����������� ������ 54 (2002), 93–110

UDK 517.521

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MIXED NORM SPACES OF DIFFERENCE SEQUENCES AND MATRIX TRANSFORMATIONS

A. M. Jarrah and E. Malkowsky

Abstract. In this paper, we generalise the definition of mixed norm spaces, define mixed norm spaces of difference sequences, determine their β-duals, and characterise matrix transformations on them. We obtain many known results as special cases.

1. Introduction

Let 1 ≤ p < ∞. By ω we denote the set of all complex sequences x = (x_{k})_{k=1}^{∞}. In 1968, Maddox [5] introduced and studied the sets

w_{0}^{p} = \{x ∈ ω : lim_{n→∞} \frac{1}{n} ∑_{k=1}^{n} |x_{k}|^{p} = 0\} and w_{∞}^{p} = \{x ∈ ω : sup_{n} \frac{1}{n} ∑_{k=1}^{n} |x_{k}|^{p} < ∞\}

of sequences that are strongly summable and bounded, respectively, with index p by the Cesàro method of order 1. He also observed that the sections 1/n ∑_{k=1}^{n} can be replaced by the blocks 1/2^{ν+1} ∑_{k=2^{ν}}^{2^{ν+1}−1}, and that the section and block norms

∥x∥ = sup_{n}(\frac{1}{n} ∑_{k=1}^{n} |x_{k}|^{p})^{1/p} and ∥x∥′ = sup_{ν≥0}(\frac{1}{2^{ν+1}} ∑_{k=2^{ν}}^{2^{ν+1}−1} |x_{k}|^{p})^{1/p}

are equivalent.

In 1974, Jagers [3] studied the Cesàro sequence spaces

ces(p) = \{x ∈ ω : ∑_{n=1}^{∞}(\frac{1}{n} ∑_{k=1}^{n} |x_{k}|)^{p} < ∞\}

which are Banach spaces with the norm

∥x∥_{ces(p)} = ( ∑_{n=1}^{∞}(\frac{1}{n} ∑_{k=1}^{n} |x_{k}|)^{p})^{1/p}.

AMS Subject Classification: 40 H 05, 46 A 45

Keywords and phrases: Mixed norm spaces, difference sequences, matrix transformations.

Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niška banja, Yugoslavia, October, 2–6, 2002.

Work of the second author supported by the DAAD foundation (German Academic Exchange Service) and the Serbian Ministry of Science, Technology and Development, Research Project # 1232, Operator Equations, Approximations and Applications.

It can be found in [1] that an equivalent norm on ces(p) is

∥x∥ = ( ∑_{ν=0}^{∞} 2^{ν(1−p)}( ∑_{k=2^{ν}}^{2^{ν+1}−1} |x_{k}|)^{p})^{1/p}.

In 1969, Hedlund [2] introduced the mixed norm spaces

ℓ(p, q) = {x ∈ ω : ∑_{ν=0}^{∞}( ∑_{k=2^{ν}}^{2^{ν+1}−1} |x_{k}|^{p})^{q/p} < ∞} see also Kellogg [4];

obviously the Cesàro sequence spaces ces(p) are weighted ℓ(p, 1) mixed norm spaces. Results on the equivalence of block and section norms on mixed norm spaces can also be found in [1].

In this paper, we generalise the definition of mixed norm spaces, define mixed norm spaces of difference sequences, determine their β-duals, and characterise matrix transformations on them. We obtain many known results as special cases.

2. Notations and Definitions

Let ℓ_{∞}, c, c_{0} and ϕ be the sets of all bounded, convergent, null and finite sequences, cs and bs be the sets of all convergent and bounded series, and ℓ_{p} = {x ∈ ω : ∑_{k=1}^{∞}|x_{k}|^{p} < ∞} for 1 ≤ p < ∞.

By e and e^{(n)} (n = 1, 2,…), we denote the sequences with e_{k} = 1 for all k, and e_{n}^{(n)} = 1 and e_{k}^{(n)} = 0 for k ≠ n.

An FK space X is a complete linear metric sequence space with continuous coordinates P_{k} : X → ℂ where P_{k}(x) = x_{k} for all x ∈ X and k = 1, 2,…; a BK space is a normed FK space. We say that an FK space X ⊃ ϕ has AK if x^{[m]} = ∑_{k=1}^{m} x_{k}e^{(k)} → x (m → ∞) for every sequence x = (x_{k})_{k=1}^{∞} ∈ X; x^{[m]} is called the m-section of the sequence x.

If X and Y are subsets of ω, and z is a sequence, we write z^{−1} ∗ Y = {x ∈ ω : xz = (x_{k}z_{k})_{k=1}^{∞} ∈ Y } and M(X, Y) = ⋂_{x∈X} x^{−1} ∗ Y = {z ∈ ω : zx ∈ Y for all x ∈ X} for the multiplier of X and Y. In the special cases when Y = ℓ_{1} or Y = cs, we write z^{α} = z^{−1} ∗ ℓ_{1} or z^{β} = z^{−1} ∗ cs, and the sets X^{α} = M(X, ℓ_{1}) and X^{β} = M(X, cs) are called the α- or Köthe-Toeplitz- and β-duals of X.

Let A = (a_{nk})_{n,k=1}^{∞} be an infinite matrix of complex numbers, x be a sequence and X be a subset of ω. Then we write A_{n} = (a_{nk})_{k=1}^{∞} and A^{k} = (a_{nk})_{n=1}^{∞} for the sequences in the n-th row and the k-th column of A, respectively, A^{T} for the transpose of A, A_{n}(x) = ∑_{k=1}^{∞}a_{nk}x_{k} (n = 1, 2,…) and A(x) = (A_{n}(x))_{n=1}^{∞}, provided A_{n} ∈ x^{β} for all n. The set X_{A} = {z ∈ ω : A(z) ∈ X} is called the matrix domain of A in X. Given any subsets X and Y of ω, then (X, Y) denotes the class of all matrices A that map X into Y, that is for which A_{n} ∈ X^{β} for all n and A(x) ∈ Y for all x ∈ X, or equivalently A ∈ (X, Y) if and only if X ⊂ Y_{A}.

Throughout, let (k(ν))_{ν=0}^{∞} be a strictly increasing sequence of integers with k(0) = 1 and I_{ν} be the set of all integers k with k(ν) ≤ k ≤ k(ν + 1) − 1 (ν = 0, 1,…). Given any sequence x, then, for each ν = 0, 1,…, x^{⟨ν⟩} = _{∑k∈I_{ν}} x_{k}e^{(k)} is the ν-block of the sequence x. Let X, Y ⊃ ϕ be sequence spaces, normed with ∥ · ∥_{X} and ∥ · ∥_{Y}. We define the generalised mixed norm spaces

Z = [Y, X]^{⟨k(ν)⟩} = {z ∈ ω : ( ∥z^{⟨ν⟩}∥_{X} )_{ν=0}^{∞} ∈ Y }

and put

g(z) = ∥ ( ∥z^{⟨ν⟩}∥_{X} )_{ν=0}^{∞} ∥_{Y} (z ∈ Z).  (2.1)

Since ϕ ⊂ X, ∥z^{⟨ν⟩}∥_{X} is defined for every z ∈ ω and for all ν = 0, 1,…. Hence the sequence y = (y_{ν})_{ν=0}^{∞} with y_{ν} = ∥z^{⟨ν⟩}∥_{X} (ν = 0, 1,…) is defined. Furthermore, since ϕ ⊂ X, Y, we obviously have ϕ ⊂ Z.

Finally, let Δ = (δ_{nk})_{n,k=1}^{∞} be the matrix with δ_{nn} = 1, δ_{n,n−1} = −1 and δ_{nk} = 0 otherwise. Then we define the mixed norm spaces of difference sequences

Z_{Δ} = ([Y, X]^{⟨k(ν)⟩})_{Δ} .

We consider a few special cases.

Example 2.1. (a) Let 1 ≤ p < ∞ and 1 ≤ r < ∞. Then we obtain

[ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩} = {z ∈ ω : ∑_{ν=0}^{∞}( ∑_{k∈I_{ν}} |z_{k}|^{p})^{r/p} < ∞},
[l_{r}, ℓ_{∞}]^{⟨k(ν)⟩} = {z ∈ ω : ∑_{ν=0}^{∞}(max_{k∈I_{ν}} |z_{k}|)^{r} < ∞},
[c_{0}, ℓ_{p}]^{⟨k(ν)⟩} = {z ∈ ω : lim_{ν→∞} ∑_{k∈I_{ν}} |z_{k}|^{p} = 0} and
[ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩} = {z ∈ ω : sup_{ν≥0} ∑_{k∈I_{ν}} |z_{k}|^{p} < ∞}.

In the special case of r = p and 1 ≤ p ≤ ∞, we have [ℓ_{p}, ℓ_{p}]^{⟨k(ν)⟩} = ℓ_{p}.

If k(ν) = 2^{ν} for ν = 0, 1,…, then [ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩} = ℓ(r, p), the mixed norm spaces in [2, 4].

If k(ν) = ν + 1 for ν = 0, 1,…, then we also obtain the classical sequence spaces [ℓ_{r}, ℓ_{1}]^{⟨k(ν)⟩} = ℓ_{r}, [c_{0}, ℓ_{1}]^{⟨k(ν)⟩} = c_{0} and [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩} = ℓ_{∞}.

(b) Let 1 ≤ p < ∞ and k(ν) = 2^{ν} for all ν. If d_{ν} = (1/k(ν + 1))^{1/p} for ν = 0, 1,… then

[d^{−1} ∗ c_{0}, ℓ_{p}]^{⟨k(ν)⟩} = w_{0}^{p} and [d^{−1} ∗ ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩} = w_{∞}^{p} [5].

If d_{ν} = 2^{ν(1/p−1)} for ν = 0, 1,… then we obtain the Cesàro sequence spaces or weighted mixed norm spaces [d^{−1} ∗ ℓ_{p}, ℓ_{1}]^{⟨k(ν)⟩} = ces(p) [3].

Example 2.2. (a) Let 1 ≤ p < ∞ and 1 ≤ r < ∞. Then we obtain

([ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ} = {z ∈ ω : ∑_{ν=0}^{∞}( ∑_{k∈I_{ν}} |z_{k} − z_{k−1}|^{p})^{r/p} < ∞} etc.

If k(ν) = ν + 1 for ν = 0, 1,… then we obtain the sets of sequences of bounded variation

bv^{p} = ([ℓ_{p}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ} = {z ∈ ω : ∑_{ν=0}^{∞} |z_{ν+1} − z_{ν}|^{p} < ∞} [12],

and the sets of difference sequences that are convergent to zero or bounded ([c_{0}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ} = (c_{0})_{Δ} = c_{0}(Δ) and ([ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ} = (ℓ_{∞})_{Δ} = ℓ_{∞}(Δ) [9].

(b) Let (μ_{k})_{k=0}^{∞} be an increasing sequence of positive reals tending to infinity and d_{ν} = 1/μ_{k(ν+1)} for ν = 0, 1,…. Then we obtain the sets of sequences that are μ-strongly convergent to zero or bounded, respectively, with index p

c_{0}(μ) = μ^{−1} ∗ ([d^{−1} ∗ c_{0}, ℓ_{p}]^{<k(ν)>})_{Δ}

= \{z ∈ ω : lim_{ν→∞} \frac{1}{μ_{k(ν+1)}^{p}} ∑_{k∈I_{ν}} |μ_{k}z_{k} − μ_{k−1}z_{k−1}|^{p} = 0\}

and c_{∞}(μ) = μ^{−1} ∗ ([d^{−1} ∗ ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ} [10].

3. The topological properties of the spaces Z and Z_{Δ}

Here we study the topological properties of Z = [Y, X]^{⟨k(ν)⟩} and Z_{Δ} = ([Y, X]^{⟨k(ν)⟩})_{Δ}.

A norm ∥ · ∥ on a sequence space X is said to be monotonous if |x_{k}| ≤ |x̃_{k}| (k = 1, 2,…) for x, x̃ ∈ X implies ∥x∥ ≤ ∥x̃∥. A subset X of ω is called normal if x ∈ X and |y_{k}| ≤ |x_{k}| (k = 1, 2,…) for a sequence y together imply y ∈ X.

Given z ∈ ω, we write y = (y_{ν})_{ν=0}^{∞} for the sequence with y_{ν} = ∥z^{⟨ν⟩}∥_{X} (ν = 0, 1,…).

Proposition 3.1. Let X ⊃ ϕ and Y ⊃ ϕ be normed sequence spaces and Z = [Y, X]^{⟨k(ν)⟩}.

(a) If Y is normal and ∥ · ∥_{X} is monotonous then Z is normal.

(b) If ∥ · ∥_{Y} is monotonous then Z is normed with respect to g defined in (2.1). If, however, ∥·∥_{Y} is not monotonous, then g does not satisfy the triangle inequality in general.

Proof. (a) If z ∈ Z and z̃ ∈ ω with |z̃_{k}| ≤ |z_{k}| for all k, then the monotony of ∥ · ∥_{X} implies |ỹ_{ν}| ≤ |y_{ν}| for all ν. Since Y is normal, it follows that z̃ ∈ Z.

(b) We show that g satisfies the triangle inequality, since it obviously satisfies the other properties of a norm. Let z, z̃ ∈ Z. Then ∥(z + z̃)^{<ν>}∥_{X} = ∥z^{⟨ν⟩} + z̃^{<ν>}∥_{X} ≤ ∥z^{⟨ν⟩}∥_{X} + ∥z̃^{<ν>}∥_{X} = y_{ν} + ỹ_{ν} (ν = 0, 1,…), so z + z̃ ∈ Z, since Y is normal. Furthermore, by the monotony of ∥ · ∥_{Y}, we have g(z + z̃) ≤ ∥y + ỹ∥_{Y} ≤ ∥y∥_{Y} + ∥ỹ∥_{Y} = g(z) + g(z̃).

To prove the last part, we choose Y = (ℓ_{1})_{Δ}, ∥y∥_{bv} = ∥Δ(y)∥_{1}, k(ν) = ν + 1 (ν = 0, 1,…) and X = ℓ_{1} with its natural norm. Then obviously ∥ · ∥_{Y} is not monotonous. If we choose z = e^{(1)} + e^{(2)} + e^{(3)} and z̃ = e^{(1)} − e^{(2)} + e^{(3)} then g(z + z̃) = 2g(e^{(1)} + e^{(3)}) = 8 > 4 = g(z) + g(z̃).

Theorem 3.2. Let X ⊃ ϕ be a normed sequence space, Y ⊃ ϕ be a normal BK space and ∥ · ∥_{Y} be monotonous. Then Z is a BK space with ∥ · ∥_{Z} = g where g is defined in (2.1). Furthermore, if Y has AK and ∥ · ∥_{X} is monotonous then Z also has AK.

Proof. By Proposition 3.1, ∥ · ∥_{Z} = g is a norm. We write ∥ · ∥ = ∥ · ∥_{Z} for short. First, since Y is a BK space, ∥z^{(m)} −z∥ → 0 (m → ∞) implies ∥(z^{(m)})^{<ν>} − z^{⟨ν⟩}∥_{X} → 0 (m → ∞) for each ν, and it follows that |z_{k}^{(m)} − z_{k}| → 0 (m → ∞) for each k ∈ I_{ν} (ν = 0, 1,…), since for each ν there are only finitely many k ∈ I_{ν}. Thus the norm ∥ · ∥ is stronger than the metric of ω on Z.

To show that Z is complete with ∥ · ∥, let (z^{(m)})_{m=1}^{∞} be a Cauchy sequence in Z, hence in ω by what we have just shown. Thus there exists z ∈ ω such that

z^{(m)} → z (m → ∞) in ω.  (3.1)

Furthermore, by the completeness of Y, there is y ∈ Y such that

y^{(m)} = (∥(z^{(m)})^{<ν>}∥_{X})_{ν=0}^{∞} → y (m → ∞) in Y.  (3.2)

From (3.1), we conclude z_{k}^{(m)} → z_{k} (m → ∞) for each k, hence (z^{(m)})^{<ν>} → z^{⟨ν⟩} (m → ∞) for each ν, and so

y_{ν}^{(m)} = ∥(z^{(m)})^{<ν>}∥_{X} → ∥z^{⟨ν⟩}∥_{X} (m → ∞) or each ν.  (3.3)

Since Y is a BK space, (3.2) implies y_{ν}^{(m)} → y_{ν} (m → ∞) for each ν, and so, by (3.3), y_{ν} = ∥z^{⟨ν⟩}∥_{X} for each ν and y = (∥z^{<ν>}∥_{X})_{ν=0}^{∞} ∈ Y, hence z ∈ Z. This shows that Z is complete.

Finally, let Y have AK and ∥·∥_{X} be monotonous. We show that Z as AK. Let z = (z_{k})_{k=1}^{∞} ∈ Z and ε > 0 be given. For each m ∈ IN let ν_{m} be the uniquely defined integer for which m ∈ I_{ν_{m}}. We define the sequence y = (y_{ν})_{ν=0}^{∞} by y_{ν} = ∥z^{⟨ν⟩}∥_{X} for ν = 0, 1,…, and write y^{[μ]} = ∑_{ν=0}^{μ} y_{ν}e^{(ν)} for μ = 0, 1,…. Since Y has AK, there exists an integer μ_{0} such that ∥y − y^{[μ]}∥_{Y} < ε for all μ ≥ μ_{0}.We choose m_{0} = k(μ_{0} + 1). Let m ≥ m_{0} be given. Then ν_{m} ≥ μ_{0} + 1 and

ỹ_{ν} = ∥(z − z^{[m]})^{⟨ν⟩}∥_{X} = 0 = y_{ν} − y_{ν}^{[ν_{m}−1]} for 0 ≤ ν ≤ ν_{m} − 1
ỹ_{ν_{m}} = ∥(z − z^{[m]})^{⟨ν_{m}⟩}∥_{X} = ∥(0,…, 0, z_{m+1},…)^{⟨ν_{m}⟩}∥_{X} ≤ ∥z^{⟨ν_{m}⟩}∥_{X} = y_{ν_{m}}

since ∥ · ∥_{X} is monotonous, and

ỹ_{ν} = ∥(z − z^{[m]})^{⟨ν⟩}∥_{X} = ∥z^{⟨ν⟩}∥_{X} = y_{ν} for all ν ≥ ν_{m} + 1.

Thus |ỹ_{ν}| ≤ |y_{ν} − y_{ν}^{[ν_{m}−1]}| for all ν, and so

∥ỹ∥_{Y} = ∥(∥(z − z^{[m]})^{<ν>}∥_{X})_{ν=0}^{∞}∥_{Y} ≤ ∥y − y^{[ν_{m}−1]}∥_{Y} < ε,

since ∥ · ∥_{Y} is monotonous. Therefore z^{[m]} → z (m → ∞).

As an immediate consequence of Theorem 3.2 and [14, Theorem 4.3.12, p. 63], we obtain

Corollary 3.3. Let X ⊃ ϕ be a normed sequence space, Y ⊃ ϕ be a normal BK space and ∥·∥_{Y} be monotonous. Then Z_{Δ} is a BK space with ∥z∥_{Δ} = g(Δ(z)) (z ∈ Z_{Δ}) where g is defined in (2.1).

We close this section with a few examples.

Example 3.4. (a) Let 1 ≤ p < ∞ and 1 ≤ r < ∞. Then [ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩} and [c_{0}, ℓ_{p}]^{⟨k(ν)⟩} are BK spaces with AK with

∥z∥_{(r,p)} = ( ∑_{ν=0}^{∞}( ∑_{k∈I_{ν}} |z_{k}|^{p})^{r/p})^{1/r} and ∥z∥_{(∞,p)} = sup_{ν≥0}( ∑_{k∈I_{ν}} |z_{k}|^{p})^{1/p},

and [ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩} is a BK space with ∥ · ∥_{(∞,p)}; moreover, [c_{0}, ℓ_{p}]^{⟨k(ν)⟩} is a closed subspace of [ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩} by [14, Corollary 4.2.4, p. 56]. The spaces [l_{r}, ℓ_{∞}]^{⟨k(ν)⟩} are BK spaces with AK with

∥z∥_{(r,∞)} = ( ∑_{ν=0}^{∞}(max_{k∈I_{ν}} |z_{k}|)^{r})^{1/r}.

(b) Let 1 ≤ p < ∞ and the sequences (k(ν))_{ν=0}^{∞} and d = (d_{ν})_{ν=0}^{∞} be defined as in Example 2.1(b). Since c_{0} and ℓ_{∞} are BK spaces and c_{0} has AK, and since d_{ν} ≠ 0 for all ν, the sets Y_{0} = d^{−1} ∗ c_{0} and Y_{∞} = d^{−1} ∗ ℓ_{∞} are BK spaces with ∥y∥_{Y_{∞}} = ∥yd∥_{∞}, and Y_{0} has AK (cf. [14, Theorems 4.3.6 and 4.3.12, pp. 62 and 63]. Furthermore, obviously ∥ · ∥_{Y_{∞}} and ∥ · ∥_{p} are monotonous. Therefore w_{0}^{p} and w_{∞}^{p} are BK spaces with

∥z∥′ = sup_{ν=0}(\frac{1}{2^{ν+1}} ∑_{k=2^{ν}}^{2^{ν+1}−1} |x_{k}|^{p})^{1/p},

and w_{0}^{p} has AK; moreover w_{0}^{p} is a closed subspace of w_{∞}^{p} by [14, Corollary 4.2.4, p. 56].

Example 3.5. (a) Let 1 ≤ p < ∞ and 1 ≤ r < ∞. Then ([ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ}, ([c_{0}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ} and ([ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ} are BK spaces with

∥z∥_{(r,p)_{Δ}} = ( ∑_{ν=0}^{∞}( ∑_{k∈I_{ν}} |z_{k} − z_{k−1}|^{p})^{r/p})^{1/r}  and

∥z∥_{(∞,p)_{Δ}} = sup_{ν≥0}( ∑_{k∈I_{ν}} |z_{k} − z_{k−1}|^{p})^{1/p},

and ([c_{0}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ} is a closed subspace of ([ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ} by Example 3.4(a) and [14, Theorem 4.3.14, p. 64].

(b) Let the sequences (μ_{k})_{k=0}^{∞} and d = (d_{ν})_{ν=0}^{∞} be as in Example 2.2(b). Then, by a similar argument as that used in Example 3.4(b), c_{0}(μ) and c_{∞}(μ) are BK spaces with

∥z∥_{c_{∞}(μ)} = sup_{ν≥0} \frac{1}{μ_{k(ν+1)}} ( ∑_{k∈I_{ν}} |μ_{k}x_{k} − μ_{k−1}x_{k−1}|^{p})^{1/p},

and c_{0}^{p}(μ) is a closed subspace of c_{∞}^{p}(μ) by Example 3.4 and [14, Theorem 4.3.14, p. 64].

4. The β-duals of the spaces Z and matrix transformations

In this section, we determine the β-duals of the spaces Z and characterise some classes of matrix transformations between them.

We denote the closed unit ball in a normed space X by B_{X} = {x ∈ X : ∥x∥ ≤ 1}. If X is a normed sequence space and a ∈ ω, we write ∥a∥_{X,α} = sup_{x∈B_{X}} ∑_{k=0}^{∞}|a_{k}x_{k}| and ∥a∥_{X,β} = sup_{x∈B_{X}} |∑_{k=0}^{∞}a_{k}x_{k}| provided the expressions exist and are finite which is the case whenever X is a BK space and a ∈ X^{α} or a ∈ X^{β} (cf. [14, Theorems 4.3.15 and 7.2.9, pp. 64 and 107].

A norm on a sequence space X is said to be KB if the set P = {P_{k} : X → ℂ : P_{k}(x) = x_{k} (x ∈ X) k = 1, 2,… } of coordinates is equicontinuous, that is if there is a constant K such that |x_{k}| ≤ K∥x∥ (k = 1, 2,…) for all x ∈ X. If X is a Banach sequence space with a norm which is KB then it is obviously a BK space. Conversely the norm of a BK space need not be KB in general. To see this, we choose X = (ℓ_{∞})_{Δ} with ∥x∥ = sup_{k} |x_{k} − x_{k−1}|, a BK space, and the sequence x with x_{k} = k for k = 1, 2,….

If X is a normed sequence space then we write X^{δ} = {a ∈ ω : ∥a∥_{X,α} < ∞}.

Theorem 4.1. Let X and Y be normed sequence spaces with X, Y ⊃ ϕ and ∥ · ∥_{Y} be monotonous.

(a) Then [Y^{δ}, X^{δ}]^{⟨k(ν)⟩} ⊂ ([Y, X]^{⟨k(ν)⟩})^{δ}.

(b) If, in addition, the norms ∥·∥_{X} and ∥·∥_{Y} are both KB, ∥·∥_{X} is monotonous and Y is normal then ([Y, X]^{⟨k(ν)⟩})^{δ} ⊂ [Y^{δ}, X^{δ}]^{⟨k(ν)⟩}.

Proof. We write Z = [Y, X]^{⟨k(ν)⟩} and W = [Y^{δ}, X^{δ}]^{⟨k(ν)⟩}. Since ∥ · ∥_{X,α} and ∥ · ∥_{Y,α} are norms on X^{δ} and Y^{δ}, respectively, and ϕ ⊂ X, Y, the set W = {w ∈ ω : (∥w^{⟨ν⟩}∥_{X,α})_{ν=0}^{∞} ∈ Y^{δ}} is defined.

(a) First we observe that Z is a normed space with ∥ · ∥ = g by Proposition 3.1. Let a ∈ W and z ∈ B_{Z}. Then z^{⟨ν⟩} ∈ X for ν = 0, 1,…, and, by the definition of the norm ∥ · ∥_{X,α}, we have

∑_{k=1}^{∞} |a_{k}^{<ν>}z_{k}^{<ν>}| ≤ ∥a^{⟨ν⟩}∥_{X,α} ∥z^{⟨ν⟩}∥_{X} for all ν = 0, 1,….  (4.1)

We define the sequences y and b by y_{ν} = ∥z^{⟨ν⟩}∥_{X} and b_{ν} = ∥a^{⟨ν⟩}∥_{X,α} (ν = 0, 1,…). Then y ∈ B_{Y} and b ∈ Y^{δ}, and it follows from (4.1) that ∑_{k=1}^{∞}|a_{k}z_{k}| = ∑_{ν=0}^{∞}∑_{k=1}^{∞}|a_{k}^{<ν>}z_{k}^{<ν>}| ≤ ∑_{ν=0}^{∞}|b_{ν}y_{ν}| ≤ ∥b∥_{Y,α} by the definition of the norm ∥ · ∥_{Y,α}. Therefore ∥a∥_{Z,α} = sup_{z∈B_{Z}} ∑_{k=1}^{∞}|a_{k}z_{k}| ≤ ∥b∥_{Y,α} < ∞, that is a ∈ Z^{δ}.

(b) First we observe that Z is a BK space by Theorem 3.2. Let a ∈ Z^{δ} be given. Then

∑_{k=1}^{∞} |a_{k}z_{k}| ≤ ∥a∥_{Z,α} = K_{1} < ∞ for all z ∈ B_{Z}.  (4.2)

We have to show a ∈ W, that is

sup_{y∈B_{Y}} ∑_{ν=0}^{∞} ∥a^{⟨ν⟩}∥_{X,α} |y_{ν}| < ∞.  (4.3)

We note that ∥a^{⟨ν⟩}∥_{X,α} is defined for each ν. For if x ∈ B_{X} is given then ∑_{k=1}^{∞}|a_{k}^{<ν>}x_{k}| = _{∑k∈I_{ν}} |a_{k}^{<ν>}x_{k}|, and since ∥ · ∥_{X} is KB, there is a constant K_{2} such that

∑_{k=1}^{∞} |a^{<ν>}x_{k}| ≤ K_{2} ∑_{k∈I_{ν}} |a_{k}^{<ν>}| ∥x∥_{X} ≤ K_{2} ∑_{k∈I_{ν}} |a_{k}^{<ν>}|,

hence ∥a^{⟨ν⟩}∥_{X,α} = sup_{x∈B_{X}} ∑_{k=1}^{∞}|a_{k}^{<ν>}x_{k}| ≤ K_{2}_{∑k∈I_{ν}} |a_{k}^{<ν>}| < ∞ for all ν. Now let y ∈ B_{Y} be given. By the definition of ∥ · ∥_{X,α}, for every ν, we can choose a sequence x(ν) = (x_{k}(ν))_{k=1}^{∞} ∈ B_{X} such that ∥a^{⟨ν⟩}∥_{X,α} ≤ ∑_{k=1}^{∞}|a_{k}^{<ν>}x_{k}(ν)| + 2^{−(ν+1)}, whence

∑_{ν=0}^{∞} ∥a^{⟨ν⟩}∥_{X,α} |y_{ν}| ≤ ∑_{ν=0}^{∞}( ∑_{k=1}^{∞} |a_{k}^{<ν>}x_{k}(ν)y_{ν}| + \frac{1}{2^{ν+1}} |y_{ν}|).  (4.4)

Since ∥ · ∥_{Y} is KB, there is a constant K_{3} such that |y_{ν}| ≤ K_{3}∥y∥_{Y} ≤ K_{3} for all ν = 0, 1,…, and it follows from (4.4) that

∑_{ν=0}^{∞} ∥a^{⟨ν⟩}∥_{X,α} |y_{ν}| ≤ ∑_{ν=0}^{∞}( ∑_{k=1}^{∞} |a_{k}^{<ν>}x_{k}(ν)y_{ν}|) + K_{3}.  (4.5)

We define the sequence z by z_{k} = x_{k}(ν)y_{ν} (k ∈ I_{ν}; ν = 0, 1,…). Then ∥z^{⟨ν⟩}∥_{X} = |y_{ν}|∥(x(ν))^{<ν>}∥_{X} for all ν = 0, 1,…. Since, for each ν, we have |(x_{k}(ν))^{<ν>}| ≤ |x_{k}(ν)| (k = 1, 2,…), the monotony of ∥·∥_{X} implies ∥(x(ν))^{<ν>}∥_{X} ≤ ∥x(ν)∥_{X} = 1 (ν = 0, 1,…), hence ∥z^{⟨ν⟩}∥_{X} ≤ |y_{ν}| (ν = 0, 1,…). Since Y is normal, this implies (∥z^{⟨ν⟩}∥_{X})_{ν=0}^{∞} ∈ Y, that is z ∈ Z. Furthermore, | |y_{ν}| | ≤ |y_{ν}| for ν = 0, 1,… implies (|y_{ν}|)_{ν=0}^{∞} ∈ Y, since Y is normal, and the monotony of ∥ · ∥_{Y} yields ∥z∥_{Z} ≤ ∥(|y_{ν}|)_{ν=0}^{∞}∥_{Y} ≤ ∥y∥_{Y}. Now (4.5) and (4.2) together imply

∑_{ν=0}^{∞} ∥a^{⟨ν⟩}∥_{X,α} |y_{ν}| ≤ ∑_{ν=0}^{∞} ∑_{k∈I_{ν}} |a_{k}^{<ν>}z_{k}|+K_{3} = ∑_{k=1}^{∞} |a_{k}z_{k}| ≤ K_{1}∥z∥_{Z} +K_{3} ≤ K_{1}+K_{3}.

Since y ∈ B_{Y} was arbitrary, condition (4.3) follows.

If X is a BK space then X^{α} = X^{δ} by [14, Theorem 4.3.15, p. 64], and if X is normal then X^{α} = X^{β}. Therefore we obtain from Proposition 3.1 and Theorems 3.2 and 4.1

Corollary 4.2. Let X be a normed sequence space, Y be a normal BK space and the norms ∥ · ∥_{X} and ∥ · ∥_{Y} be monotonous and KB. Then Z^{α} = ([Y, X]^{⟨k(ν)⟩})^{α} = [Y^{α}, X^{α}]^{⟨k(ν)⟩}. If, in addition, X is normal then Z^{β} = [Y^{β}, X^{β}]^{⟨k(ν)⟩}.

Example 4.3. (a) Let 1 ≤ p < ∞, 1 ≤ r < ∞, q and s be the conjugate numbers of p and r, that is q = ∞ for p = 1 and q = p/(p − 1) for 1 < p < ∞ and s defined similarly. Since the norms ∥ · ∥_{ℓ_{p},β} and ∥ · ∥_{q} and ∥ · ∥_{ℓ_{∞},α} and ∥ · ∥_{1} are equivalent on ℓ_{p}^{β} and on ℓ_{∞}^{β} = c_{0}^{β}, we have ([ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩})^{β} = [ℓ_{s}, ℓ_{q}]^{⟨k(ν)⟩} and ([c_{0}, ℓ_{p}]^{⟨k(ν)⟩})^{β} = ([ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩})^{β} = [ℓ_{1}, ℓ_{q}]^{⟨k(ν)⟩}.

(b) Let U denote the set of all sequences u with u_{k} ≠ 0 for all k. If u ∈ U then we write 1/u = (1/u_{k})_{k=1}^{∞}, and it is obvious that (u^{−1} ∗ X)^{β} = (1/u)^{−1} ∗ X^{β} for arbitrary subsets X of ω. Let the sequences k(ν) and d be defined as in Example 2.1(b). Then

(w_{0}^{p})^{β} = (w_{∞}^{p})^{β} = M_{p} = \{ \begin{matrix}\{a ∈ ω : ∑_{ν=0}^{∞} 2^{ν+1} max_{k∈Iν} |a_{k}| < ∞\} & (p = 1) \\ \{a ∈ ω : ∑_{ν=0}^{∞} 2^{ν+1}( ∑_{k∈Iν} |a_{k}|^{q})^{1/q} < ∞\} & (1 < p < ∞). \end{matrix}

Now we characterise some classes of matrix transformations between mixed norm spaces.

Let (m(μ))_{μ=0}^{∞} be a strictly increasing sequence of integers with m(0) = 1 and M_{μ} = {m ∈ IN : m(μ) ≤ m ≤ m(μ + 1) − 1} (μ = 0, 1,…). Furthermore, let T denote the set of all sequences (t_{μ})_{μ=0}^{∞} of integers such that for each μ there is one and only one t_{μ} ∈ M_{μ}.

First we give a result that characterises the classes (X, Y) where X is any BK space and Y is any of the spaces ℓ_{∞}, c_{0}, ℓ_{1}, [ℓ_{∞}, ℓ_{1}]^{⟨m(μ)⟩}, [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩} or [c_{0}, ℓ_{1}]^{⟨m(μ)⟩}.

Theorem 4.4. Let X be a BK space, or a BK space with AK in the cases marked ∗. We write sup_{N} for the supremum taken over all finite subsets N of ℕ_{0}. Then the conditions for A ∈ (X, Y) when Y is any of the spaces ℓ_{∞}, c_{0}, ℓ_{1}, [ℓ_{∞}, ℓ_{1}]^{⟨m(μ)⟩}, [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩} or [c_{0}, ℓ_{1}]^{⟨m(μ)⟩} can be read from the table

\begin{matrix}From To & ℓ_{∞} & c_{0} & ℓ_{1} & [ℓ_{∞}, ℓ_{1}]^{⟨m(μ)⟩} & [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩} & [c_{0}, ℓ_{1}]^{⟨m(μ)⟩} \\ X & (1.) & *(2.) & (3.) & (4.) & (5.) & *(6.) \end{matrix}

where

(1.) (1.1)  where (1.1) sup_{n} ∥A_{n}∥_{X,β} < ∞
(2.) (1.1) and (2.1) where (2.1) lim_{n→∞} a_{nk} = 0 for each k
(3.) (3.1)  where (3.1) sup_{N}∥ ∑_{n∈N} A_{n}∥_{X,β} < ∞
(4.) (4.1)  where (4.1) sup_{μ}( max_{M(μ)⊂M_{μ}}∥ ∑_{m∈M(μ)} A_{m}∥_{X,β}) < ∞
(5.) (5.1)  where (5.1) sup_{N}(sup_{t∈T}∥ ∑_{μ∈N} A_{t_{μ}} ∥_{X,β}) < ∞
(6.) (4.1) and (6.1) where (6.1) lim_{μ→∞} ∑_{n∈M_{μ}} |a_{nk}| = 0 for each k.

Proof. (1.) is [11, Theorem 1.23, p. 155], (2.) follows from (1.) and [14, 8.3.6, p. 123], since c_{0} is a closed subspace of ℓ_{∞}, and (3.) is [8, Satz 1].

(4.) We have A ∈ (X, [ℓ_{∞}, ℓ_{1}]^{⟨m(μ)⟩}) if and only if A_{n} ∈ X^{β} for all n and

(∥(A(x))^{⟨m(μ)⟩}∥_{1})_{μ=0}^{∞} ∈ ℓ_{∞} for all x ∈ X.  (4.6)

Since by a well-known inequality [13]

max_{M(μ)⊂M_{μ}}| ∑_{m∈M(μ)} A_{m}(x)| ≤ ∑_{m∈M_{μ}} |A_{m}(x)| = ∥(A(x))^{⟨μ⟩}∥_{1} ≤

≤ 4 · max_{M(μ)⊂M_{μ}}| ∑_{m∈M(μ)} A_{m}(x)| for all μ and all x ∈ X,

it follows by condition (1.1) that (4.6) holds if and only if condition (4.1) is satisfied.

(5.) First we assume that condition (5.1) holds. Then obviously A_{n} ∈ X^{β} for all n. Let x ∈ X be given. For each μ = 0, 1,…, let m_{μ} ∈ M_{μ} be such that |A_{m_{μ}}(x)| = max_{m∈M_{μ}} |A_{m}(x)|. Let μ_{0} be an arbitrary nonnegative integer. Then we have by the definition of the norm ∥ · ∥_{X,β}

∑_{μ=0}^{μ_{0}} ∥(A(x))^{⟨m(μ)⟩}∥_{∞} = ∑_{μ=0}^{μ_{0}} |A_{m_{μ}}(x)| ≤ 4 · sup_{{N⊂ℕ_{0} \atop Nfinite}} | ∑_{μ∈N} A_{m_{μ}}(x)|

≤ 4 · ( sup_{{N⊂ℕ_{0} \atop Nfinite}} ∥ ∑_{μ∈N} A_{m_{μ}} ∥_{X,β}) ∥x∥ ≤ 4 · sup_{{N⊂ℕ_{0} \atop Nfinite}}(sup_{t∈T}∥ ∑_{μ∈N} A_{t_{μ}} ∥_{X,β}) ∥x∥ < ∞.

Since μ_{0} was arbitrary, it follows that (∥(A(x))^{⟨m(μ⟩)}∥_{∞})_{μ=0}^{∞} ∈ ℓ_{1}, that is A(x) ∈ [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩}.

Conversely we assume A ∈ [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩}. Since X and [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩} are BK spaces, the map f_{A} : X → [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩} with f_{A}(x) = A(x) (x ∈ X) is continuous (cf. [14, Theorem 4.2.8, p. 57]. Hence there is a constant K such that

∥f_{A}(x)∥_{(1,∞)} = ∥A(x)∥_{(1,∞)} ≤ K∥x∥ for all x ∈ X.  (4.7)

We observe that A_{m} ∈ X^{β} for all m implies ∑_{μ∈N} A_{t_{μ}} ∈ X^{β} for all finite subsets N of IN_{0} and for all sequences t ∈ T, and so by (4.7), | ∑_{μ∈N} A_{t_{μ}}(x)| ≤ ∑_{μ=0}^{∞} |A_{t_{μ}}(x)| ≤ ∥f_{A}(x)∥_{(1,∞)} ≤ K∥x∥. Now condition (5.1) follows from the definition of the norm ∥ · ∥_{X,β}.

(6.) By Example 3.4(a), [c_{0}, ℓ_{1}]^{⟨m(μ)⟩} is a closed subspace of [ℓ_{∞}, ℓ_{1}]^{⟨m(μ)⟩}. Thus (6.) is an immediate consequence of (4.) and [14, 8.3.6, p. 123].

We obtain as an immediate consequence of Example 4.3 and Theorem 4.4

Corollary 4.5. Let 1 < r < ∞ and 1 < p ≤ ∞ and s and q be the conjugate numbers of r and p. Then the conditions for A ∈ ([ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩}, Y) where Y is any of the spaces in Theorem 4.4 can be read from the table

\begin{matrix}From To & ℓ_{∞} & c_{0} & ℓ_{1} & [ℓ_{∞}, ℓ_{1}]^{⟨m(μ)⟩} & [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩} & [c_{0}, ℓ_{1}]^{⟨m(μ)⟩} \\ [ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩} & (1.) & (2.) & (3.) & (4.) & (5.) & (6.) \end{matrix}

Mixed norm spaces of difference sequences 103

where

(1.) (1.1)  where (1.1) sup_{n} ∑_{ν=0}^{∞}( ∑_{k∈I_{ν}} |a_{nk}|^{q})^{s/q} < ∞
(2.) (1.1) and (2.1) where (2.1) is (2.1) in Theorem 4.4
(3.) (3.1)  where (3.1) sup_{N} ∑_{ν=0}^{∞}(_{∑k∈I_{ν}} | ∑_{n∈N} a_{nk}|^{q})^{s/q} < ∞
(4.) (4.1)  where (4.1)

sup_{μ}( max_{M(μ)⊂M_{μ}} ∑_{ν=0}^{∞}(_{∑k∈I_{ν}} | ∑_{m∈M(μ)} a_{mk}|^{q})^{s/q}) < ∞

(5.) (5.1)  where (5.1) sup_{N}(sup_{t∈T} ∑_{ν=0}^{∞}(_{∑k∈I_{ν}} | ∑_{μ∈N} a_{t_{μ},k}|^{q})^{s/q}) < ∞
(6.) (4.1) and (6.1) where (6.1) is (6.1) in Theorem 4.4.

If r = 1 or p = 1 replace ∑_{ν=0}^{∞} or _{∑k∈I_{ν}} by sup_{ν≥0} or max_{k∈I_{ν}} in conditions (1.1), (3.1), (4.1) and (5.1) in (1.)–(6.). The conditions for A ∈ ([c_{0}, ℓ_{p}]^{⟨k(ν)⟩}), Y) are those in (1.)–(6.) with s = 1 in (1.1), (3.1), (4.1) and (5.1). Finally ([ℓ_{∞}, ℓ_{p}]^{⟨k(ν)⟩}, Y) = ([c_{0}, ℓ_{p}]^{⟨k(ν)⟩}, Y) for Y ≠ c_{0}, [c_{0}, ℓ_{1}]^{⟨m(μ)⟩}.

Now we give the dual result of Theorem 4.4. We write T′ for the set of all strictly increasing sequences t = (t_{ν})_{ν=0}^{∞} of integers such that for each ν there is one and only one t_{ν} ∈ I_{ν}.

Theorem 4.6. Let W be a BK space with AK and Y = W^{β}. Then the conditions for A ∈ (X, Y) where X is any of the spaces ℓ_{∞}, c_{0}, ℓ_{1}, [ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩}, [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩} or [c_{0}, ℓ_{1}]^{⟨k(ν)⟩} can be read from the table

\begin{matrix}To From & ℓ_{∞} & c_{0} & ℓ_{1} & [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩} & [ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩} & [c_{0}, ℓ_{1}]^{⟨k(ν)⟩} \\ Y & (1.) & (2.) & (3.) & (4.) & (5.) & (6.) \end{matrix}

where

(1.) (1.1) where (1.1) sup_{N} ∥ ∑_{n∈N} A^{n}∥_{Y} < ∞
(2.) (1.1)
(3.) (3.1) where (3.1) sup_{n} ∥A^{n}∥_{Y} < ∞
(4.) (4.1) where (4.1) sup_{N}(sup_{t∈T′} ∥ ∑_{ν∈N} A^{t_{ν}} ∥_{Y} ) < ∞
(5.) (5.1) where (5.1) sup_{N}( max_{K(ν)⊂K_{ν}} ∥ ∑_{m∈K(ν)} A^{m}∥_{Y} ) < ∞
(6.) (4.1).

Proof. Since X is a BK space with AK when X is any of the spaces c_{0}, ℓ_{1}, [ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩} and [c_{0}, ℓ_{1}]^{⟨k(ν)⟩}, we have A ∈ (X, Y) if and only if A^{T} ∈ (W, X^{β}) by [14, Theorem 8.3.9, p. 124], and (2.), (3.), (5.) and (6.) are immediate consequences of Theorem 4.4 (3.), (1.), (4.) and (5.). Furthermore, since c_{0}^{ββ} = ℓ_{∞} and ([c_{0}, ℓ_{1}]^{⟨k(ν)⟩})^{ββ} = [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩}, and (X, Y) = (X^{ββ}, Y) by [14, Theorem 8.3.9, p. 124], (1.) and (4.) follow from (2.) and (6.).

We obtain as an immediate consequence of Theorem 4.6

Corollary 4.7. Let 1 < r < ∞ and 1 < p < ∞. Then the conditions for A ∈ (X, [ℓ_{r}, ℓ_{p}]^{⟨m(μ)⟩}) where X is any of the spaces in Theorem 4.4 can be read from the table

\begin{matrix}To From & ℓ_{∞} & c_{0} & ℓ_{1} & [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩} & [ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩} & [c_{0}, ℓ_{1}]^{⟨k(ν)⟩} \\ [ℓ_{r}, ℓ_{p}]^{⟨m(μ)⟩} & (1.) & (2.) & (3.) & (4.) & (5.) & (6.) \end{matrix}

where

(1.) (1.1) where (1.1) sup_{N} ∑_{μ=0}^{∞}( ∑_{k∈M_{μ}}| ∑_{n∈N} a_{kn}|^{p})^{r/p} < ∞
(2.) (1.1)
(3.) (3.1) where (3.1) sup_{n} ∑_{μ=0}^{∞}( ∑_{k∈M_{μ}} |a_{kn}|^{p})^{r/p} < ∞

(4.) (4.1) where (4.1) sup_{N}(sup_{t∈T′} ∑_{μ=0}^{∞}( ∑_{k∈M_{μ}}| ∑_{ν∈N} a_{k,t_{ν}} |^{p})^{r/p}) < ∞

(5.) (5.1) where (5.1) sup_{N}( max_{k(ν)∈K_{ν}} ∑_{ν=0}^{∞}( ∑_{k∈M_{μ}}| ∑_{m∈K(ν)} a_{km}|^{p})^{r/p}) < ∞
(6.) (4.1).

5. The β-duals of the spaces Z_{Δ} and matrix transformations

In this section, we determine the β-duals of the sets Z_{Δ} and characterise some matrix transformations between them.

First we prove a general result which reduces the determination of (X_{Δ})^{β} for arbitrary BK spaces with AK to that of X^{β} and the characterisation of the class (X, c_{0}).

If X is a normed space, we write X^{∗} its continuous dual, that is the set of all continuous linear functionals f on X with the norm ∥f∥ = sup_{x∈B_{X}} |f(x)|.

Let Σ = (σ_{nk})_{n,k=1}^{∞} be the matrix with σ_{nk} = 1 for 1 ≤ k ≤ n and σ_{nk} = 0 for k > n (n = 1, 2,…). Then x = Δ(Σ(x)) = Σ(Δ(x)) for all x ∈ ω. Let X ⊂ ω and Y = X_{Δ}. Then x ∈ X if and only if y = Σ(x) ∈ Y, and y ∈ Y if and only if x = Δ(y) ∈ X. If X is a BK space then so is Y and B_{X} = B_{Y} by [14, Theorem 4.3.12, p. 63].

Given any sequence a, we write B^{a} for the matrix with the rows B_{n}^{a} = a_{n}e^{[n]} (n = 1, 2,…). Then B_{n}^{a}(x) = a_{n}Σ_{n}(x) = a_{n}y_{n} for all x ∈ X, y = Σ(x) and all n, that is a ∈ M(X_{Δ}, W) if and only if B^{a} ∈ (X, W) for arbitrary subsets X and W of ω.

(5.1)

Theorem 5.1. Let E = Σ^{T}. If X is a BK space with AK then a ∈ (X_{Δ})^{β} if and only if a ∈ (X^{β})_{E} and V^{a} ∈ (X, c_{0}) where V^{a} is the matrix with the rows V_{n}^{a} = E_{n}(a)e^{[n]} (n = 1, 2,…). Furthermore if a ∈ (X_{Δ})^{β} then

∑_{k=1}^{∞} a_{k}y_{k} = ∑_{k=1}^{∞} E_{k}(a)Δ_{k}(y) for all y ∈ X_{Δ}.  (5.2)

Proof. We write Y = X_{Δ} and V = V^{a} for short.

First we assume a ∈ Y^{β}. Then B^{a} ∈ (X, cs) by (5.1), and so C = ΣB^{a} ∈ (X, c) by [11, Theorem 3.8, p. 180]. Since c is a closed subspace of ℓ_{∞}, we have by [14, 8.3.6, p. 123]

lim_{n→∞} c_{nk} = ∑_{j=k}^{∞} a_{j} = E_{k}(a) exists for all k  (5.3)

and

C ∈ (X, ℓ_{∞}).  (5.4)

From (5.3), we obtain that the matrix V is defined and

lim_{n→∞} v_{nk} = lim_{n→∞} ∑_{j=n}^{∞} a_{j} = 0.  (5.5)

We also have

∑_{k=1}^{m−1} a_{k}y_{k} = ∑_{k=1}^{m} E_{k}(a)Δ_{k}(y) − ∑_{k=1}^{m} v_{mk}Δ_{k}(y) for all m and all y.  (5.6)

Since X is a BK space with AK, condition (5.4) implies C^{T} ∈ (ℓ_{1}, X^{β}) by [14, Theorem 8.3.9, p. 124]. Now X^{β} is a BK space with

∥b∥^{β} = sup_{m} sup_{x∈B_{X}} | ∑_{k=1}^{m} b_{k}x_{k}| = sup_{m} ∥b^{[m]}∥_{X,β} (b ∈ X^{β})

by [14, Example 4.3.16, p. 65]. Therefore, by [14, Example 8.4.1, p. 126], the columns of the matrix C^{T}, that is the rows of C are a bounded set in X^{β}. Thus there is a constant K_{1} such that

| ∑_{k=1}^{m} c_{nk}x_{k}| ≤ K_{1} for all m and n and for all x ∈ B_{X}.  (5.7)

Now (5.3) implies | ∑_{k=1}^{m} E_{k}(a)x_{k}| ≤ K_{1} for all m and all x ∈ B_{X}. It follows from this and (5.6) that

|V_{m}(x)| ≤ K_{1} + | ∑_{k=1}^{m−1} a_{k}y_{k}| for all x ∈ B_{X}, y ∈ B_{Y} and all m.  (5.8)

We define the linear functionals f_{m} (m = 1, 2,…) on Y by f_{m}(y) = ∑_{k=1}^{m−1} a_{k}y_{k} (y ∈ Y). We note that f_{m} ∈ Y^{∗} for all m, since Y is a BK space. Furthermore a ∈ Y^{β} implies that f(y) = lim_{m→∞} f_{m}(y) exists for every y ∈ Y, that is the sequence (f_{m})_{m=1}^{∞} is pointwise convergent, hence pointwise bounded, and so uniformly bounded by the uniform boundedness principle. Thus there exists a constant K_{2} such that |f_{m}(y)| = | ∑_{k=1}^{m−1} a_{k}y_{k}| ≤ K_{2} for all y ∈ B_{Y} and all m, and it follows from (5.8) that |V_{m}(x)| ≤ K_{1} + K_{2} for all m and for all x ∈ B_{X}, hence sup_{m} ∥V_{m}∥_{X,β} < ∞. This and (5.5) imply V ∈ (X, c_{0}) by Theorem 4.4(2.); and then (5.6) implies E(a) ∈ X^{β}, that is a ∈ (X^{β})_{E}.

If a ∈ Y^{β} then E(a) ∈ X^{β} and V ∈ (X, c_{0}), as we have just shown, and so (5.2) follows from (5.6).

Conversely, if a ∈ (X^{β}) and V ∈ (X, c_{0}) then a ∈ Y^{β} by (5.6).

Now we give the (Z_{Δ})^{β} in some special cases.

Example 5.2. (a) Let 1 ≤ p ≤ ∞, 1 ≤ r ≤ ∞ and q and s be the conjugate numbers of p and r. The conditions for E(a) ∈ ([ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩})^{β} and E(a) ∈ ([c_{0}, ℓ_{p}]^{⟨k(ν)⟩})^{β} are given in Example 4.3(a). Corollary 4.5 yields the conditions for V^{a} ∈ ([ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩}, c_{0}) and V^{a} ∈ ([c_{0}, ℓ_{p}]^{⟨k(ν)⟩}, c_{0}), the condition lim_{n→∞} v_{nk}^{a} = 0 for each k being redundant. For each positive integer n, let ν(n) denote the uniquely defined integer such that n ∈ I_{ν(n)}. We define the sequence b^{s,q} by

b_{n}^{s,q} = \{ \begin{matrix}( ∑_{ν=0}^{ν(n)−1}((k(ν + 1) − k(ν))^{s/q} + (n + 1 − k(ν(n))^{s/q})^{1/s} & (1 < r ≤ ∞, 1 < p ≤ ∞) \\ (ν(n) + 1)^{1/s} & (1 < r ≤ ∞, p = 1) \\ max\{ max_{0≤ν≤ν(n)−1}(k(ν + 1) − k(ν))^{1/q}, (n + 1 − k(ν(n)))^{1/q}\} & (r = 1, 1 < p ≤ ∞). \end{matrix}

It is easy to see that condition (1.1) for A = V^{a} in Corollary 4.5 is equivalent to E(a) ∈ (b^{s,q})^{−1} ∗ ℓ_{∞}; in the case of [c_{0}, ℓ_{p}]^{⟨k(ν)⟩} (1 ≤ p < ∞), we use the sequence b^{1,q}.

Let us mention that the condition E(a) ∈ (b^{s,q})^{−1} ∗ ℓ_{∞} becomes redundant is some cases. As in Example 2.2, let k(ν) = ν+1 (ν = 0, 1,…). Then, for 1 < p < ∞, we have bv^{p} = ([ℓ_{p}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ}, and a ∈ (bv^{p})^{β} if and only if ∑_{ν=0}^{∞}| ∑_{k=ν}^{∞} a_{k}|^{q} < ∞ and sup_{n}(n + 1)^{1/q} |∑_{k=n}^{∞} a_{k}| < ∞, and it is easy to see that, in general neither condition implies the other. If, however, p = 1, then bv = ([ℓ_{1}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ} = ([ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩})_{Δ}, and the conditions E(a) ∈ [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩} and E(a) ∈ b^{∞,1} ∗ ℓ_{∞} are the same, namely sup_{n} | ∑_{k=n}^{∞} a_{k}| < ∞ that is a ∈ cs.

(b) Let the sequences μ and d be defined as in Example 2.2(b). First we observe that a ∈ (c_{0}^{p}(μ))^{β} if and only if a/u = (a_{k}/u_{k})_{k=1}^{∞} ∈ (([d^{−1} ∗ c_{0}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ})^{β}. Also E(a/u) ∈ [(1/d)^{−1} ∗ ℓ_{1}, ℓ_{q}]^{⟨k(ν)⟩} if and only if

c ∈ [ℓ_{1}, ℓ_{q}]^{⟨k(ν)⟩} where c_{k} = 1/d_{ν}E_{k}(a/μ) = μ_{k(ν+1)} ∑_{j=k}^{∞} \frac{a_{j}}{μ_{j}} (k ∈ I_{ν}; ν = 0, 1,…).

(5.10)

Since obviously, for all u ∈ U and for all X, Y ⊂ ω, we have A ∈ (u^{−1} ∗ X, Y) if and only if B ∈ (X, Y) where b_{nk} = a_{nk}/u_{k} for all n and k, it follows that V^{a} ∈ (μ^{−1} ∗ [d^{−1} ∗ c_{0}, ℓ_{p}]^{⟨k(ν)⟩}, c_{0}) if and only if Ṽ^{a} ∈ ([d^{−1} ∗ c_{0}, ℓ_{p}]^{⟨k(ν)⟩}, c_{0}), where ṽ_{nk}^{a} = E_{n}(a/u) for 1 ≤ k ≤ n and ṽ_{nk}^{a} = 0 for k > n (n = 1, 2,…). Finally, since z ∈ [d^{−1} ∗ c_{0}, ℓ_{p}]^{⟨k(ν)⟩} if and only if y ∈ [c_{0}, ℓ_{p}]^{⟨k(ν)⟩} where y_{k} = d_{ν}z_{k} (k ∈ I_{ν}; ν = 0, 1,…), we have Ṽ^{a} ∈ ([d^{−1} ∗ c_{0}, ℓ_{p}]^{⟨k(ν)⟩}, c_{0}) if and only if W^{a} ∈ ([c_{0}, ℓ_{p}]^{⟨k(ν)⟩}, c_{0}) where w_{nk}^{a} = ṽ_{nk}^{a}1/d_{ν} (k ∈ I_{ν}; ν = 0, 1,…) for all n = 1, 2,…. Again, the condition lim_{n→∞} w_{nk} = 0 is redundant, and we need

sup_{n} ∑_{ν=0}^{∞} ∥(W_{n}^{a})^{⟨ν⟩}∥_{q} < ∞.  (5.11)

We define the sequence b^{1,q}(μ) by

b_{n}^{1,q}(μ) = \{ \begin{matrix}∑_{ν=0}^{ν(n)−1} μ_{k(ν+1)}(k(ν + 1) − k(ν))^{1/q} − μ_{k(ν(n)+1)}(n + 1 − k(ν(n)))^{1/q} & (1 < p < ∞) \\ ∑_{ν=0}^{ν(n)} μ_{k(ν+1)} & (p = 1). \end{matrix}

Condition (5.10) is equivalent to

∑_{ν=0}^{∞} μ_{k(ν+1)}( ∑_{k∈I_{ν}} | ∑_{j=k}^{∞} \frac{a_{j}}{μ_{j}} |^{q})^{1/q} < ∞ (1 < p < ∞),

∑_{ν=0}^{∞} μ_{k(ν+1)} max_{k∈I_{ν}}| ∑_{j=k} \frac{a_{j}}{μ_{j}} | < ∞ (p = 1),

and it is easy to see that condition (5.11) is equivalent to E(a/μ) ∈ (b^{1,q})^{−1} ∗ ℓ_{∞} for 1 < p < ∞ and E(a/u) ∈ (b^{1,∞}(μ))^{−1} ∗ ℓ_{∞} for p = 1, this condition being redundant, if there are reals s and t with 0 < s ≤ μ_{k(ν)}/μ_{k(ν+1)} ≤ t < 1 for all ν.

The next result reduces the the characterisation of (X_{Δ}, Y) to that of (X, Y) and (X, c_{0}).

Theorem 5.3. Let X ⊃ ϕ be a BK space with AK and Y be a subset of ω. Then A ∈ (X_{Δ}, Y) if and only if

E^{A} ∈ (X, Y) where e_{nk}^{A} = ∑_{j=k}^{∞} a_{nj} for all n and k  (5.12)

and

V^{A_{n}} ∈ (X, c_{0}) for all n  (5.13)

where V^{A_{n}} is the matrix with the rows V_{m}^{A_{n}} = E_{m}(A_{n})e^{[m]} (m = 1, 2,…).

Proof. First we assume A ∈ (X_{Δ}, Y). Then A_{n} ∈ (X_{Δ})^{β} for all n, hence condition (5.13) holds and

E(A_{n}) ∈ X^{β} for all n  (5.14)

by Theorem 5.1. Let x ∈ X be given. Then A_{n} ∈ (X_{Δ})^{β} implies

(E^{A})_{n}(x) = A_{n}(Σ(x)) for all n  (5.15)

by (5.2). Since Σ(x) ∈ X_{Δ}, it follows that A(Σ(x)) ∈ Y, hence E^{A}(x) ∈ Y. Thus (5.12) also holds.

Conversely we assume that conditions (5.12) and (5.13) are satisfied. Then (5.14) holds, and this and (5.13) imply A_{n} ∈ (X_{Δ})^{β} for all n by Theorem 5.1. Again (5.15) holds and then A ∈ (X_{Δ}, Y).

Now we give some characterisations of matrix transformations between Z and Z_{Δ}.

We obtain as an immediate consequence of Theorems 5.3 and 4.4 and of [11, Theorem 3.8, p. 180]

Theorem 5.4. Let X be a BK space with AK and Y be any of the spaces ℓ_{∞}, c_{0}, ℓ_{1}, [ℓ_{∞}, ℓ_{1}]^{⟨m(μ)⟩}, [ℓ_{1}, ℓ_{∞}]^{⟨m(μ)⟩} or [c_{0}, ℓ_{1}]^{⟨m(μ)⟩}.

(a) Then A ∈ (X_{Δ}, Y) holds if and only if condition (5.13) holds in addition to the respective conditions in Theorem 4.4 with the A replaced by E^{A}.

(b) Let C = ΔA, that is c_{nk} = a_{nk}−a_{n−1,k} for all n and k. Then A ∈ (X_{Δ}, Y_{Δ}) if and only if condition (5.13) with V^{A_{n}} replaced by V^{C_{n}} holds in addition to the respective conditions of Theorem 4.4 with A replaced by E^{C}.

In particular, we have, applying Corollary 4.5

Corollary 5.5. Let 1 ≤ r < ∞ and 1 ≤ p ≤ ∞, s and q be the conjugate numbers of r and p, and Y be any of the spaces in Theorem 5.4. Finally, let the sequences b^{s,q} be defined as in Example 5.2(a).

(a) Then (A ∈ ([ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ}, Y) if and only if E(A_{n}) ∈ (b^{(s,q)})^{−1} ∗ ℓ_{∞} for all n, and the respective conditions in Corollary 4.5 hold with A replaced by E^{A}. Furthermore, A ∈ (([c_{0}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ}, Y) for 1 ≤ p < ∞ if and only if E(A_{n}) ∈ (b^{1,q})^{−1} ∗ ℓ_{∞} for all n, and the respective conditions in Corollary 4.5 hold with A replaced by E^{A}.

(b) The conditions for A ∈ (([ℓ_{r}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ}, Y_{Δ}) and (([c_{0}, ℓ_{p}]^{⟨k(ν)⟩})_{Δ}, Y_{Δ}) are obtained from the respective ones in Part (a) by replacing A by C throughout.

For the next result, we need to know the β-duals of ℓ_{∞} and [ℓ_{∞}, ℓ_{1}] which cannot be determined by Theorem 5.1, since they do not have AK.

Lemma 5.6. Let E = Σ^{T}. Then

(a) a ∈ ((ℓ_{∞})_{Δ})^{β} if and only if a ∈ (ℓ_{1} ∩ ((n)_{n=1}^{∞})^{−1} ∗ c_{0})_{E};

(b) a ∈ (([ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ})^{β} if and only if a ∈ ([ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩} ∩ (b^{1,∞})^{−1} ∗ c_{0})_{E} where the sequence b^{1,∞} is defined as in Example 5.2(a).

In both parts, if a ∈ ((ℓ_{∞})_{Δ})^{β} or a ∈ (([ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ})^{β} then (5.2) holds.

Proof. (a) This follows from [9, Theorem 2, Corollary 2].

(b) We write X_{∞} = (([ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ} and X_{0} = ([c_{0}, ℓ_{1}]^{⟨k(ν)⟩})_{Δ}, for short.

First X_{0} ⊂ X_{∞} implies X_{∞}^{β} ⊂ X_{0}^{β}, hence X_{∞}^{β} ⊂ ([ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩})_{E} by Example 4.3(a). Now we assume a ∈ X_{∞}^{β}. Since e ∈ X_{∞}, the sequence E(a) is defined. Let y ∈ X_{∞} be given. Then, by (5.6), a ∈ X_{∞}^{β} and E(a) ∈ [ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩} together yield V^{a} ∈ ([ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩}, c), that is E(a) ∈ (X_{∞}, c) by (5.1). Conversely, if a ∈ ([ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩} ∩ M(X_{∞}, c))_{E}, then E(a) ∈ [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩} and V^{a} ∈ (X_{∞}, c), hence a ∈ X_{∞}^{β} by (5.6). Thus we have shown X_{∞}^{β} = ([ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩} ∩ M(X_{∞}, c))_{E}. We will prove

M(X_{∞}, c) = M(X_{∞}, c_{0}) = (b^{1,∞})^{−1} ∗ c_{0}.  (5.16)

We write b = b^{1,∞} and observe that a ∈ M(X_{∞}, c) if and only if B^{a} ∈ ([ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩}, c) by (5.1).

First we assume B^{a} ∈ ([ℓ_{∞}, c]^{⟨k(ν)⟩}, c). Then, by [7, Satz 4.8],
∑_{ν=0}^{∞} max_{k∈I_{ν}} |b_{nk}^{a}| converges uniformly in n  (5.17)

and lim_{n→∞} b_{nk}^{a} = α_{k} exists for each k. Since [c_{0}, ℓ_{1}]^{⟨k(ν)⟩} ⊂ [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩} implies ([ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩}, c) ⊂ ([c_{0}, ℓ_{1}]^{⟨k(ν)⟩}, c), we have sup_{n} ∥B_{n}^{a}∥_{1,∞} < ∞ by Corollary 4.5(2.), and this is equivalent to a ∈ b^{−1} ∗ ℓ_{∞}, by Example 5.2(a). Thus there is a constant K such that sup_{n} |a_{n}|b_{n} ≤ K, whence |a_{n}| ≤ K/b_{n} → 0 (n → ∞), that is a ∈ c_{0}. By (5.17), given ε > 0 there is ν_{0} ∈ IN_{0} such that

∑_{ν=ν_{0}}^{∞} max_{k∈I_{ν}} |b_{nk}^{a}| ≤ |a_{n}| (b_{n} − b_{k(ν_{0})−1}) < ε/2 for all n.

Furthermore, since a ∈ c_{0}, we can choose n_{0} ∈ IN such that |a_{n}|b_{k(ν_{0})−1} < ε/2 for all n ≥ n_{0}. Then |a_{n}|b_{n} < ε for all n ≥ n_{0}, that is a ∈ b^{−1} ∗ c_{0}.

Conversely we assume a ∈ b^{−1} ∗ c_{0}. Then obviously a ∈ c_{0}. Furthermore a ∈ b^{−1} ∗ c_{0} implies

∥B_{n}^{a}∥_{(1,∞)} = ∑_{ν=0}^{∞} max_{k∈I_{ν}} |b_{nk}^{a}| → 0 (n → ∞) and sup ∥B_{n}^{a}∥_{(1,∞)} < ∞

By [6, Lemma, p. 168], these two conditions together imply (5.17). From this and lim_{n→∞} b_{nk}^{a} = lim_{n→∞} a_{n} = 0, we conclude B^{a} ∈ ([ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩}, c_{0}) by [7, Satz 4.8], hence a ∈ M(X_{∞}, c_{0}).

We obtain as an immediate consequence of Theorems 5.3, 4.6, Example 5.2(a) and Lemma 5.6

Theorem 5.7. Let W be a BK space with AK and Y = W^{β} and X be any of the spaces ℓ_{∞}, c_{0}, ℓ_{1}, [ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩}, [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩} or [c_{0}, ℓ_{1}]^{⟨k(ν)⟩}.

(a) Then A ∈ (X_{Δ}, Y) if and only if the respective conditions in Theorem 4.6 hold with A replaced by E^{A} and, in addition for all m, E(A_{m}) ∈ ((n)_{n=1}^{∞})^{−1} ∗ c_{0} when X = ℓ_{∞}, E(A_{m}) ∈ ((n)_{n=1}^{∞})^{−1} ∗ ℓ_{∞} when X = c_{0}, E(A_{m}) ∈ (b^{1,∞})^{−1} ∗ c_{0} when X = [ℓ_{∞}, ℓ_{1}]^{⟨k(ν)⟩}, E(A_{m}) ∈ (b^{∞,1})^{−1} ∗ ℓ_{∞} when X = [ℓ_{1}, ℓ_{∞}]^{⟨k(ν)⟩}, and E(A_{m}) ∈ (b^{1,∞})^{−1} ∗ ℓ_{∞} when X = [c_{0}, ℓ_{1}]^{⟨k(ν)⟩}; no additional condition is needed when X = ℓ_{1} by Example 5.2(a).

(b) Then A ∈ (X_{Δ}, Y_{Δ}) if and only if the respective conditions in Part (a) hold with A replaced by C = ΔA.

We obtain from Corollary 4.7

Corollary 5.8. Let 1 < r < ∞ and 1 < p < ∞ ad X be any of the spaces in Theorem 5.7.

(a) Then A ∈ (X_{Δ}, [ℓ_{r}, ℓ_{p}]^{⟨m(μ)⟩}) if and only if the conditions in Corollary 4.7 with A replaced by E^{A} and the additional conditions of Theorem 5.7(a) hold.

(b) Then A ∈ (X_{Δ}, ([ℓ_{r}, ℓ_{p}]^{⟨m(μ)⟩})_{Δ}) if and only if the conditions of Part (a) hold with A replaced by C = ΔA.

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(received 25.10.2002)

A. Jarrah, Department of Mathematics, Qatar University Doha, Qatar E-mail: jarrahahed@hotmail.com E. Malkowsky, Mathematisches Institut, Universität Giessen, Arndtstrasse 2, D-35392 Giessen, Germany Department of Mathematics, Faculty of Science and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Yugoslavia E-mail: eberhard.malkowsky@math.uni-giessen.de ema@bankerinter.net