Criteria of boundedness and compactness of a class of matrix operators

Ryskul Oinarov; Zhanar Taspaganbetova

doi:10.1186/1029-242X-2012-53

Research

Open Access

Criteria of boundedness and compactness of a class of matrix operators

Ryskul Oinarov¹Email author and
Zhanar Taspaganbetova¹

Journal of Inequalities and Applications20122012:53

https://doi.org/10.1186/1029-242X-2012-53

Received: 15 September 2011

Accepted: 2 March 2012

Published: 2 March 2012

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 ${(A^{+} f)}_{i} : = \sum_{j = 1}^{i} a_{i, j} f_{j}$ , i ≥ 1 and ${(A^{-} f)}_{j} : = \sum_{i = j}^{\infty} a_{i, j} f_{i}$ , 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.

Keywords

inequalitiesdiscrete Hardy-type inequalitiesweightsmatrix operatorsboundednesscompactness

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

{(A^{+} f)}_{i} : = \sum_{j = 1}^{i} a_{i, j} f_{j}, i \geq 1,

(1)

{(A^{-} f)}_{j} : = \sum_{i = j}^{\infty} a_{i, j} f_{i}, j \geq 1,

(2)

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

{∥A^{\pm} f∥}_{q, u} \leq 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^{'}} = 1

and

u = {u_{i}}_{i = 1}^{\infty}, v = {v_{i}}_{i = 1}^{\infty}

are positive sequences of real numbers. l_p,v is the space of sequences

f = {f_{i}}_{i = 1}^{\infty}

of real numbers such that

| | f | |_{p, v} : = {(\sum_{i = 1}^{\infty} | v_{i} f_{i} |^{p})}^{\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 ${(A_{0}^{+} f)}_{i} : = \sum_{j = 1}^{i} f_{j}, {(A_{0}^{-} f)}_{j} : = \sum_{i = j}^{\infty} f_{i}$ , 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} (a_{i, k} + a_{k, j}) \leq a_{i, j} \leq d (a_{i, k} + a_{k, j}), i \geq k \geq j \geq 1

(4)

hold.

A sequence ${a_{i}}_{i = 1}^{\infty}$ 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

{ω_{k}}_{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} (b_{i, k} ω_{j} + a_{k, j}) \leq a_{i, j} \leq d (b_{i, k} ω_{j} + a_{k, j})

(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

{ω_{k}}_{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} (a_{i, k} + b_{k, j} ω_{i}) \leq a_{i, j} \leq d (a_{i, k} + b_{k, j} ω_{i})

(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 $O_{n}^{+}$ and $O_{n}^{-}$ of matrices (a_i,j ). We assume that $a_{i, j} \equiv a_{i, j}^{(n)}$ if $(a_{i, j}) \in O_{n}^{+}$ or $(a_{i, j}) \in O_{n}^{-}$ .

We define the classes

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}^{(0)} = α_{j}

, ∀i. ≥ j ≥ 1 belong to the class

O_{0}^{+}

. Let the classes

O_{γ}^{+}

, γ = 0, 1,..., n - 1, n ≥ 1 be defined. By definition, the matrix

(a_{i, j}) \equiv (a_{i, j}^{(n)})

belongs to the class

O_{n}^{+}

if and only if there exist matrices

(a_{i, j}^{(γ)}) \in O_{γ}^{+}

, γ = 0, 1,..., n - 1 and a number r_n > 0 such that

a_{i, j}^{(n)} \leq r_{n} \sum_{γ = 0}^{n} b_{i, k}^{n, γ} a_{k, j}^{(γ)}

(7)

for all i ≥ k ≥ j ≥ 1, where

b_{i, k}^{n, n} \equiv 1

and

b_{i, k}^{n, γ} = inf_{1 \leq j \leq k} \frac{a_{i, j}^{(n)}}{a_{k, j}^{(γ)}}, γ = 0, 1, \dots, n - 1 .

(8)

From (8) it follows that entries of the matrices

(b_{i, k}^{n, γ})

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}^{(n)} \geq b_{i, k}^{n, γ} a_{k, j}^{(γ)}

(9)

i \geq k \geq j \geq 1, γ = 0, 1, \dots, n, n = 0, 1, \dots

Then for

(a_{i, j}^{(n)}) \in O_{n}^{+}

we have

a_{i, j}^{(n)} \approx \sum_{γ = 0}^{n} b_{i, k}^{n, γ} a_{k, j}^{(γ)}, n \geq 0

(10)

for all i ≥ k ≥ j ≥ 1.

REMARK 1. It is easy to show that if for the matrix $(a_{i, j}^{(n)})$ , n ≥ 0 there exist matrices $(a_{i, j}^{(γ)}) \in O_{γ}^{+}$ , γ = 0, 1,..., n - 1, and matrices $({\tilde{b}}_{i, k}^{n, γ})$ , γ = 0, 1,..., n such that the equivalence (10) is valid for all i ≥ k ≥ j ≥ 1, then $(a_{i, j}^{(n)}) \in O_{n}^{+}$ and ${\tilde{b}}_{i, k}^{n, γ} \approx b_{i, k}^{n, γ}$ . Hence we may assume that the matrices $(b_{i, k}^{n, γ})$ 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, γ} \geq b_{i, s}^{n, l} \cdot b_{s, k}^{l, γ} \forall i \geq s \geq k \geq 1 .

(11)

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

b_{i, k}^{n, γ} = inf_{1 \leq j \leq k} \frac{a_{i, j}^{(n)}}{a_{k, j}^{(γ)}} \geq b_{i, s}^{n, l} \cdot inf_{1 \leq j \leq k} \frac{a_{s, j}^{(l)}}{a_{k, j}^{(γ)}} = b_{i, s}^{n, l} \cdot b_{s, k}^{l, γ} .

As above, we introduce the classes

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

(a_{i, j}) = (a_{i, j}^{(0)})

belongs to the class

O_{0}^{-}

if and only if it has the form

a_{i, j}^{(0)} = β_{i}

for all i ≥ j ≥ 1. Let the classes

O_{γ}^{-}

, γ = 0, 1,..., m - 1, m ≥ 1 be defined. A matrix

(a_{i, j}) = (a_{i, j}^{(m)})

belongs to the class

O_{m}^{-}

if and only if there exist matrices

(a_{i, j}^{(γ)}) \in O_{γ}^{-}

, γ = 0, 1,..., m - 1 and a number h_m > 0 such that

a_{i, j}^{(m)} \leq h_{m} \sum_{γ = 0}^{m} a_{i, k}^{(γ)} d_{k, j}^{γ, m},

(12)

for all i ≥ k ≥ j ≥ 1, where

d_{k, j}^{m, m} \equiv 1

and

d_{k, j}^{γ, m} = inf_{k \leq i \leq \infty} \frac{a_{i, j}^{(m)}}{a_{i, k}^{(γ)}}, γ = 0, 1, \dots, m - 1 .

(13)

From the definition of the matrix

(d_{k, j}^{γ, m})

, γ = 0, 1,..., m - 1, m = 0, 1,..., it is obvious that the entries of the matrix

(d_{k, j}^{γ, m})

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}^{γ, m} \geq d_{k, s}^{γ, l} \cdot d_{s, j}^{l, m} .

(14)

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

a_{i, j}^{(m)} \geq a_{i, k}^{(γ)} d_{k, j}^{γ, m}, γ = 0, 1, \dots, m - 1 .

(15)

As in (10) every class

O_{m}^{-}

, m ≥ 0 of matrices

(a_{i, j}^{(m)})

is characterized by the following relation

a_{i, j}^{(m)} \approx \sum_{γ = 0}^{m} a_{i, k}^{(γ)} d_{k, j}^{γ, m},

(16)

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

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

REMARK 3. By the definitions of the classes $O_{n}^{\pm}$ , n ≥ 0 we have $O_{0}^{\pm} \subset O_{1}^{\pm} \subset \dots \subset O_{n}^{\pm} \subset \dots$

In particular, the matrices of the classes

O_{1}^{+}

and

O_{1}^{-}

are characterized by the following relations, respectively,

\begin{array}{l} a_{i, j}^{(1)} \approx b_{i, k}^{1, 0} a_{k, j}^{(0)} + a_{k, j}^{(1)} \forall i \geq k \geq j \geq 1, \\ a_{i, j}^{(1)} \approx a_{i, k}^{(1)} + a_{i, k}^{(0)} d_{k, j}^{0, 1} \forall i \geq k \geq j \geq 1 . \end{array}

It is easy to see that the class $O_{1}^{+}$ 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 $O_{1}^{-}$ . This implies that the classes $O_{n}^{+}$ , n ≥ 1 and $O_{m}^{-}$ , m ≥ 1 are wider than the classes of matrices which have been used in this connection before.

The matrices of the classes

O_{2}^{+}

and

O_{2}^{-}

are described by the following relations, respectively,

\begin{array}{l} a_{i, j}^{(2)} \approx b_{i, k}^{2, 0} a_{k, j}^{(0)} + b_{i, k}^{2, 1} a_{k, j}^{(1)} + a_{k, j}^{(2)} \forall i \geq k \geq j \geq 1, \\ a_{i, j}^{(2)} \approx a_{i, k}^{(2)} + a_{i, k}^{(1)} d_{k, j}^{1, 2} + a_{i, k}^{(0)} d_{k, j}^{0, 2} \forall i \geq k \geq j \geq 1 . \end{array}

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

We set

w_{i, k} = \sum_{j = k}^{i} a_{i, j} σ_{j, k} .

Then we have the following

Lemma 2.1. Let $(a_{i, j}) \in O_{n}^{+}, (σ_{j, k}) \in O_{m}^{+}$ . Then $(w_{i, k}) \in O_{m + n + 1}^{+}$ .

PROOF OF LEMMA 2.1. Since

(a_{i, j}) \in O_{n}^{+}

, there exist matrices

(a_{i, j}^{(γ)}) \in O_{γ}^{+}

, γ = 0, 1,..., n-1, and matrices

(δ_{i, l}^{n, γ})

such that

a_{i, j} \equiv a_{i, j}^{(n)} \approx \sum_{γ = 0}^{n} δ_{i, l}^{n, γ} a_{l, k}^{(γ)}, n = 0, 1, \dots, δ_{i, l}^{n, n} \equiv 1

for all i ≥ l ≥ k ≥ 1.

Since

(σ_{j, k}) \in O_{m}^{+}

, there exist matrices

(σ_{j, k}^{(μ)}) \in O_{μ}^{+}

, μ = 0, 1,..., m - 1, and matrices

(e_{k, l}^{m, μ})

such that

σ_{j, k} \equiv σ_{j, k}^{(m)} \approx \sum_{μ = 0}^{m} e_{j, l}^{m, μ} σ_{l, k}^{(μ)}, 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}^{(n)} σ_{j, k}^{(m)} .

First, we consider the case when m ≥ 0, n = 0. In this case

a_{i, j}^{(0)} = α_{j}

, ∀i ≥ j ≥ 1. For ∀i ≥ l ≥ k we obtain

w_{i, k}^{o, m} = \sum_{j = k}^{i} α_{j} σ_{j, k}^{(m)} \approx \sum_{j = k}^{l} α_{j} σ_{j, k}^{(m)} + \sum_{j = l}^{i} α_{j} σ_{j, k}^{(m)} \approx w_{l, k}^{o, m} + \sum_{μ = 0}^{m} σ_{l, k}^{(μ)} \sum_{j = l}^{i} α_{j} e_{j, l}^{m, μ} = w_{l, k}^{o, m} + \sum_{μ = 0}^{m} ẽ_{i, l}^{m + 1, μ} σ_{l, k}^{(μ)},

where

ẽ_{i, l}^{m + 1, μ} = \sum_{j = l}^{i} α_{j} e_{j, l}^{m, μ}

, μ = 0, 1,..., m. Suppose that

ẽ_{i, l}^{m + 1, m + 1} \equiv 1

. Since

(σ_{l, k}^{(μ)}) \in O_{μ}^{+}

, μ = 0, 1,..., m, by definition we easily see that

w_{i, k}^{o, m} \in O_{m + 1}^{+}

. By induction, we assume that for n = 0, 1,..., r - 1,

r \geq 1 (w_{i, k}^{n, m})

belongs to the class

O_{n + m + 1}^{+}

. For i ≥ l ≥ k we have

\begin{array}{l} w_{i, k}^{r, m} & = \sum_{j = k}^{i} a_{i, j}^{(r)} σ_{j, k}^{(m)} \approx \sum_{j = k}^{l} a_{i, j}^{(r)} σ_{j, k}^{(m)} + \sum_{j = l}^{i} a_{i, j}^{(r)} σ_{j, k}^{(m)} \\ \approx \sum_{j = k}^{l} (\sum_{γ = 0}^{r} δ_{i, l}^{r, γ} a_{l, j}^{(γ)}) σ_{j, k}^{(m)} + \sum_{j = l}^{i} a_{i, j}^{(r)} (\sum_{μ = 0}^{m} e_{j, l}^{m, μ} σ_{l, k}^{(μ)}) \\ = \sum_{γ = 0}^{r} δ_{i, l}^{r, γ} \sum_{j = k}^{l} a_{l, j}^{(γ)} σ_{j, k}^{(m)} + \sum_{μ = 0}^{m} σ_{l, k}^{(μ)} \sum_{j = l}^{i} a_{i, j}^{(r)} e_{j, l}^{m, μ} \\ = \sum_{j = k}^{l} a_{l, j}^{(r)} σ_{j, k}^{(m)} + \sum_{γ = 0}^{r - 1} δ_{i, l}^{r, γ} \sum_{j = k}^{l} a_{l, j}^{(γ)} σ_{j, k}^{(m)} + \sum_{μ = 0}^{m} σ_{l, k}^{(μ)} \sum_{j = l}^{i} a_{i, j}^{(r)} e_{j, l}^{m, μ} \\ = w_{l, k}^{r, m} + \sum_{γ = 0}^{r - 1} δ_{i, l}^{r, γ} {\tilde{σ}}_{l, k}^{(γ + m + 1)} + \sum_{μ = 0}^{m} ẽ_{i, l}^{m + 1, μ} σ_{l, k}^{(μ)}, \end{array}

where

{\tilde{σ}}_{l, k}^{(γ + m + 1)} \equiv \sum_{j = k}^{l} a_{l, j}^{(γ)} σ_{j, k}^{(m)}

, γ = 0,..., r - 1 and

ẽ_{i, l}^{m + 1, μ} \equiv \sum_{j = l}^{i} a_{i, j}^{(r)} e_{j, l}^{m, μ}

μ = 0,..., m. We denote γ + m + 1 by μ. Then we have

\begin{array}{l} w_{i, k}^{r, m} & \approx w_{l, k}^{r, m} + \sum_{μ = m + 1}^{r + m} δ_{i, l}^{r, μ - m - 1} {\tilde{σ}}_{l, k}^{(μ)} + \sum_{μ = 0}^{m} ẽ_{i, l}^{m + 1, μ} σ_{l, k}^{(μ)} \\ = w_{l, k}^{r, m} + \sum_{μ = 0}^{r + m} {\tilde{δ}}_{i, l}^{r + m, μ} {\tilde{σ}}_{l, k}^{(μ)}, \end{array}

where

{\tilde{δ}}_{i, l}^{r + m, μ} = \{\begin{gathered} ẽ_{i, l}^{m + 1, μ}, 0 \leq μ \leq m, \\ δ_{i, l}^{r, μ - m - 1}, m + 1 \leq μ \leq r + m, \end{gathered}

and

{\tilde{σ}}_{l, k}^{(μ)} = \{\begin{gathered} σ_{l, k}^{(μ)}, 0 \leq μ \leq m, \\ {\tilde{σ}}_{l, k}^{(μ)}, m + 1 \leq μ \leq r + m . \end{gathered}

Since ${\tilde{σ}}_{l, k}^{(μ)} \in O_{μ}^{+}$ , μ = 0, 1,..., r + m we obtain that $w_{i, k}^{r, m} \in O_{r + m + 1}^{+}$ . The proof is complete.

Now we set

φ_{k, j} = \sum_{i = j}^{k} σ_{k, i} a_{i, j} .

Then we have the following lemma.

Lemma 2.2. Let $(a_{i, j}) \in O_{n}^{-}$ , $(σ_{k, i}) \in O_{m}^{-}$ . Then $(φ_{k, j}) \in O_{m + n + 1}^{-}$ .

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

{(\sum_{j = 1}^{\infty} {(\sum_{i = j}^{\infty} ω_{i} f_{i})}^{q} u_{j}^{q})}^{\frac{1}{q}} \leq C {(\sum_{i = 1}^{\infty} | v_{i} f_{i} |^{p})}^{\frac{1}{p}}, 0 \leq f \in l_{p, v}

(17)

holds if and only if

H : = sup_{n \geq 1} {(\sum_{j = 1}^{n} u_{j}^{q})}^{\frac{1}{q}} {(\sum_{i = n}^{\infty} ω_{i}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}} < \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 = {x_{i}}_{i = 1}^{\infty} \in T

the inequality

\sum_{i = N}^{\infty} | x_{i} |^{p} < ε

holds.

3 Main results

We define

\begin{gathered} {(B_{p, q}^{+})}_{k} = {(\sum_{j = 1}^{k} v_{j}^{- p^{'}} {(\sum_{i = k}^{\infty} a_{i, j}^{q} u_{i}^{q})}^{\frac{p^{'}}{q}})}^{\frac{1}{p^{'}}}, \\ {(B_{p, q}^{-})}_{k} = {(\sum_{i = k}^{\infty} u_{i}^{q} {(\sum_{j = 1}^{k} a_{i, j}^{p^{'}} v_{j}^{- p^{'}})}^{\frac{q}{p^{'}}})}^{\frac{1}{q}}, \\ {(A_{p, q}^{+})}_{k} = {(\sum_{j = 1}^{k} u_{j}^{q} {(\sum_{i = k}^{\infty} a_{i, j}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{q}{p^{'}}})}^{\frac{1}{q}}, \\ {(A_{p, q}^{-})}_{k} = {(\sum_{i = k}^{\infty} v_{i}^{- p^{'}} {(\sum_{j = 1}^{k} a_{i, j}^{q} u_{j}^{q})}^{\frac{p^{'}}{q}})}^{\frac{1}{p^{'}}} . \end{gathered}

We set $B^{+} = sup_{k \geq 1} {(B_{p, q}^{+})}_{k}, B^{-} = sup_{k \geq 1} {(B_{p, q}^{-})}_{k}, A^{+} = sup_{k \geq 1} {(A_{p, q}^{+})}_{k}$ and $A^{-} = sup_{k \geq 1} {(A_{p, q}^{-})}_{k} .$

Theorem 3.1. Suppose that 1 < p ≤ q < ∞. Let the matrix (a_i,j) in (1) belong to the class $O_{m}^{+} \cup O_{m}^{-}$ , 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 $B^{+} < \infty$ and $B^{-} < \infty$ holds. Moreover $B^{+} \approx B^{-} \approx C,$ 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 $lim_{k \to \infty} {(B_{p, q}^{+})}_{k} = 0 and lim_{k \to \infty} {(B_{p, q}^{-})}_{k} = 0$ holds.

Theorem 3.2. Suppose that 1 < p ≤ q < ∞. Let the matrix (a_i,j ) in (2) belong to the class $O_{m}^{+} \cup O_{m}^{-}, m \geq 0 .$ 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 $A^{+} < \infty$ and $A^{-} < \infty$ holds. Moreover $A^{+} \approx A^{-} \approx C,$ 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 $lim_{k \to \infty} {(A_{p, q}^{+})}_{k} = 0$ and $lim_{k \to \infty} {(A_{p, q}^{-})}_{k} = 0$ holds.

Before proving our main theorems we establish the boundedness and compactness of the operators ${(A^{+} f)}_{i} : = \sum_{j = 1}^{i} a_{i, j} f_{j}, i \geq 1$ and ${(A^{-} f)}_{j} : = \sum_{i = j}^{\infty} a_{i, j} f_{i}, j \geq 1$ 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 $O_{n}^{+}, n \geq 0 .$ Then the estimate (3) for the operator defined by (1) holds if and only if one of the conditions $B^{+} < \infty$ and $B^{-} < \infty$ holds. Moreover $B^{+} \approx B^{-} \approx C,$ 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 $O_{m}^{-}, m \geq 0 .$ Then the estimate (3) for the operator defined by (2) holds if and only if one of the conditions $A^{+} < \infty$ and $A^{-} < \infty$ holds. Moreover $A^{+} \approx A^{-} \approx C,$ 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

O_{m}^{-}, m \geq 0 .

Then we have the following equivalence

{(A_{p, q}^{+})}_{k} \approx {(A_{m})}_{k} \equiv max_{0 \leq γ \leq m} {(A_{γ, m})}_{k} \approx {(A_{p, q}^{-})}_{k},

(18)

where

{(A_{γ, m})}_{k} = {(\sum_{j = 1}^{k} {(d_{k, j}^{γ, m})}^{q} u_{j}^{q})}^{\frac{1}{q}} {(\sum_{i = k}^{\infty} {(a_{i, k}^{(γ)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}} .

By (18) it follows that

A^{+} \approx A_{m} = sup_{k \geq 1} {(A_{m})}_{k} \approx A^{-}, \forall m \geq 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 $O_{m}^{-}, m \geq 0$ and (3) holds.

For k > 1 we assume that $g = {g_{i}}_{i = 1}^{\infty} : g_{i} = \{\begin{gathered} u_{i}, 1 \leq i \leq k \\ 0, i > k . \end{gathered}$

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

| | A^{*} g | |_{p^{'}, v^{- 1}} \leq C | | g | |_{q^{'}, u^{- 1}}, g \in l_{q^{'}, 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 = {g_{i}}_{i = 1}^{\infty}

in (20) and using (15) we obtain

\begin{array}{l} C k^{\frac{1}{q^{'}}} & \geq | | A^{*} g | |_{p^{'}, v^{- 1}} \geq {(\sum_{i = k}^{\infty} {(\sum_{j = 1}^{k} a_{i, j}^{(m)} u_{j})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}} \\ ≫ {(\sum_{i = k}^{\infty} {(a_{i, k}^{(γ)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}} (\sum_{j = 1}^{k} d_{k, j}^{γ, m} u_{j}), γ = 0, 1, \dots, m . \end{array}

Therefore ${a_{i, k}^{(γ)}}_{i = 1}^{\infty} \in l_{p^{'}, v^{- 1}} .$

Now for 1≤ r < M < ∞, we assume that

f = {f_{s}}_{s = 1}^{\infty},

where

\begin{gathered} f_{s} = \{\begin{matrix} {(a_{s, r}^{(γ)})}^{p^{'} - 1} v_{s}^{- p^{'}}, & r \leq s \leq M \\ 0, & s < r or s > M . \end{matrix} \end{gathered}

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

\begin{array}{l} C {(\sum_{s = r}^{M} {(a_{s, r}^{(γ)})}^{p^{'}} v_{s}^{- p^{'}})}^{\frac{1}{p}} & \geq | | A^{-} f | |_{q, u} = {(\sum_{j = 1}^{\infty} {(\sum_{s = j}^{\infty} a_{s, j}^{(m)} f_{s})}^{q} u_{j}^{q})}^{\frac{1}{q}} \\ ≫ {(\sum_{j = 1}^{r} {(d_{r, j}^{γ, m})}^{q} u_{j}^{q})}^{\frac{1}{q}} (\sum_{s = r}^{M} {(a_{s, r}^{(γ)})}^{p^{'}} v_{s}^{- p^{'}}), \end{array}

which implies

C ≫ {(\sum_{j = 1}^{r} {(d_{r, j}^{γ, m})}^{q} u_{j}^{q})}^{\frac{1}{q}} {(\sum_{s = r}^{M} {(a_{s, r}^{(γ)})}^{p^{'}} v_{s}^{- p^{'}})}^{\frac{1}{p^{'}}} .

(21)

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

sup_{k \geq 1} {(A_{m})}_{k} ≪ C .

(22)

By using Lemma 4.1 we obtain

A^{+} \approx A^{-} ≪ C .

(23)

The proof of the necessity is complete.

Sufficiency. Let the matrix (a_i,j ) of the operator (2) belong to the class

O_{m}^{-}, m \geq 0 .

Let

0 \leq f \in l_{p, v}

and at least one of the conditions

A^{+} < \infty

and

A^{-} < \infty

hold. Assume that m = 0. By the definition of

O_{0}^{-},

the matrix of the operator (2) has the form

a_{i, j}^{(0)} = β_{i} \forall i \geq j \geq 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

| | A_{0}^{-} f | |_{q, u} ≪ A_{0} | | f | |_{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 ≪ A^{+} \approx 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

| | A_{m}^{-} f | |_{q, u} ≪ A_{m} | | f | |_{p, v}, \forall f \in l_{p, v},

(25)

where $A_{m}^{-}$ is given by (2) with the matrix $(a_{i, j}^{(m)}) \in O_{m}^{-} .$

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} = {k \in ℤ : {(h + 1)}^{- k} \leq {(A_{n}^{-} f)}_{j}},

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

{(A_{n}^{-} f)}_{1} \neq 0 .

Since

a_{i, j}^{(n)}

is non-increasing in j, we have k_j ≤ k_j₊₁. If k_j < ∞, then

{(h + 1)}^{- k_{j}} \leq {(A_{n}^{-} f)}_{j} < {(h + 1)}^{- (k_{j} - 1)}, j \geq 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 $< \dots <$ 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 = ${k_{m_{s}}}_{+ 1}} .$

Let N₀ = {s ∊ ℕ: m_s <∞}. Further, we assume that

{k_{m_{s}}}_{+ 1}

= n_s₊₁, s ∊ N₀. From the definition of m_s and from (26) it follows that, for s ∊ N₀,

{(h + 1)}^{- n_{s + 1}} \leq {(A_{n}^{-} f)}_{j} < {(h + 1)}^{- n_{s + 1} + 1}, m_{s} + 1 \leq j \leq m_{s + 1}

(27)

and

ℕ = ⋃_{s \in N_{0}} [m_{s} + 1, m_{s + 1}], where [m_{s} + 1, m_{s + 1}] \cap [m_{l} + 1, m_{l + 1}] = \emptyset, s \neq l .

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

| | A_{n}^{-} f | |_{q, u}^{q} = \sum_{s \in N_{0}} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{^{j}}^{q} u_{j}^{q} .

(28)

We assume that $\sum_{j = m_{s} + 1}^{m_{s + 1}} = 0, if m_{s} = \infty .$

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}{l} {(h + 1)}^{- n_{s + 1} - 1} & = {(h + 1)}^{- n_{s + 1}} - h {(h + 1)}^{- n_{s + 1} - 1} \\ \leq {(h + 1)}^{- n_{s + 1}} - h {(h + 1)}^{- n_{s + 3} + 1} < {(A_{n}^{-} f)}_{m_{s + 1}} - h {(A_{n}^{-} f)}_{m_{s + 3}} \\ = \sum_{i = m_{s + 1}}^{\infty} a_{i, m_{s + 1}}^{(n)} f_{i} - h \sum_{i = m_{s + 3}}^{\infty} a_{i, m_{s + 3}}^{(n)} f_{i} \\ \leq \sum_{i = m_{s + 1}}^{m_{s + 3}} a_{i, m_{s + 1}}^{(n)} f_{i} + \sum_{i = m_{s + 3}}^{\infty} [a_{i, m_{s + 1}}^{(n)} - h a_{i, m_{s + 3}}^{(n)}] f_{i} \\ \leq \sum_{i = m_{s + 1}}^{m_{s + 3}} a_{i, m_{s + 1}}^{(n)} f_{i} + \sum_{i = m_{s + 3}}^{\infty} [h \sum_{γ = 0}^{n} a_{i, m_{s + 3}}^{(γ)} d_{m_{s + 3}, m_{s + 1}}^{γ, n} - h a_{i, m_{s + 3}}^{(n)}] f_{i} \\ = \sum_{i = m_{s + 1}}^{m_{s + 3}} a_{i, m_{s + 1}}^{(n)} f_{i} + h \sum_{i = m_{s + 3}}^{\infty} \sum_{γ = 0}^{n - 1} a_{i, m_{s + 3}}^{(γ)} d_{m_{s + 3}, m_{s + 1}}^{γ, n} f_{i} . \end{array}

(29)

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

\begin{gathered} \sum_{s \in N} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} < \sum_{s \in N} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(h + 1)}^{(- n_{s + 1} + 1) q} u_{j}^{q} \\ = {(h + 1)}^{2 q} \sum_{s \in N} {(h + 1)}^{(- n_{s + 1} - 1) q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} \\ ≪ \sum_{s \in N} {(\sum_{i = m_{s + 1}}^{m_{s + 3}} a_{i, m_{s + 1}}^{(n)} f_{i} + h \sum_{γ = 0}^{n - 1} \sum_{i = m_{s + 3}}^{\infty} a_{i, m_{s + 3}}^{(γ)} d_{m_{s + 3}, m_{s + 1}}^{γ, n} f_{i})}^{q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} \\ ≪ \sum_{s \in N} {(\sum_{i = m_{s + 1}}^{m_{s + 3}} a_{i, m_{s + 1}}^{(n)} f_{i})}^{q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} \\ + \sum_{γ = 0}^{n - 1} \sum_{s \in N} {(d_{m_{s + 3}, m_{s + 1}}^{γ, n})}^{q} {(\sum_{i = m_{s + 3}}^{\infty} a_{i, m_{s + 3}}^{(γ)} f_{i})}^{q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} : = I_{n} + \sum_{γ = 0}^{n - 1} I_{γ}, \end{gathered}

(30)

where

I_{n} = \sum_{s \in N} {(\sum_{i = m_{s + 1}}^{m_{s + 3}} a_{i, m_{s + 1}}^{(n)} f_{i})}^{q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q}

and

I_{γ} = \sum_{s \in N} {(d_{m_{s + 3}, m_{s + 1}}^{γ, n})}^{q} {(\sum_{i = m_{s + 3}}^{\infty} a_{i, m_{s + 3}}^{(γ)} f_{i})}^{q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q}, 0 \leq γ \leq n - 1 .

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

\begin{gathered} I_{n} \leq \sum_{s \in N} {(\sum_{i = m_{s + 1}}^{m_{s + 3}} {(a_{i, m_{s + 1}}^{(n)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{q}{p^{'}}} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} {(\sum_{i = m_{s + 1}}^{m_{s + 3}} | f_{i} v_{i} |^{p})}^{\frac{q}{p}} \\ \leq {[sup_{k \geq 1} {(\sum_{j = 1}^{k} u_{j}^{q})}^{\frac{1}{q}} {(\sum_{i = k}^{\infty} {(a_{i, k}^{(n)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}}]}^{q} \sum_{s \in N} {(\sum_{j = m_{s + 1}}^{m_{s + 3}} | f_{i} v_{i} |^{p})}^{\frac{q}{p}} \\ \leq {[sup_{k \geq 1} {(\sum_{j = 1}^{k} {(d_{k, j}^{n, n})}^{q} u_{j}^{q})}^{\frac{1}{q}} {(\sum_{i = k}^{\infty} {(a_{i, k}^{(n)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}}]}^{q} {(\sum_{s \in N} \sum_{i = m_{s + 1}}^{m_{s + 3}} | f_{i} v_{i} |^{p})}^{\frac{q}{p}} ≪ A_{n}^{q} | | f | |_{p, v}^{q} . \end{gathered}

(31)

We introduce the sequence

{Δ_{j}}_{j = 1}^{\infty}

defined by

Δ_{j} = {(d_{m_{s + 3}, m_{s + 1}}^{γ, n})}^{q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q}

, j = m_s₊₃ and

Δ_{j} = 0, j \neq m_{s + 3}

, s ∊ N. Hence, we can rewrite I_γ, γ = 0,..., n - 1 in the following form:

\begin{gathered} I_{γ} = \sum_{s \in N} {(\sum_{i = m_{s + 3}}^{\infty} a_{i, m_{s + 3}}^{(γ)} f_{i})}^{q} {(d_{m_{s + 3}, m_{s + 1}}^{γ, n})}^{q} \sum_{i = m_{s} + 1}^{m_{s + 1}} u_{i}^{q} \\ = \sum_{j = 1}^{\infty} {(\sum_{i = j}^{\infty} a_{i, j}^{(γ)} f_{i})}^{q} Δ_{j} . \end{gathered}

(32)

By the assumptions on

a_{i, j}^{(γ)}, γ = 0, \dots, n - 1, i \geq j \geq 1,

we have the validity of (25). Therefore,

I_{γ} ≪ {\tilde{A}}_{γ}^{q} | | f | |_{p}^{q}, γ = 0, \dots, n - 1,

(33)

where

{\tilde{A}}_{γ} = max_{0 \leq l \leq γ} sup_{k \geq 1} {(\sum_{j = 1}^{k} {(d_{k, j}^{l, γ})}^{q} Δ_{j})}^{\frac{1}{q}} {(\sum_{i = k}^{\infty} {(a_{i, k}^{(l)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}} .

(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{gathered} \sum_{j = 1}^{k} {(d_{k, j}^{l, γ})}^{q} Δ_{j} = \sum_{m_{s + 3} \leq k} {(d_{k, m_{s + 3}}^{l, γ})}^{q} {(d_{m_{s + 3}, m_{s + 1}}^{γ, n})}^{q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} \\ ≪ \sum_{m_{s + 3} \leq k} \sum_{i = m_{s} + 1}^{m_{s + 1}} {(d_{k, i}^{l, n})}^{q} u_{i}^{q} \leq \sum_{i = 1}^{k} {(d_{k, i}^{l, n})}^{q} u_{i}^{q} . \end{gathered}

(35)

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

I_{γ} ≪ A_{n}^{q} | | f | |_{p, v}^{q} .

(36)

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

| | A_{n}^{-} f | |_{q, u} ≪ A_{n} | | f | |_{p, v}, f \geq 0,

(37)

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

C ≪ A_{n} \approx A^{+} \approx 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} + 1} = \infty .

We assume that

\sum_{s = k}^{n} = 0, if k > n

and

\sum_{s = k}^{n} = \sum_{s = 1}^{n}, if k \leq 0 .

We have two possible cases:

{n_{s}}_{_{0} + 1}

< ∞ and

{n_{s}}_{_{0} + 1}

= ∞. We consider these cases separately:

1)

If ${n_{s}}_{_{0} + 1}$ < ∞, then from (28) it follows that
$\begin{gathered} {∥A_{n}^{-} f∥}_{q, u}^{q} = \sum_{s \in N_{0}} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} = \sum_{s = 1}^{s_{0}} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} \\ = \sum_{s = 1}^{s_{0} - 3} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} + \sum_{s = s_{0} - 2}^{s_{0} - 1} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} + \sum_{j = m_{s_{0}} + 1}^{\infty} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} = J_{1} + J_{2} + J_{3} . \end{gathered}$

(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} ≪ A_{n}^{q} | | 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{gathered} J_{2} = \sum_{s = s_{0} - 2}^{s_{0} - 1} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} < \sum_{s = s_{0} - 2}^{s_{0} - 1} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(h + 1)}^{(- n_{s + 1} + 1) q} u_{j}^{q} \\ = {(h + 1)}^{q} \sum_{s = s_{0} - 2}^{s_{0} - 1} {(h + 1)}^{- n_{s + 1} q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} ≪ \sum_{s = s_{0} - 2}^{s_{0} - 1} {(A_{n}^{-} f)}_{m_{s + 1}}^{q} \sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} \\ = \sum_{s = s_{0} - 2}^{s_{0} - 1} {(\sum_{i = m_{s + 1}}^{\infty} a_{i, m_{s + 1}}^{(n)} f_{i})}^{q} \sum_{J = m_{s} + 1}^{m_{s + 1}} u_{j}^{q} \\ \leq \sum_{s = s_{0} - 2}^{s_{0} - 1} {[{(\sum_{i = m_{s + 1}}^{\infty} {(a_{i, m_{s + 1}}^{(n)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}} {(\sum_{j = m_{s} + 1}^{m_{s + 1}} u_{j}^{q})}^{\frac{1}{q}}]}^{q} {(\sum_{j = m_{s + 1}}^{\infty} | v_{i} f_{i} |^{p})}^{\frac{q}{p}} \\ \leq {[sup_{k \geq 1} {(\sum_{i = k}^{\infty} {(a_{i, k}^{(n)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}} {(\sum_{j = 1}^{k} u_{j}^{q})}^{\frac{1}{q}}]}^{q} {(\sum_{s = s_{0} - 2}^{s_{0} - 1} \sum_{j = m_{s + 1}}^{\infty} | v_{i} f_{i} |^{p})}^{\frac{q}{p}} \\ \leq 2 A_{n}^{q} | | f | |_{p, v}^{q} ≪ A_{n}^{q} | | f | |_{p, v}^{q} . \end{gathered}

(41)

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

\begin{gathered} J_{3} = \sum_{j = m_{s_{0}} + 1}^{\infty} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} \leq sup_{t \geq m_{s_{0}} + 1} \sum_{j = m_{s_{0}} + 1}^{t} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} \\ \leq {(h + 1)}^{q} sup_{t \geq m_{s_{0} + 1}} {(h + 1)}^{- {q n}_{s_{0} + 1}} \sum_{j = m_{s_{0}} + 1}^{t} u_{j}^{q} \\ ≪ sup_{t \geq m_{s_{0}} + 1} {(A_{n}^{-} f)}_{t}^{q} \sum_{j = m_{s_{0}} + 1}^{t} u_{j}^{q} = sup_{t \geq m_{s_{0}} + 1} {(\sum_{i = t}^{\infty} a_{i, t}^{(n)} f_{i})}^{q} \sum_{j = m_{s_{0}} + 1}^{t} u_{j}^{q} \\ \leq sup_{t \geq m_{s_{0} + 1}} {[{(\sum_{i = t}^{\infty} {(a_{i, t}^{(n)})}^{p^{'}} v_{i}^{- p^{'}})}^{\frac{1}{p^{'}}} {(\sum_{j = m_{s_{0}} + 1}^{t} u_{j}^{q})}^{\frac{1}{q}}]}^{q} | | f | |_{p, v}^{q} \leq A_{n}^{q} | | f | |_{p, v}^{q} . \end{gathered}

(42)

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

2)

If ${n_{s}}_{_{0} + 1}$ = ∞, which means that $k_{{m_{s}_{_{0}}}_{+ 1}} = \infty,$ then by the definition of $m_{s_{0} + 1}$ we have $k_{j} = \infty$ and $T_{j} = \emptyset, if j \geq m_{s_{0}} + 1, i .e ., {(A_{n}^{-} f)}_{j} = 0, if j \geq m_{s_{0}} + 1 .$ By the assumption that ${(A_{n}^{-} f)}_{1} \neq 0$ it follows that $s_{0} > 1 .$ Therefore, m ₂ < ∞ and s ₀ ≥ 2. Thus by (28) we have
$\begin{gathered} | | A_{n}^{-} f | |_{q, u}^{q} = \sum_{s \in N_{0}} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} \\ = \sum_{s = 1}^{s_{0} - 3} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} + \sum_{s = s_{0} - 2}^{s_{0} - 1} \sum_{j = m_{s} + 1}^{m_{s + 1}} {(A_{n}^{-} f)}_{j}^{q} u_{j}^{q} = {J^{'}}_{1} + {J^{'}}_{2} . \end{gathered}$

(43)

By estimating $J_{1}^{'}$ and $J_{2}^{'}$ 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 $C \approx A_{n} .$ 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

O_{n}^{+}, n \geq 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

lim_{k \to \infty} {(B_{p, q}^{+})}_{k} = 0,

(44)

lim_{k \to \infty} {(B_{p, q}^{-})}_{k} = 0 .

(45)

Theorem 5.2. Let 1 < p ≤ q < ∞. Let the matrix (a_i,j ) of (2) belong to the class

O_{m}^{-}, m \geq 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

lim_{k \to \infty} {(A_{p, q}^{+})}_{k} = 0,

(46)

lim_{k \to \infty} {(A_{p, q}^{-})}_{k} = 0 .

(47)

Now we give the proof of compactness for the class $O_{n}^{+}, n \geq 0 .$

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

{(B_{p, q}^{+})}_{k} \approx {(B_{n})}_{k} \equiv max_{0 \leq γ \leq n} {(B_{γ, n})}_{k} \approx {(B_{p, q}^{-})}_{k},

(48)

where

{(B_{γ, n})}_{k} = {(\sum_{i = k}^{\infty} {(b_{i, k}^{n, γ})}^{q} u_{i}^{q})}^{\frac{1}{q}} {(\sum_{j = 1}^{k} {(a_{k, j}^{(γ)})}^{p^{'}} v_{j}^{- p^{'}})}^{\frac{1}{p^{'}}} .

The equivalence directly follows from (10).

Necessity. Suppose that the matrix of operator (1) belongs to the class $O_{n}^{+}, n \geq 0 .$ Let the operator (1) be compact from l_p,v into l_q,u .

For r ≥ 1, we introduce the following sequence:

φ_{r} = {φ_{r, j}}_{j = 1}^{\infty} : φ_{r, j} = \frac{f_{r, j}}{| | f_{r} | |_{p, v}},

where $f_{r} = {f_{r, j}}_{j = 1}^{\infty} : f_{r, j} = \{\begin{matrix} {(a_{r, j}^{(γ)})}^{p^{'} - 1} v_{j}^{- p^{'}}, 1 \leq j \leq r, \\ 0, j > r . \end{matrix}$

It is obvious that

{φ_{r}}_{p, v} = 1 .

Since the operator (1) is compact from l_p,v into l_q,u , the set

{u A^{+} φ, {∥ φ ∥}_{p, v} = 1}

is precompact in l_q . Hence from criteria on precompactness of the sets in l_p (see Theorem B) we conclude that

lim_{r \to \infty} sup_{| | φ | |_{p, v} = 1} {(\sum_{i = r}^{\infty} u_{i}^{q} {(A^{+} φ)}_{i}^{q})}^{\frac{1}{q}} = 0 .

(49)

Moreover, by using (9) we have that

\begin{gathered} sup_{| | φ | |_{p, v} = 1} {(\sum_{i = r}^{\infty} u_{i}^{q} {(A^{+} φ)}_{i}^{q})}^{\frac{1}{q}} \geq {(\sum_{i = r}^{\infty} u_{i}^{q} {(A^{+} φ_{r})}_{i}^{q})}^{\frac{1}{q}} \\ = {(\sum_{i = r}^{\infty} u_{i}^{q} {(\sum_{j = 1}^{i} a_{i, j}^{(n)} \frac{f_{r, j}}{| | f_{r} | |_{p, v}})}^{q})}^{\frac{1}{q}} \geq {(\sum_{i = r}^{\infty} u_{i}^{q} {(\sum_{j = 1}^{r} a_{i, j}^{(n)} \frac{f_{r, j}}{| | f_{r} | |_{p, v}})}^{q})}^{\frac{1}{q}} \\ = (\sum_{i = r}^{\infty} u_{i}^{q} (\sum_{j = 1}^{r} a_{i, j}^{(})) \end{gathered}