,j}^{\left(n\right)}$ if $\left(a_{i,j}\right)\in {\mathcal{O}}_n^{+}$ or $\left(a_{i,j}\right)\in {\mathcal{O}}_n^{-}$.

We define the classes ${\mathcal{O}}_n^{+}$, n ≥ 0 by induction. Let (a _i,j ) be a matrix which is non-negative and non-decreasing in the first index for all i ≥ j ≥ 1. By definition matrices of the type $a_{i,j}^{\left(0\right)}={\alpha }_j$, ∀i. ≥ j ≥ 1 belong to the class ${\mathcal{O}}_0^{+}$. Let the classes ${\mathcal{O}}_{\gamma }^{+}$, γ = 0, 1,..., n - 1, n ≥ 1 be defined. By definition, the matrix $\left(a_{i,j}\right)\equiv \left(a_{i,j}^{\left(n\right)}\right)$ belongs to the class ${\mathcal{O}}_n^{+}$ if and only if there exist matrices $\left(a_{i,j}^{\left(\gamma \right)}\right)\in {\mathcal{O}}_{\gamma }^{+}$, γ = 0, 1,..., n - 1 and a number r _n > 0 such that

$$a_{i,j}^{\left(n\right)}\le r_n\sum _{\gamma =0}^n{b}_{i,k}^{n,\gamma }{a}_{k,j}^{\left(\gamma \right)}$$

(7)

for all i ≥ k ≥ j ≥ 1, where $b_{i,k}^{n,n}\equiv 1$and

$$b_{i,k}^{n,\gamma }=\underset{1\le j\le k}{\text{inf}}\frac{a_{i,j}^{\left(n\right)}}{a_{k,j}^{\left(\gamma \right)}},\gamma =0,1,\dots ,n-1.$$

(8)

From (8) it follows that entries of the matrices $\left(b_{i,k}^{n,\gamma }\right)$ do not decrease in the first index and do not increase in the second index. And (8) provides the validity of the following inequality

$$a_{i,j}^{\left(n\right)}\ge b_{i,k}^{n,\gamma }{a}_{k,j}^{\left(\gamma \right)}$$

(9)

$$i\ge k\ge j\ge 1,\gamma =0,1,\dots ,n,n=0,1,\dots$$

Then for $\left(a_{i,j}^{\left(n\right)}\right)\in {\mathcal{O}}_n^{+}$ we have

$$a_{i,j}^{\left(n\right)}\approx \sum _{\gamma =0}^n{b}_{i,k}^{n,\gamma }{a}_{k,j}^{\left(\gamma \right)},n\ge 0$$

(10)

for all i ≥ k ≥ j ≥ 1.

REMARK 1. It is easy to show that if for the matrix$\left(a_{i,j}^{\left(n\right)}\right)$, n ≥ 0 there exist matrices$\left(a_{i,j}^{\left(\gamma \right)}\right)\in {\mathcal{O}}_{\gamma }^{+}$, γ = 0, 1,..., n - 1, and matrices$\left({\tilde{b}}_{i,k}^{n,\gamma }\right)$, γ = 0, 1,..., n such that the equivalence (10) is valid for all i ≥ k ≥ j ≥ 1, then$\left(a_{i,j}^{\left(n\right)}\right)\in {\mathcal{O}}_n^{+}$and${\tilde{b}}_{i,k}^{n,\gamma }\approx b_{i,k}^{n,\gamma }$. Hence we may assume that the matrices$\left(b_{i,k}^{n,\gamma }\right)$are arbitrary non-negative matrices which satisfy (10).

For the proof of our main results we also need the following inequality. Let n ≥ l ≥ γ. Then we have

$$b_{i,k}^{n,\gamma }\ge b_{i,s}^{n,l}\cdot b_{s,k}^{l,\gamma }\forall i\ge s\ge k\ge 1.$$

(11)

Indeed, using (9), for i ≥ s ≥ k ≥ 1, n ≥ l ≥ γ we obtain

$$b_{i,k}^{n,\gamma }=\underset{1\le j\le k}{\text{inf}}\frac{a_{i,j}^{\left(n\right)}}{a_{k,j}^{\left(\gamma \right)}}\ge b_{i,s}^{n,l}\cdot \underset{1\le j\le k}{\text{inf}}\frac{a_{s,j}^{\left(l\right)}}{a_{k,j}^{\left(\gamma \right)}}=b_{i,s}^{n,l}\cdot b_{s,k}^{l,\gamma }.$$

As above, we introduce the classes ${\mathcal{O}}_m^{-}$, m ≥ 0. Let (a _i,j ) be a matrix which is non-negative and non-increasing in the second index for all i ≥ j ≥ 1. By definition a matrix $\left(a_{i,j}\right)=\left(a_{i,j}^{\left(0\right)}\right)$ belongs to the class ${\mathcal{O}}_0^{-}$ if and only if it has the form $a_{i,j}^{\left(0\right)}={\beta }_i$ for all i ≥ j ≥ 1. Let the classes ${\mathcal{O}}_{\gamma }^{-}$, γ = 0, 1,..., m - 1, m ≥ 1 be defined. A matrix $\left(a_{i,j}\right)=\left(a_{i,j}^{\left(m\right)}\right)$ belongs to the class ${\mathcal{O}}_m^{-}$ if and only if there exist matrices $\left(a_{i,j}^{\left(\gamma \right)}\right)\in {\mathcal{O}}_{\gamma }^{-}$, γ = 0, 1,..., m - 1 and a number h _m > 0 such that

$$a_{i,j}^{\left(m\right)}\le h_m\sum _{\gamma =0}^m{a}_{i,k}^{\left(\gamma \right)}{d}_{k,j}^{\gamma ,m},$$

(12)

for all i ≥ k ≥ j ≥ 1, where $d_{k,j}^{m,m}\equiv 1$ and

$$d_{k,j}^{\gamma ,m}=\underset{k\le i\le \infty }{\text{inf}}\frac{a_{i,j}^{\left(m\right)}}{a_{i,k}^{\left(\gamma \right)}},\gamma =0,1,\dots ,m-1.$$

(13)

From the definition of the matrix $\left(d_{k,j}^{\gamma ,m}\right)$, γ = 0, 1,..., m - 1, m = 0, 1,..., it is obvious that the entries of the matrix $\left(d_{k,j}^{\gamma ,m}\right)$ do not decrease in the first index and do not increase in the second index and for m ≥ l ≥ γ, k ≥ s ≥ j satisfy the following inequality

$$d_{k,j}^{\gamma ,m}\ge d_{k,s}^{\gamma ,l}\cdot d_{s,j}^{l,m}.$$

(14)

From (13) it follows that for all i ≥ k ≥ j ≥ 1

$$a_{i,j}^{\left(m\right)}\ge a_{i,k}^{\left(\gamma \right)}{d}_{k,j}^{\gamma ,m},\gamma =0,1,\dots ,m-1.$$

(15)

As in (10) every class ${\mathcal{O}}_m^{-}$, m ≥ 0 of matrices $\left(a_{i,j}^{\left(m\right)}\right)$ is characterized by the following relation

$$a_{i,j}^{\left(m\right)}\approx \sum _{\gamma =0}^m{a}_{i,k}^{\left(\gamma \right)}{d}_{k,j}^{\gamma ,m},$$

(16)

for all i ≥ k ≥ j, where $d_{k,j}^{\gamma ,m}$, γ = 0, 1,..., m are defined by the formula (13).

REMARK 2. As mentioned before we may assume that the matrices$\left(d_{k,j}^{\gamma ,m}\right)$, γ = 0, 1,..., m, m ≥ 0 are arbitrary non-negative matrices which satisfy (16).

REMARK 3. By the definitions of the classes${\mathcal{O}}_n^{\pm }$, n ≥ 0 we have${\mathcal{O}}_0^{\pm }\subset {\mathcal{O}}_1^{\pm }\subset \cdots \subset {\mathcal{O}}_n^{\pm }\subset \cdots$

In particular, the matrices of the classes ${\mathcal{O}}_1^{+}$ and ${\mathcal{O}}_1^{-}$ are characterized by the following relations, respectively,

$$\begin{array}{l}{a}_{i,j}^{\left(1\right)}\approx b_{i,k}^{1,0}{a}_{k,j}^{\left(0\right)}+a_{k,j}^{\left(1\right)}\forall i\ge k\ge j\ge 1,\\ a_{i,j}^{\left(1\right)}\approx a_{i,k}^{\left(1\right)}+a_{i,k}^{\left(0\right)}{d}_{k,j}^{0,1}\forall i\ge k\ge j\ge 1.\end{array}$$

It is easy to see that the class ${\mathcal{O}}_1^{+}$ include the matrices, whose entries satisfy conditions (4) and (5). Also it should be noted that the matrices with conditions (4) and (6) belong to the class ${\mathcal{O}}_1^{-}$. This implies that the classes ${\mathcal{O}}_n^{+}$, n ≥ 1 and ${\mathcal{O}}_m^{-}$, m ≥ 1 are wider than the classes of matrices which have been used in this connection before.

The matrices of the classes ${\mathcal{O}}_2^{+}$and ${\mathcal{O}}_2^{-}$are described by the following relations, respectively,

$$\begin{array}{l}{a}_{i,j}^{\left(2\right)}\approx b_{i,k}^{2,0}{a}_{k,j}^{\left(0\right)}+b_{i,k}^{2,1}{a}_{k,j}^{\left(1\right)}+a_{k,j}^{\left(2\right)}\forall i\ge k\ge j\ge 1,\\ a_{i,j}^{\left(2\right)}\approx a_{i,k}^{\left(2\right)}+a_{i,k}^{\left(1\right)}{d}_{k,j}^{1,2}+a_{i,k}^{\left(0\right)}{d}_{k,j}^{0,2}\forall i\ge k\ge j\ge 1.\end{array}$$

Next, we show properties of the classes of matrices ${\mathcal{O}}_n^{+}$ and ${\mathcal{O}}_n^{-}$, n ≥ 0.

We set

$$w_{i,k}=\sum _{j=k}^i{a}_{i,j}{\sigma }_{j,k}.$$

Then we have the following

Lemma 2.1. Let$\left(a_{i,j}\right)\in {\mathcal{O}}_n^{+},\left({\sigma }_{j,k}\right)\in {\mathcal{O}}_m^{+}$. Then$\left(w_{i,k}\right)\in {\mathcal{O}}_{m+n+1}^{+}$.

PROOF OF LEMMA 2.1. Since $\left(a_{i,j}\right)\in {\mathcal{O}}_n^{+}$, there exist matrices $\left(a_{i,j}^{\left(\gamma \right)}\right)\in {\mathcal{O}}_{\gamma }^{+}$, γ = 0, 1,..., n-1, and matrices $\left({\delta }_{i,l}^{n,\gamma }\right)$ such that

$$a_{i,j}\equiv a_{i,j}^{\left(n\right)}\approx \sum _{\gamma =0}^n{\delta }_{i,l}^{n,\gamma }{a}_{l,k}^{\left(\gamma \right)},n=0,1,\dots ,{\delta }_{i,l}^{n,n}\equiv 1$$

for all i ≥ l ≥ k ≥ 1.

Since $\left({\sigma }_{j,k}\right)\in {\mathcal{O}}_m^{+}$, there exist matrices $\left({\sigma }_{j,k}^{\left(\mu \right)}\right)\in {\mathcal{O}}_{\mu }^{+}$, μ = 0, 1,..., m - 1, and matrices $\left(e_{k,l}^{m,\mu }\right)$ such that

$${\sigma }_{j,k}\equiv {\sigma }_{j,k}^{\left(m\right)}\approx \sum _{\mu =0}^m{e}_{j,l}^{m,\mu }{\sigma }_{l,k}^{\left(\mu \right)},m=0,1,...,e_{j,l}^{m,m}\equiv 1$$

for all j ≥ l ≥ k ≥ 1.

We set

$$w_{i,k}\equiv w_{i,k}^{n,m}=\sum _{j=k}^i{a}_{i,j}^{\left(n\right)}{\sigma }_{j,k}^{\left(m\right)}.$$

First, we consider the case when m ≥ 0, n = 0. In this case $a_{i,j}^{\left(0\right)}={\alpha }_j$, ∀i ≥ j ≥ 1. For ∀i ≥ l ≥ k we obtain

$$w_{i,k}^{o,m}=\sum _{j=k}^i{\alpha }_j{\sigma }_{j,k}^{\left(m\right)}\approx \sum _{j=k}^l{\alpha }_j{\sigma }_{j,k}^{\left(m\right)}+\sum _{j=l}^i{\alpha }_j{\sigma }_{j,k}^{\left(m\right)}\approx w_{l,k}^{o,m}+\sum _{\mu =0}^m{\sigma }_{l,k}^{\left(\mu \right)}\sum _{j=l}^i{\alpha }_j{e}_{j,l}^{m,\mu }=w_{l,k}^{o,m}+\sum _{\mu =0}^m{ẽ}_{i,l}^{m+1,\mu }{\sigma }_{l,k}^{\left(\mu \right)},$$

where ${ẽ}_{i,l}^{m+1,\mu }=\sum _{j=l}^i{\alpha }_j{e}_{j,l}^{m,\mu }$, μ = 0, 1,..., m. Suppose that ${ẽ}_{i,l}^{m+1,m+1}\equiv 1$. Since $\left({\sigma }_{l,k}^{\left(\mu \right)}\right)\in {\mathcal{O}}_{\mu }^{+}$, μ = 0, 1,..., m, by definition we easily see that $w_{i,k}^{o,m}\in {\mathcal{O}}_{m+1}^{+}$. By induction, we assume that for n = 0, 1,..., r - 1, $r\ge 1\left(w_{i,k}^{n,m}\right)$ belongs to the class ${\mathcal{O}}_{n+m+1}^{+}$. For i ≥ l ≥ k we have

$$\begin{array}{ll}\hfill w_{i,k}^{r,m}& =\sum _{j=k}^i{a}_{i,j}^{\left(r\right)}{\sigma }_{j,k}^{\left(m\right)}\approx \sum _{j=k}^l{a}_{i,j}^{\left(r\right)}{\sigma }_{j,k}^{\left(m\right)}+\sum _{j=l}^i{a}_{i,j}^{\left(r\right)}{\sigma }_{j,k}^{\left(m\right)}\\ \approx \sum _{j=k}^l\left(\sum _{\gamma =0}^r{\delta }_{i,l}^{r,\gamma }{a}_{l,j}^{\left(\gamma \right)}\right){\sigma }_{j,k}^{\left(m\right)}+\sum _{j=l}^i{a}_{i,j}^{\left(r\right)}\left(\sum _{\mu =0}^m{e}_{j,l}^{m,\mu }{\sigma }_{l,k}^{\left(\mu \right)}\right)\\ =\sum _{\gamma =0}^r{\delta }_{i,l}^{r,\gamma }\sum _{j=k}^l{a}_{l,j}^{\left(\gamma \right)}{\sigma }_{j,k}^{\left(m\right)}+\sum _{\mu =0}^m{\sigma }_{l,k}^{\left(\mu \right)}\sum _{j=l}^i{a}_{i,j}^{\left(r\right)}{e}_{j,l}^{m,\mu }\\ =\sum _{j=k}^l{a}_{l,j}^{\left(r\right)}{\sigma }_{j,k}^{\left(m\right)}+\sum _{\gamma =0}^{r-1}{\delta }_{i,l}^{r,\gamma }\sum _{j=k}^l{a}_{l,j}^{\left(\gamma \right)}{\sigma }_{j,k}^{\left(m\right)}+\sum _{\mu =0}^m{\sigma }_{l,k}^{\left(\mu \right)}\sum _{j=l}^i{a}_{i,j}^{\left(r\right)}{e}_{j,l}^{m,\mu }\\ =w_{l,k}^{r,m}+\sum _{\gamma =0}^{r-1}{\delta }_{i,l}^{r,\gamma }{\tilde{\sigma }}_{l,k}^{\left(\gamma +m+1\right)}+\sum _{\mu =0}^m{ẽ}_{i,l}^{m+1,\mu }{\sigma }_{l,k}^{\left(\mu \right)},\end{array}$$

where ${\tilde{\sigma }}_{l,k}^{\left(\gamma +m+1\right)}\equiv \sum _{j=k}^l{a}_{l,j}^{\left(\gamma \right)}{\sigma }_{j,k}^{\left(m\right)}$, γ = 0,..., r - 1 and ${ẽ}_{i,l}^{m+1,\mu }\equiv \sum _{j=l}^i{a}_{i,j}^{\left(r\right)}{e}_{j,l}^{m,\mu }$μ = 0,..., m. We denote γ + m + 1 by μ. Then we have

$$\begin{array}{ll}\hfill w_{i,k}^{r,m}& \approx w_{l,k}^{r,m}+\sum _{\mu =m+1}^{r+m}{\delta }_{i,l}^{r,\mu -m-1}{\tilde{\sigma }}_{l,k}^{\left(\mu \right)}+\sum _{\mu =0}^m{ẽ}_{i,l}^{m+1,\mu }{\sigma }_{l,k}^{\left(\mu \right)}\\ =w_{l,k}^{r,m}+\sum _{\mu =0}^{r+m}{\tilde{\delta }}_{i,l}^{r+m,\mu }{\tilde{\sigma }}_{l,k}^{\left(\mu \right)},\end{array}$$

where

$${\tilde{\delta }}_{i,l}^{r+m,\mu }=\left\{\begin{array}{c}{ẽ}_{i,l}^{m+1,\mu },0\le \mu \le m,\\ {\delta }_{i,l}^{r,\mu -m-1},m+1\le \mu \le r+m,\end{array}\right.$$

and

$${\tilde{\sigma }}_{l,k}^{\left(\mu \right)}=\left\{\begin{array}{c}{\sigma }_{l,k}^{\left(\mu \right)},0\le \mu \le m,\\ {\tilde{\sigma }}_{l,k}^{\left(\mu \right)},m+1\le \mu \le r+m.\end{array}\right.$$

Since ${\tilde{\sigma }}_{l,k}^{\left(\mu \right)}\in {\mathcal{O}}_{\mu }^{+}$, μ = 0, 1,..., r + m we obtain that $w_{i,k}^{r,m}\in {\mathcal{O}}_{r+m+1}^{+}$. The proof is complete.

Now we set

$${\phi }_{k,j}=\sum _{i=j}^k{\sigma }_{k,i}{a}_{i,j}.$$

Then we have the following lemma.

Lemma 2.2. Let$\left(a_{i,j}\right)\in {\mathcal{O}}_n^{-}$, $\left({\sigma }_{k,i}\right)\in {\mathcal{O}}_m^{-}$. Then$\left({\phi }_{k,j}\right)\in {\mathcal{O}}_{m+n+1}^{-}$.

Lemma 2.2 can be proved in the same way as Lemma 2.1.

For the proof of our main theorem we will need the following well-known result for the discrete weighted Hardy inequality (see [1, 9]) and the criteria of precompactness of sets in l _p (see [10, p. 32]).

Theorem A. Let 1 < p ≤ q < ∞. Then the inequality

$${\left(\sum _{j=1}^{\infty }{\left(\sum _{i=j}^{\infty }{\omega }_i{f}_i\right)}^q{u}_j^q\right)}^{\frac{1}{q}}\le C{\left(\sum _{i=1}^{\infty }|v_i{f}_i{|}^p\right)}^{\frac{1}{p}},0\le f\in l_{p,v}$$

(17)

holds if and only if

$$H:=\underset{n\ge 1}{\text{sup}}{\left(\sum _{j=1}^n{u}_j^q\right)}^{\frac{1}{q}}{\left(\sum _{i=n}^{\infty }{\omega }_i^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}<\infty .$$

Moreover, H ≈ C, where C is the best constant in (17).

Theorem B. Let T be a set in l _p , 1 ≤ p < ∞. The set T is compact if and only if T is bounded and for all ε > 0 there exists N = N(ε) such that for all$x={\left\{x_i\right\}}_{i=1}^{\infty }\in T$the inequality

$$\sum _{i=N}^{\infty }|x_i{|}^p<\epsilon$$

holds.

3 Main results

We define

$$\begin{array}{c}{\left({\mathcal{B}}_{p,q}^{+}\right)}_k={\left(\sum _{j=1}^k{v}_j^{-p^{\prime }}{\left(\sum _{i=k}^{\infty }{a}_{i,j}^q{u}_i^q\right)}^{\frac{p^{\prime }}{q}}\right)}^{\frac{1}{p^{\prime }}},\\ {\left({\mathcal{B}}_{p,q}^{-}\right)}_k={\left(\sum _{i=k}^{\infty }{u}_i^q{\left(\sum _{j=1}^k{a}_{i,j}^{p^{\prime }}{v}_j^{-p^{\prime }}\right)}^{\frac{q}{p^{\prime }}}\right)}^{\frac{1}{q}},\\ {\left({\mathcal{A}}_{p,q}^{+}\right)}_k={\left(\sum _{j=1}^k{u}_j^q{\left(\sum _{i=k}^{\infty }{a}_{i,j}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{q}{p^{\prime }}}\right)}^{\frac{1}{q}},\\ {\left({\mathcal{A}}_{p,q}^{-}\right)}_k={\left(\sum _{i=k}^{\infty }{v}_i^{-p^{\prime }}{\left(\sum _{j=1}^k{a}_{i,j}^q{u}_j^q\right)}^{\frac{p^{\prime }}{q}}\right)}^{\frac{1}{p^{\prime }}}.\end{array}$$

We set ${\mathcal{B}}^{+}=\underset{k\ge 1}{\text{sup}}{\left({\mathcal{B}}_{p,q}^{+}\right)}_k,{\mathcal{B}}^{-}=\underset{k\ge 1}{\text{sup}}{\left({\mathcal{B}}_{p,q}^{-}\right)}_k,{\mathcal{A}}^{+}=\underset{k\ge 1}{\text{sup}}{\left({\mathcal{A}}_{p,q}^{+}\right)}_k$and ${\mathcal{A}}^{-}=\underset{k\ge 1}{\text{sup}}{\left({\mathcal{A}}_{p,q}^{-}\right)}_k.$

Theorem 3.1. Suppose that 1 < p ≤ q < ∞. Let the matrix (a _i,j ) in (1) belong to the class${\mathcal{O}}_m^{+}\cup {\mathcal{O}}_m^{-}$, m ≥ 0. Let A⁺be the operator defined in (1). Then the following statements hold:

(i) A⁺ is bounded from l _p,v into l _q,u if and only if one of the conditions${\mathcal{B}}^{+}<\infty$and${\mathcal{B}}^{-}<\infty$holds. Moreover${\mathcal{B}}^{+}\approx {\mathcal{B}}^{-}\approx C,$where C is the best constant in (3).

( ii ) A⁺is compact from l _p,v into l _q,u if and only if one of the conditions$\underset{k\to \infty }{\text{lim}}{\left({\mathcal{B}}_{p,q}^{+}\right)}_k=0\mathsf{\text{and}}\underset{k\to \infty }{\text{lim}}{\left({\mathcal{B}}_{p,q}^{-}\right)}_k=0$holds.

Theorem 3.2. Suppose that 1 < p ≤ q < ∞. Let the matrix (a _i,j ) in (2) belong to the class${\mathcal{O}}_m^{+}\cup {\mathcal{O}}_m^{-},m\ge 0.$Let A^- be the operator defined in (2). Then the following statements hold:

(j) A^- is bounded from l _p,v into l _q,u if and only if one of the conditions${\mathcal{A}}^{+}<\infty$and${\mathcal{A}}^{-}<\infty$holds. Moreover${\mathcal{A}}^{+}\approx {\mathcal{A}}^{-}\approx C,$where C is the best constant in (3).

(jj) A^- is compact from l _p,v into l _q,u if and only if one of the conditions$\underset{k\to \infty }{\text{lim}}{\left({\mathcal{A}}_{p,q}^{+}\right)}_k=0$and$\underset{k\to \infty }{\text{lim}}{\left({\mathcal{A}}_{p,q}^{-}\right)}_k=0$holds.

Before proving our main theorems we establish the boundedness and compactness of the operators ${\left(A^{+}f\right)}_i:=\sum _{j=1}^i{a}_{i,j}{f}_j,i\ge 1$and ${\left(A^{-}f\right)}_j:=\sum _{i=j}^{\infty }{a}_{i,j}{f}_i,j\ge 1$ from the weighted l _p,v space into the weighted l _q,u space in particular cases.

4 Boundedness of the matrix operators

Theorem 4.1. Let 1 < p ≤ q < ∞. Let the matrix (a _i,j ) in (1) belong to the class${\mathcal{O}}_n^{+},n\ge 0.$Then the estimate (3) for the operator defined by (1) holds if and only if one of the conditions${\mathcal{B}}^{+}<\infty$and${\mathcal{B}}^{-}<\infty$holds. Moreover${\mathcal{B}}^{+}\approx {\mathcal{B}}^{-}\approx C,$where C is the best constant in (3).

Theorem 4.2. Let 1 < p ≤ q < ∞. Let the matrix (a _i,j ) in (2) belong to the class${\mathcal{O}}_m^{-},m\ge 0.$Then the estimate (3) for the operator defined by (2) holds if and only if one of the conditions${\mathcal{A}}^{+}<\infty$and${\mathcal{A}}^{-}<\infty$holds. Moreover${\mathcal{A}}^{+}\approx {\mathcal{A}}^{-}\approx C,$where C is the best constant in (3).

Here we present only the proof of Theorem 4.2, since the proof of Theorem 4.1 is very similar.

For the proof of Theorem 4.2 we need the following.

Lemma 4.1. Let the matrix of (2) belongs to the class${\mathcal{O}}_m^{-},m\ge 0.$Then we have the following equivalence

$${\left({\mathcal{A}}_{p,q}^{+}\right)}_k\approx {\left({\mathcal{A}}_m\right)}_k\equiv \underset{0\le \gamma \le m}{\text{max}}{\left({\mathcal{A}}_{\gamma ,m}\right)}_k\approx {\left({\mathcal{A}}_{p,q}^{-}\right)}_k,$$

(18)

where

$${\left({\mathcal{A}}_{\gamma ,m}\right)}_k={\left(\sum _{j=1}^k{\left(d_{k,j}^{\gamma ,m}\right)}^q{u}_j^q\right)}^{\frac{1}{q}}{\left(\sum _{i=k}^{\infty }{\left(a_{i,k}^{\left(\gamma \right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}.$$

By (18) it follows that

$${\mathcal{A}}^{+}\approx {\mathcal{A}}_m=\underset{k\ge 1}{\text{sup}}{\left({\mathcal{A}}_m\right)}_k\approx {\mathcal{A}}^{-},\forall m\ge 0.$$

(19)

Indeed, this equivalence easily follows from (16).

PROOF OF THEOREM 4.2. Necessity. Suppose that the matrix of the operator (2) belongs to the class ${\mathcal{O}}_m^{-},m\ge 0$and (3) holds.

For k > 1 we assume that $g={\left\{g_i\right\}}_{i=1}^{\infty }:g_i=\left\{\begin{array}{c}{u}_i,1\le i\le k\\ 0,i>k.\end{array}\right.$

It is known that inequality (3) holds if and only if the following dual inequality

$$\left|\right|A^{*}g|{|}_{p^{\prime },v^{-1}}\le C|\left|g\right|{|}_{q^{\prime },u^{-1}},g\in l_{q^{\prime },u^{-1}}$$

(20)

holds for the conjugate operator A*, which coincides with operator defined by (1). Moreover, the best constants in (3) and (20) coincide (see e.g., [3]).

Hence applying $g={\left\{g_i\right\}}_{i=1}^{\infty }$in (20) and using (15) we obtain

$$\begin{array}{ll}\hfill C{k}^{\frac{1}{q^{\prime }}}& \ge \left|\right|A^{*}g|{|}_{p^{\prime },v^{-1}}\ge {\left(\sum _{i=k}^{\infty }{\left(\sum _{j=1}^k{a}_{i,j}^{\left(m\right)}{u}_j\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}\\ \gg {\left(\sum _{i=k}^{\infty }{\left(a_{i,k}^{\left(\gamma \right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}\left(\sum _{j=1}^k{d}_{k,j}^{\gamma ,m}{u}_j\right),\gamma =0,1,\dots ,m.\end{array}$$

Therefore ${\left\{a_{i,k}^{\left(\gamma \right)}\right\}}_{i=1}^{\infty }\in l_{p^{\prime },v^{-1}}.$

Now for 1≤ r < M < ∞, we assume that $f={\left\{f_s\right\}}_{s=1}^{\infty },$where

$$\begin{array}{c}{f}_s=\left\{\begin{array}{cc}\hfill {\left(a_{s,r}^{\left(\gamma \right)}\right)}^{p^{\prime }-1}{v}_s^{-p^{\prime }},\hfill & \hfill r\le s\le M\hfill \\ \hfill 0,\hfill & \hfill s<r\mathsf{\text{or}}s>M.\hfill \end{array}\right.\end{array}$$

Applying f to inequality (3) and using (15) we find that

$$\begin{array}{ll}\hfill C{\left(\sum _{s=r}^M{\left(a_{s,r}^{\left(\gamma \right)}\right)}^{p^{\prime }}{v}_s^{-p^{\prime }}\right)}^{\frac{1}{p}}& \ge \left|\right|A^{-}f|{|}_{q,u}={\left(\sum _{j=1}^{\infty }{\left(\sum _{s=j}^{\infty }{a}_{s,j}^{\left(m\right)}{f}_s\right)}^q{u}_j^q\right)}^{\frac{1}{q}}\\ \gg {\left(\sum _{j=1}^r{\left(d_{r,j}^{\gamma ,m}\right)}^q{u}_j^q\right)}^{\frac{1}{q}}\left(\sum _{s=r}^M{\left(a_{s,r}^{\left(\gamma \right)}\right)}^{p^{\prime }}{v}_s^{-p^{\prime }}\right),\end{array}$$

which implies

$$C\gg {\left(\sum _{j=1}^r{\left(d_{r,j}^{\gamma ,m}\right)}^q{u}_j^q\right)}^{\frac{1}{q}}{\left(\sum _{s=r}^M{\left(a_{s,r}^{\left(\gamma \right)}\right)}^{p^{\prime }}{v}_s^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}.$$

(21)

Since inequality (21) holds for all γ = 0,1,..., m and r ≥ 1 is arbitrary, passing to the limit as M → ∞ we have

$$\underset{k\ge 1}{\text{sup}}{\left({\mathcal{A}}_m\right)}_k\ll C.$$

(22)

By using Lemma 4.1 we obtain

$${\mathcal{A}}^{+}\approx {\mathcal{A}}^{-}\ll C.$$

(23)

The proof of the necessity is complete.

Sufficiency. Let the matrix (a _i,j ) of the operator (2) belong to the class ${\mathcal{O}}_m^{-},m\ge 0.$ Let $0\le f\in l_{p,v}$and at least one of the conditions ${\mathcal{A}}^{+}<\infty$and ${\mathcal{A}}^{-}<\infty$hold. Assume that m = 0. By the definition of ${\mathcal{O}}_0^{-},$the matrix of the operator (2) has the form $a_{i,j}^{\left(0\right)}={\beta }_i\forall i\ge j\ge 1.$ Then the estimate (3) coincides with the estimate (17) and the operator (2) is the matrix operator $A_0^{-}.$ Hence from Theorem A it follows that

$$\left|\right|A_0^{-}f|{|}_{q,u}\ll {\mathcal{A}}_0|\left|f\right|{|}_{p,v},\forall f\in l_{p,v}.$$

Based on Lemma 4.1 it follows that the inequality (3) holds for m = 0 and for the best constant in (3) the following estimate is valid

$$C\ll {\mathcal{A}}^{+}\approx {\mathcal{A}}^{-}.$$

(24)

Now we assume that the inequality (3) holds for m = 0,1,..., n - 1, n ≥ 1 and for the best constant in (3) the estimate (24) is valid. We consider the inequality

$$\left|\right|A_m^{-}f|{|}_{q,u}\ll {\mathcal{A}}_m|\left|f\right|{|}_{p,v},\forall f\in l_{p,v},$$

(25)

where $A_m^{-}$ is given by (2) with the matrix $\left(a_{i,j}^{\left(m\right)}\right)\in {\mathcal{O}}_m^{-}.$

Now our aim is to show that the inequality (25) holds for m = n with the estimate (24).

Let $h\equiv h_n,$where h _n is the constant in (12) with m = n. For all j ≥ 1we define the following set:

$$T_j=\left\{k\in \mathbb{Z}:{\left(h+1\right)}^{-k}\le {\left(A_n^{-}f\right)}_j\right\},$$

where ℤ is the set of integers. We assume that k _j = inf T _j , if T _j ≠ ∅ and k _j = ∞, if T _j = ∅. In order to avoid trivial cases we directly suppose that ${\left(A_n^{-}f\right)}_1\ne 0.$ Since $a_{i,j}^{\left(n\right)}$ is non-increasing in j, we have k _j ≤ k _{j +1}. If k _j < ∞, then

$${\left(h+1\right)}^{-k_j}\le {\left(A_n^{-}f\right)}_j<{\left(h+1\right)}^{-\left(k_j-1\right)},j\ge 1.$$

(26)

Let m₁ = 0, $k_1=k_{m_1+1}$and M₁ = {j ∈ ℕ: k _j = k₁ = $k_{m_1+1}\}$, where ℕ is the set of natural numbers. Suppose that m₂ is such that sup M₁ = m₂. Obviously m₂> m₁ and if the set M₁ is bounded from above, then m₂< ∞ and m₂ = max M₁. We now define the numbers 0 = m₁< m₂$<\cdots <$m _s < ∞, s ≥ 1 by induction. To define m _{s +1} we assume that m _{s +1} = sup M _s , where M _s = {j ∊ ℕ: k _j = ${k_{m_s}}_{+\mathsf{\text{1}}}\}.$

Let N₀ = {s ∊ ℕ: m _s <∞}. Further, we assume that ${k_{m_s}}_{+\mathsf{\text{1}}}$= n _{s +1}, s ∊ N₀. From the definition of m _s and from (26) it follows that, for s ∊ N₀,

$${\left(h+1\right)}^{-n_{s+1}}\le {\left(A_n^{-}f\right)}_j<{\left(h+1\right)}^{-n_{s+1}+1},m_s+1\le j\le m_{s+1}$$

(27)

and

$$\mathbb{N}=\bigcup _{s\in N_0}\left[m_s+1,m_{s+1}\right],\mathsf{\text{where}}\left[m_s+1,m_{s+1}\right]\cap \left[m_l+1,m_{l+1}\right]=\varnothing ,s\ne l.$$

Therefore for 0 ≤ f ∊ l _p,v the left-hand side of (3) has the following form

$$\left|\right|A_n^{-}f|{|}_{q,u}^q=\sum _{s\in N_0}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_{{}^j}^q{u}_j^q.$$

(28)

We assume that $\sum _{j=m_s+1}^{m_{s+1}}=0,\mathsf{\text{if}}{m}_s=\infty .$

There are two possible cases: N₀ = ℕ and N₀ ≠ ℕ.

1. If N₀ = ℕ, then we estimate (28) in the following way.

Clearly inequalities n _{s +1}< n _{s +2}< n _{s +3} imply that -n _{s +3} + 1 ≤ -n _{s +1} - 1 for all s ∊ ℕ. Hence, (27), (16) imply that

$$\begin{array}{ll}\hfill {\left(h+1\right)}^{-n_{s+1}-1}& ={\left(h+1\right)}^{-n_{s+1}}-h{\left(h+1\right)}^{-n_{s+1}-1}\\ \le {\left(h+1\right)}^{-n_{s+1}}-h{\left(h+1\right)}^{-n_{s+3}+1}<{\left(A_n^{-}f\right)}_{m_{s+1}}-h{\left(A_n^{-}f\right)}_{m_{s+3}}\\ =\sum _{i=m_{s+1}}^{\infty }{a}_{i,m_{s+1}}^{\left(n\right)}{f}_i-h\sum _{i=m_{s+3}}^{\infty }{a}_{i,m_{s+3}}^{\left(n\right)}{f}_i\\ \le \sum _{i=m_{s+1}}^{m_{s+3}}{a}_{i,m_{s+1}}^{\left(n\right)}{f}_i+\sum _{i=m_{s+3}}^{\infty }\left[a_{i,m_{s+1}}^{\left(n\right)}-h{a}_{i,m_{s+3}}^{\left(n\right)}\right]f_i\\ \le \sum _{i=m_{s+1}}^{m_{s+3}}{a}_{i,m_{s+1}}^{\left(n\right)}{f}_i+\sum _{i=m_{s+3}}^{\infty }\left[h\sum _{\gamma =0}^n{a}_{i,m_{s+3}}^{\left(\gamma \right)}{d}_{m_{s+3},m_{s+1}}^{\gamma ,n}-h{a}_{i,m_{s+3}}^{\left(n\right)}\right]f_i\\ =\sum _{i=m_{s+1}}^{m_{s+3}}{a}_{i,m_{s+1}}^{\left(n\right)}{f}_i+h\sum _{i=m_{s+3}}^{\infty }\sum _{\gamma =0}^{n-1}{a}_{i,m_{s+3}}^{\left(\gamma \right)}{d}_{m_{s+3},m_{s+1}}^{\gamma ,n}{f}_i.\end{array}$$

(29)

Now, by using (27) and (29), we can estimate (28) in the following way.

$$\begin{array}{c}\sum _{s\in N}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q<\sum _{s\in N}\sum _{j=m_s+1}^{m_{s+1}}{\left(h+1\right)}^{\left(-n_{s+1}+1\right)q}{u}_j^q\\ ={\left(h+1\right)}^{2q}\sum _{s\in N}{\left(h+1\right)}^{\left(-n_{s+1}-1\right)q}\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q\\ \ll \sum _{s\in N}{\left(\sum _{i=m_{s+1}}^{m_{s+3}}{a}_{i,m_{s+1}}^{\left(n\right)}{f}_i+h\sum _{\gamma =0}^{n-1}\sum _{i=m_{s+3}}^{\infty }{a}_{i,m_{s+3}}^{\left(\gamma \right)}{d}_{m_{s+3},m_{s+1}}^{\gamma ,n}{f}_i\right)}^q\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q\\ \ll \sum _{s\in N}{\left(\sum _{i=m_{s+1}}^{m_{s+3}}{a}_{i,m_{s+1}}^{\left(n\right)}{f}_i\right)}^q\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q\\ +\sum _{\gamma =0}^{n-1}\sum _{s\in N}{\left(d_{m_{s+3},m_{s+1}}^{\gamma ,n}\right)}^q{\left(\sum _{i=m_{s+3}}^{\infty }{a}_{i,m_{s+3}}^{\left(\gamma \right)}{f}_i\right)}^q\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q:=I_n+\sum _{\gamma =0}^{n-1}{I}_{\gamma },\end{array}$$

(30)

where

$$I_n=\sum _{s\in N}{\left(\sum _{i=m_{s+1}}^{m_{s+3}}{a}_{i,m_{s+1}}^{\left(n\right)}{f}_i\right)}^q\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q$$

and

$$I_{\gamma }=\sum _{s\in N}{\left(d_{m_{s+3},m_{s+1}}^{\gamma ,n}\right)}^q{\left(\sum _{i=m_{s+3}}^{\infty }{a}_{i,m_{s+3}}^{\left(\gamma \right)}{f}_i\right)}^q\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q,0\le \gamma \le n-1.$$

To estimate I _n we apply Hölder's and Jensen's inequalities and find that

$$\begin{array}{c}{I}_n\le \sum _{s\in N}{\left(\sum _{i=m_{s+1}}^{m_{s+3}}{\left(a_{i,m_{s+1}}^{\left(n\right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{q}{p^{\prime }}}\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q{\left(\sum _{i=m_{s+1}}^{m_{s+3}}|f_i{v}_i{|}^p\right)}^{\frac{q}{p}}\\ \le {\left[\underset{k\ge 1}{\text{sup}}{\left(\sum _{j=1}^k{u}_j^q\right)}^{\frac{1}{q}}{\left(\sum _{i=k}^{\infty }{\left(a_{i,k}^{\left(n\right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}\right]}^q\sum _{s\in N}{\left(\sum _{j=m_{s+1}}^{m_{s+3}}|f_i{v}_i{|}^p\right)}^{\frac{q}{p}}\\ \le {\left[\underset{k\ge 1}{\text{sup}}{\left(\sum _{j=1}^k{\left(d_{k,j}^{n,n}\right)}^q{u}_j^q\right)}^{\frac{1}{q}}{\left(\sum _{i=k}^{\infty }{\left(a_{i,k}^{\left(n\right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}\right]}^q{\left(\sum _{s\in N}\sum _{i=m_{s+1}}^{m_{s+3}}|f_i{v}_i{|}^p\right)}^{\frac{q}{p}}\ll {\mathcal{A}}_n^q\left|\right|f|{|}_{p,v}^q.\end{array}$$

(31)

We introduce the sequence ${\left\{{\Delta }_j\right\}}_{j=1}^{\infty }$ defined by ${\Delta }_j={\left(d_{m_{s+3},m_{s+1}}^{\gamma ,n}\right)}^q\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q$, j = m _{s +3} and ${\Delta }_j=0,j\ne m_{s+3}$, s ∊ N. Hence, we can rewrite I _γ , γ = 0,..., n - 1 in the following form:

$$\begin{array}{c}{I}_{\gamma }=\sum _{s\in N}{\left(\sum _{i=m_{s+3}}^{\infty }{a}_{i,m_{s+3}}^{\left(\gamma \right)}{f}_i\right)}^q{\left(d_{m_{s+3},m_{s+1}}^{\gamma ,n}\right)}^q\sum _{i=m_s+1}^{m_{s+1}}{u}_i^q\\ =\sum _{j=1}^{\infty }{\left(\sum _{i=j}^{\infty }{a}_{i,j}^{\left(\gamma \right)}{f}_i\right)}^q{\Delta }_j.\end{array}$$

(32)

By the assumptions on $a_{i,j}^{\left(\gamma \right)},\gamma =0,\dots ,n-1,i\ge j\ge 1,$we have the validity of (25). Therefore,

$$I_{\gamma }\ll {\tilde{\mathcal{A}}}_{\gamma }^q\left|\right|f|{|}_p^q,\gamma =0,\dots ,n-1,$$

(33)

where

$${\tilde{\mathcal{A}}}_{\gamma }=\underset{0\le l\le \gamma }{\text{max}}\underset{k\ge 1}{\text{sup}}{\left(\sum _{j=1}^k{\left(d_{k,j}^{l,\gamma }\right)}^q{\Delta }_j\right)}^{\frac{1}{q}}{\left(\sum _{i=k}^{\infty }{\left(a_{i,k}^{\left(l\right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}.$$

(34)

Using (14) and taking into account that $d_{i,j}^{l,n}$is non-decreasing in i and non-increasing in j, we find that

$$\begin{array}{c}\sum _{j=1}^k{\left(d_{k,j}^{l,\gamma }\right)}^q{\Delta }_j=\sum _{m_{s+3}\le k}{\left(d_{k,m_{s+3}}^{l,\gamma }\right)}^q{\left(d_{m_{s+3},m_{s+1}}^{\gamma ,n}\right)}^q\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q\\ \ll \sum _{m_{s+3}\le k}\sum _{i=m_s+1}^{m_{s+1}}{\left(d_{k,i}^{l,n}\right)}^q{u}_i^q\le \sum _{i=1}^k{\left(d_{k,i}^{l,n}\right)}^q{u}_i^q.\end{array}$$

(35)

By combining (33), (34), and (35), we obtain that

$$I_{\gamma }\ll {\mathcal{A}}_n^q\left|\right|f|{|}_{p,v}^q.$$

(36)

Thus, from (30), (31), and (36) it follows that

$$\left|\right|A_n^{-}f|{|}_{q,u}\ll {\mathcal{A}}_n|\left|f\right|{|}_{p,v},f\ge 0,$$

(37)

i.e., inequality (3) is valid and by Lemma 4.1, we obtain that

$$C\ll {\mathcal{A}}_n\approx {\mathcal{A}}^{+}\approx {\mathcal{A}}^{-}.$$

(38)

2. If N₀ ≠ N, i.e., max N₀< ∞ and N₀ = {1, 2,..., s₀}, s₀ ≥1. Therefore, ${m_s}_{{}_0}$< ∞ and ${m_s}_{{}_0+\mathsf{\text{1}}}=\infty .$ We assume that $\sum _{s=k}^n=0,\mathsf{\text{if}}k>n$and $\sum _{s=k}^n=\sum _{s=1}^n,\mathsf{\text{if}}k\le 0.$We have two possible cases: ${n_s}_{{}_0+\mathsf{\text{1}}}$< ∞ and ${n_s}_{{}_0+\mathsf{\text{1}}}$= ∞. We consider these cases separately:

• 1) If ${n_s}_{{}_0+\mathsf{\text{1}}}$ < ∞, then from (28) it follows that

  $$\begin{array}{c}{∥A_n^{-}f∥}_{q,u}^q=\sum _{s\in N_0}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q=\sum _{s=1}^{s_0}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q\\ =\sum _{s=1}^{s_0-3}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q+\sum _{s=s_0-2}^{s_0-1}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q+\sum _{j=m_{s_0}+1}^{\infty }{\left(A_n^{-}f\right)}_j^q{u}_j^q=J_1+J_2+J_3.\end{array}$$

  (39)

If J₁ ≠ 0 then for s₀> 3, we estimate J₁ using (29) and the previous proof for the case N₀ = ℕ as in estimate I _γ . Hence we get

$$J_1\ll {\mathcal{A}}_n^q\left|\right|f|{|}_{p,v}^q.$$

(40)

If J₂ ≠ 0 then by using (27) and applying Hölder's and Jensen's inequalities, we obtain the following estimate

$$\begin{array}{c}{J}_2=\sum _{s=s_0-2}^{s_0-1}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q<\sum _{s=s_0-2}^{s_0-1}\sum _{j=m_s+1}^{m_{s+1}}{\left(h+1\right)}^{\left(-n_{s+1}+1\right)q}{u}_j^q\\ ={\left(h+1\right)}^q\sum _{s=s_0-2}^{s_0-1}{\left(h+1\right)}^{-n_{s+1}q}\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q\ll \sum _{s=s_0-2}^{s_0-1}{\left(A_n^{-}f\right)}_{m_{s+1}}^q\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q\\ =\sum _{s=s_0-2}^{s_0-1}{\left(\sum _{i=m_{s+1}}^{\infty }{a}_{i,m_{s+1}}^{\left(n\right)}{f}_i\right)}^q\sum _{J=m_s+1}^{m_{s+1}}{u}_j^q\\ \le \sum _{s=s_0-2}^{s_0-1}{\left[{\left(\sum _{i=m_{s+1}}^{\infty }{\left(a_{i,m_{s+1}}^{\left(n\right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}{\left(\sum _{j=m_s+1}^{m_{s+1}}{u}_j^q\right)}^{\frac{1}{q}}\right]}^q{\left(\sum _{j=m_{s+1}}^{\infty }|v_i{f}_i{|}^p\right)}^{\frac{q}{p}}\\ \le {\left[\underset{k\ge 1}{\text{sup}}{\left(\sum _{i=k}^{\infty }{\left(a_{i,k}^{\left(n\right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}{\left(\sum _{j=1}^k{u}_j^q\right)}^{\frac{1}{q}}\right]}^q{\left(\sum _{s=s_0-2}^{s_0-1}\sum _{j=m_{s+1}}^{\infty }|v_i{f}_i{|}^p\right)}^{\frac{q}{p}}\\ \le 2{\mathcal{A}}_n^q\left|\right|f|{|}_{p,v}^q\ll {\mathcal{A}}_n^q|\left|f\right|{|}_{p,v}^q.\end{array}$$

(41)

Using (27) and applying Hölder's inequality we estimate J₃ in the following way.

$$\begin{array}{c}{J}_3=\sum _{j=m_{s_0}+1}^{\infty }{\left(A_n^{-}f\right)}_j^q{u}_j^q\le \underset{t\ge m_{s_0}+1}{\text{sup}}\sum _{j=m_{s_0}+1}^t{\left(A_n^{-}f\right)}_j^q{u}_j^q\\ \le {\left(h+1\right)}^q\underset{t\ge m_{s_0+1}}{\text{sup}}{\left(h+1\right)}^{-{qn}_{s_0+1}}\sum _{j=m_{s_0}+1}^t{u}_j^q\\ \ll \underset{t\ge m_{s_0}+1}{\text{sup}}{\left(A_n^{-}f\right)}_t^q\sum _{j=m_{s_0}+1}^t{u}_j^q=\underset{t\ge m_{s_0}+1}{\text{sup}}{\left(\sum _{i=t}^{\infty }{a}_{i,t}^{\left(n\right)}{f}_i\right)}^q\sum _{j=m_{s_0}+1}^t{u}_j^q\\ \le \underset{t\ge m_{s_0+1}}{\text{sup}}{\left[{\left(\sum _{i=t}^{\infty }{\left(a_{i,t}^{\left(n\right)}\right)}^{p^{\prime }}{v}_i^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}{\left(\sum _{j=m_{s_0}+1}^t{u}_j^q\right)}^{\frac{1}{q}}\right]}^q\left|\right|f|{|}_{p,v}^q\le {\mathcal{A}}_n^q|\left|f\right|{|}_{p,v}^q.\end{array}$$

(42)

By (39), (40), (41), and (42) we obtain (37) and, consequently (38).

• 2) If ${n_s}_{{}_0+\mathsf{\text{1}}}$= ∞, which means that $k_{{m_s{}_{{}_0}}_{+1}}=\infty ,$then by the definition of $m_{s_0+1}$we have $k_j=\infty$and $T_j=\varnothing ,\mathsf{\text{if}}j\ge m_{s_0}+1,\mathsf{\text{i}}\mathsf{\text{.e}}.,{\left(A_n^{-}f\right)}_j=0,\mathsf{\text{if}}j\ge m_{s_0}+1.$ By the assumption that ${\left(A_n^{-}f\right)}_1\ne 0$it follows that ${\mathsf{\text{s}}}_{\mathsf{\text{0}}}>1\mathsf{\text{.}}$ Therefore, m ₂ < ∞ and s ₀ ≥ 2. Thus by (28) we have

  $$\begin{array}{c}\left|\right|A_n^{-}f|{|}_{q,u}^q=\sum _{s\in N_0}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q\\ =\sum _{s=1}^{s_0-3}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q+\sum _{s=s_0-2}^{s_0-1}\sum _{j=m_s+1}^{m_{s+1}}{\left(A_n^{-}f\right)}_j^q{u}_j^q={J^{\prime }}_1+{J^{\prime }}_2.\end{array}$$

  (43)

By estimating $J_1^{\prime }$ and $J_2^{\prime }$ as J₁ and J₂, respectively, from (43) we obtain (37) and, consequently (38). Therefore, we see that inequality (25) holds for m = n and the estimate (24) is valid. This means that inequality (25) holds for all m ≥ 0 with the estimate (24), which together with (23) gives $C\approx {\mathcal{A}}_n.$ The proof is complete.

5 Compactness of the matrix operators

Theorem 5.1. Let 1 < p ≤ q < ∞. Let the matrix (a _i,j ) of (1) belong to the class${\mathcal{O}}_n^{+},n\ge 0.$Then the operator defined by (1) is compact from l _p,v into l _q,u if and only if one of the following conditions holds

$$\underset{k\to \infty }{\text{lim}}{\left({\mathcal{B}}_{p,q}^{+}\right)}_k=0,$$

(44)

$$\underset{k\to \infty }{\text{lim}}{\left({\mathcal{B}}_{p,q}^{-}\right)}_k=0.$$

(45)

Theorem 5.2. Let 1 < p ≤ q < ∞. Let the matrix (a _i,j ) of (2) belong to the class${\mathcal{O}}_m^{-},m\ge 0.$Then the operator defined by (2) is compact from l _p,v into l _q,u if and only if one of the following conditions holds

$$\underset{k\to \infty }{\text{lim}}{\left({\mathcal{A}}_{p,q}^{+}\right)}_k=0,$$

(46)

$$\underset{k\to \infty }{\text{lim}}{\left({\mathcal{A}}_{p,q}^{-}\right)}_k=0.$$

(47)

Now we give the proof of compactness for the class ${\mathcal{O}}_n^{+},n\ge 0.$

PROOF OF THEOREM 5.1. For the proof of Theorem 5.1, we need the following equivalence

$${\left({\mathcal{B}}_{p,q}^{+}\right)}_k\approx {\left({\mathcal{B}}_n\right)}_k\equiv \underset{0\le \gamma \le n}{\text{max}}{\left({\mathcal{B}}_{\gamma ,n}\right)}_k\approx {\left({\mathcal{B}}_{p,q}^{-}\right)}_k,$$

(48)

where

$${\left(B_{\gamma ,n}\right)}_k={\left(\sum _{i=k}^{\infty }{\left(b_{i,k}^{n,\gamma }\right)}^q{u}_i^q\right)}^{\frac{1}{q}}{\left(\sum _{j=1}^k{\left(a_{k,j}^{\left(\gamma \right)}\right)}^{p^{\prime }}{v}_j^{-p^{\prime }}\right)}^{\frac{1}{p^{\prime }}}.$$

The equivalence directly follows from (10).

Necessity. Suppose that the matrix of operator (1) belongs to the class ${\mathcal{O}}_n^{+},n\ge 0.$Let the operator (1) be compact from l _p,v into l _q,u .

For r ≥ 1, we introduce the following sequence:

$${\phi }_r={\left\{{\phi }_{r,j}\right\}}_{j=1}^{\infty }:{\phi }_{r,j}=\frac{f_{r,j}}{\left|\right|f_r|{|}_{p,v}},$$

where $f_r={\left\{f_{r,j}\right\}}_{j=1}^{\infty }:f_{r,j}=\left\{\begin{array}{c}\hfill {\left(a_{r,}^{}\right)}^{}\hfill \end{array}\right.$