����������� ������ 49 (1997), 187–196

UDK 517.521

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RECENT RESULTS IN THE THEORY OF MATRIX TRANSFORMATIONS IN SEQUENCE SPACES

Eberhard Malkowsky

Abstract. In this paper we give a survey of recent results in the theory of matrix transfomrations between sequence spaces. We shall deal with sequence spaces that are closely related to various concepts of summability, study their topological structures, find their Schauder-bases and determine their β-duals. Further we give necessary and sufficient conditions for matrix transformations between them.

1. Introduction and well-known results

We shall write ω for the set of all complex sequences x = (x_{k})_{k=0}^{∞} and ϕ, l_{∞}, c and c_{0} for the sets of all finite, bounded, convergent sequences and sequences convergent to naught, respectively; and finally, for 1 ⩽ p < ∞, l_{p} = { x ∈ ω : ∑_{k=0}^{∞} |x_{k}|^{p} < ∞ }.

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

A BK space is a Banach sequence space with continuous coordinates. A sequence (b_{n})_{n=0}^{∞} in a linear metric space X is called a (Schauder-) basis if for each x ∈ X there exists a unique sequence (λ_{n})_{n=0}^{∞} of scalars such that x = ∑_{n=0}^{∞} λ_{n}b_{n}. A BK spaces X ⊃ ϕ is said to have AK if every x = (x_{k})_{k=0}^{∞} ∈ X has a unique representation x = ∑_{n=0}^{∞} x_{n}e^{(n)}.

Let A = (a_{nk})_{n,k=0}^{∞} be a infinite matrix of complex numbers, x ∈ ω and 1 ⩽ p < ∞. Then we shall write

A_{n}(x) = ∑_{k=0}^{∞} a_{nk}x_{k}, A_{n}(|x|^{p}) = ∑_{k=0}^{∞} a_{nk}|x_{k}|^{p} (n = 0, 1,…);

A(x) = (A_{n}(x))_{n=0}^{∞} and A(|x|^{p}) = (A_{n}(|x|^{p}))_{n=0}^{∞}.

AMS Subject Classification: 40 H 05, 46 A 45, 47 B 07

Keywords and phrases: BK spaces, bases, matrix transformations

Communicated at the 4th Symposium on Mathematical Analysis and Its Applications, Aran - delovac 1997.

For any subset X of ω, we define the sets

X_{A} = { x ∈ ω : A(x) ∈ X } and X_{[A]^{p}} = { x ∈ ω : A(|x|^{p}) ∈ X }.

If p = 1, then we omit the index p, i.e. we write X_{[A]} = A_{[A]^{1}} for short. For instance, if E is the matrix defined by e_{nk} = 1 (0 ⩽ k ⩽ n) and e_{nk} = 0 (k > n) for all n = 0, 1, …, then cs = c_{E} and bs = (l_{bb})_{E} are the sets of convergent and bounded series.

We shall be interested in sequence spaces that are closely related to the concepts of ordinary, strong and absolute summability with index p ⩾ 1 [5, pp. 185, 189, 190]. Further, we shall study the topological properties of these sequence spaces, give their β-duals and characterize matrix transformations between them.

The reader is referred to [2, 13, 19] for the results in classical summability theory, and to [18, 14, 5, 15, 19] for the theory of sequence spaces.

2. The classical BK spaces

In this section, we shall state the fundamental results. It is well known [18, 5, 14] that the spaces l_{p} (1 ⩽ p < ∞), c_{0}, c and l_{∞} are BK spaces with their natural norms, l_{p} and c have AK, every sequence x = (x_{k})_{k=0}^{∞} ∈ c has a unique representation x = le + ∑_{k=0}^{∞}(x_{k} − l)e^{(k)} where l = lim_{k→∞} x_{k} and l_{∞} has no Schauder basis.

If X and Y are arbitrary subsets of ω and z any sequence, then we shall write

z^{−1} ∗ X = {x ∈ ω : xz ∈ X} and M(X, Y) = ⋂_{x∈X} x^{−1} ∗ Y.

In the special case, where Y = cs, the set

X^{β} = M(X, cs) = {a ∈ ω : ∑_{k=0}^{∞} a_{k}x_{k} converges for all x ∈ X}

is called the β-dual of X. If X is an arbitrary normed space, then we denote its continuous dual by X^{∗}, i.e. X^{∗} is the space of all continuous linear functionals on X, with the norm ∥ · ∥^{∗} defined by

∥f∥^{∗} = sup{|f(x)| : ∥x∥ = 1} (f ∈ X^{∗}).

There is a close relation between the continuous dual and the β-dual of a BK space.

Theorem 2.1. ([18, Theorems 4.3.15, p. 64 and 7.2.9, p. 107) Let (X, ∥ · ∥) be a BK space.

(a) Then X^{β} is a BK space with ∥a∥_{β} = sup{sup_{n} | ∑_{k=0}^{n} a_{k}x_{k}| : ∥x∥ = 1}.

(b) The inclusion X^{β} ⊂ X^{∗} holds in the following sense: Let the mapˆ : X^{β} → X^{∗} be defined by ˆ (a) = â: X → C (a ∈ X^{β}) where â(x) = ∑_{k=0}^{∞} a_{k}x_{k} for all x ∈ X. Thenˆ is an isomorphism into X^{∗}. Further, if X has AK, then the mapˆ is onto X^{∗}.

The β-duals of the classical sequence spaces are well-known [16, 18, 14]:

ω^{β} = ϕ, ϕ^{β} = ω, l_{1}^{β} = l_{∞} and c_{0}^{β} = c^{β} = l_{∞}^{β} = l_{1}.

If we put q = p/(p − 1) for 1 < p < ∞, then l_{p}^{β} = l_{q}. The spaces c^{∗} and c_{0}^{∗} are norm isomorphic with l_{1}. Further, we have ∥a∥_{β} = ∥a∥_{1} for all a ∈ l_{∞}^{β}.

Let A be an infinite matrix of complex numbers and A_{n} = (a_{nk})_{k=0}^{∞} be the sequence in the n^{th}-row of A. By (X, Y) we denote the class of all matrices A that map the set X ⊂ ω into the set Y ⊂ ω. Thus

A ∈ (X, Y) if and only if \{ \begin{matrix}& A_{n} ∈ X^{β} for all n \\ and \\ & A(x) = (A_{n}(x))_{n=0}^{∞} ∈ Y for all x ∈ X. \end{matrix}

The most important result in the theory of matrix transformations is

Theorem 2.2 ([18, Theorem 4.2.8, p. 57) Matrix transformations between BK spaces are continuous.

All our results are obtained from the characterization of the class (X, l_{∞}):

Theorem 2.3. ([10, Lemma 4.1]) Let X be a BK space.

(a) Then A ∈ (X, l_{∞}) if and only if

∥A∥^{∗} = sup_{n} ∥A_{n}∥^{∗} < ∞ where ∥A_{n}∥^{∗} = sup_{n}{|A_{n}(x)| : ∥x∥ = 1}  (2.1)

for n = 0, 1,….

(b) Further, if (b^{k})_{k=0}^{∞} is a Schauder basis of X, Y and Y_{1} are BK spaces with Y_{1} a closed subspace of Y, then A ∈ (X, Y_{1}) if and only if A ∈ (X, Y) and A(b^{(k)}) ∈ Y_{1} for all k.

Let T be a triangle, i.e. t_{nk} = 0 for all k > n and t_{nn} ≠ 0 (n = 0, 1,…). Further let B be a positive triangle. A subset X of ω is called normal if x ∈ X and |y_{k}| ⩽ |x_{k}| (k = 0, 1,…) together imply y ∈ X, and a norm ∥ · ∥ on X is called monotonous if |y_{k}| ⩽ |x_{k}| (k = 0, 1,…) for x, y ∈ X implies ∥y∥ ⩽ ∥x∥.

Theorem 2.4. (a) Let X be a BK space with the norm ∥ · ∥. Then X_{T} is a BK space with

∥x||_{T} = ∥T (x)∥ for all x ∈ X_{T} ([18, Theorem 4.3.12, p. 63]).

(b) Let X be a normal BK space with monotonous norm ∥ · ∥. We put

x_{n} = (B_{n}(|y|^{p}))^{1/p} for 1 ⩽ p < ∞ (n = 0, 1,…)

and Y = {y ∈ ω : x ∈ X}. Then Y is a BK space with ∥y∥_{Y} = ∥x∥ for all y ∈ Y.

The characterizations of the classes (X, Y_{T}) and (X, Y_{[B]}) can be reduced to those of (X, Y).

Theorem 2.5. ([9, Theorem 1] and [10, Theorem 2.4]) Let X and Y be arbitrary sets of sequences, T a triangle and B a positive triangle.

(a) Then A ∈ (X, Y_{T}) if and only if T A ∈ (X, Y).

(b) For each m = 0, 1,…, let N_{m} denote a subset of the set {0,…, m}, N = (N_{m})_{m=0}^{∞} be the sequence of the sets N_{m} and N be the class of all such sequences. Further, if A is an infinite matrix, then for each sequence N ∈ N let S^{N}(B, A) be the matrix defined by

S_{m}^{N}(B, A) = ∑_{n∈N_{m}} b_{mn}A_{n}, i.e. s_{mk}^{N}(B, A) = ∑_{n∈N_{m}} b_{mn}a_{nk} (m, k = 0, 1,…).

Finally, let T be a normal set of sequences. Then A ∈ (X, Y_{[B]}) if and only if S^{N}(B, A) ∈ (X, Y) for all N ∈ N.

3. Applications

3.1. Sequences that are (\overline{N}, q)-summable or bounded. Let (q_{k})_{k=0}^{∞} be a positive sequence and Q the sequence with Q_{n} = ∑_{k=0}^{n} q_{k} (n = 0, 1,…). Further, let the matrix \overline{N}_{q} be defined by (\overline{N}_{q})_{n,k} = q_{k}/Q_{n} (0 ⩽ k ⩽ n) and (\overline{N}_{q})_{n,k} = 0 (k > n) for all n. Then we define the sets

(\overline{N}, q)_{0} = (c_{0})_{\overline{N}_{q}}, (\overline{N}, q) = c_{\overline{N}_{q}} and (\overline{N}, q)_{∞} = (l_{∞})_{\overline{N}_{q}}

of sequences that are (\overline{N}, q) summable to naught, summable and bounded.

We shall write U for the set of all sequences u such that u_{k} ≠ 0 (k = 0, 1,…). For u ∈ U, let 1/u = (1/u_{k})_{k=0}^{∞}.

Theorem 3.1. (cf. [1, Theorem 2]) (a) Let X be a BK space with basis (b^{(k)})_{k=0}^{∞}, u ∈ U and c^{(k)} = (1/u)b^{(k)} (k = 0, 1,…). Then (c^{(k)})_{k=0}^{∞} is a basis of Y = u^{−1} ∗ X.

(b) Let u ∈ U be a sequence such that

|u_{0}| ⩽ |u_{1}| ⩽ ⋯ and |u_{n}| → ∞ (n → ∞),

and T a triangle with t_{nk} = 1/u_{n} (0 ⩽ k ⩽ n) and t_{nk} = 0 (k > n) for all n. Then (c_{0})_{T} has AK.

We have by Theorems 2.4 and 3.1:

Theorem 3.2. (cf. [1, Corollary 1]) Each of the sets (\overline{N}, q)_{0}, (\overline{N}, q) and (\overline{N}, q)_{∞} is a BK space with

∥x∥_{\overline{N}_{q}} = sup_{n}|\frac{1}{Q_{n}} ∑_{k=0}^{n} q_{k}x_{k}|.

If Q_{n} → ∞ (n → ∞), then (\overline{N}, q)_{0} has AK, and every sequence x = (x_{k})_{k=0}^{∞} ∈ (\overline{N}, q) has a unique representation

x = le + ∑_{k=0}^{∞}(x_{k} − l)e^{(k)} where l ∈ C is such that x − le ∈ (\overline{N}, q)_{0}.

Matrix transformations in sequence spaces 191

Let the operator Δ^{+} : ω → ω be defined by

Δ^{+}x = ((Δ^{+}x)_{k})_{k=0}^{∞} = (x_{k} − x_{k+1})_{k=0}^{∞}.

Theorem 3.3. (cf. [1, Theorem 5]) Let T be a triangle such that

∥T ∥ = sup_{n}( ∑_{k=0}^{n} |t_{nk}|) < ∞ and lim_{n→∞} t_{nk} = 0 (k = 0, 1,…),

(c_{0})_{T} have AK and Y = (c_{0})_{T} ⊕ le. Then f ∈ Y^{∗} if and only if

f(x) = l_{χ_{f}} + ∑_{k=0}^{∞} a_{k}x_{k} with a ∈ Y^{β}, where l ∈ C is such that

x − le ∈ (c_{0})_{T} and χ_{f} = f(e) − l ∑_{k=0}^{∞} a_{k}.

Theorem 3.4. (cf. [1, Theorem 6]) Let Δ^{+} : ω → ω be defined by Δ^{∗}x = ((Δ^{+}x)_{k})_{k=0}^{∞}. We put

N_{0} = (1/q)^{−1} ∗ ((Q^{−1} ∗ l_{1})_{Δ^{+}} ∩ (Q^{−1} ∗ l_{∞})),
N = (1/q)^{−1} ∗ ((Q^{−1} ∗ l_{1})_{Δ^{+}} ∩ (Q^{−1} ∗ c)),
N_{∞} = (1/q)^{−1} ∗ ((Q^{−1} ∗ l_{1})_{Δ^{+}} ∩ (Q^{−1} ∗ c_{0})).

(a) Then (\overline{N}, q)_{0}^{β} = N_{0}, (\overline{N}, q)^{β} = N and (\overline{N}, q)_{∞}^{β} = N_{∞}.

(b) Let Q_{n} → ∞ for n → ∞. Then f ∈ (\overline{N}, q)_{0}^{∗} if and only if

f(x) = ∑_{k=0}^{∞} a_{k}x_{k} with a ∈ N_{0} and ∥f∥^{∗} = sup_{n}( ∑_{k=0}^{n−1} Q_{k}|Δ^{+} \frac{a_{k}}{q_{k}} | + |a_{n}Q_{n}/q_{n}|).

Then f ∈ (\overline{N}, q)^{∗} if and only if

f(x) = l_{χ_{f}} + ∑_{k=0}^{∞} a_{k}x_{k} with a ∈ N where l is such that

x − le ∈ (\overline{N}, q)_{0} and χ_{f} = f(e) − l ∑_{k=0}^{∞} a_{k}

and ∥f∥^{∗} = |χ_{f}| + sup_{n}( ∑_{k=0}^{n−1} Q_{k}|Δ^{+} \frac{a_{k}}{q_{k}} | + |a_{n}Q_{n}/q_{n}|).

It follows from Theorems 2.3, 3.2 and 3.4: Theorem 3.5. Let Q_{n} → ∞ (n → ∞) and consider the conditions

sup_{n}( ∑_{k=0}^{n−1} Q_{k}|Δ^{+}(a_{nk}/q_{k})| + |Q_{n}a_{nk}/q_{n}|) < ∞  (3.1)

(3.2) A_{n}Q/q ∈ c for all n  (3.3) A_{n}Q/q ∈ c_{0} for all n
(3.4) lim_{n→∞} a_{nk} = 0 for all k  (3.5) lim_{n→∞} a_{nk} = l_{k} for all k = 0
(3.6) lim_{n→∞} ∑_{k=0}^{∞} a_{nk} = 0  (3.7) lim_{n→∞} ∑_{k=0}^{∞} a_{nk} = l

Then

A ∈ ((\overline{N}, q)_{0}, l_{∞}) if and only if (3.1);
A ∈ ((\overline{N}, q), l_{∞}) if and only if (3.1) and (3.2);
A ∈ ((\overline{N}, q)_{∞}, l_{∞}) if and only if (3.1) and (3.3);
A ∈ ((\overline{N}, q)_{0}, c_{0}) if and only if (3.1) and (3.4);
A ∈ ((\overline{N}, q)_{0}, c)  if and only if (3.1) and (3.5);
A ∈ ((\overline{N}, q), c_{0})  if and only if (3.1), (3.2), (3.4) and (3.6);
A ∈ ((\overline{N}, q), c)  if and only if (3.1), (3.2), (3.5) and (3.7).

The conditions for A ∈ ((\overline{N}, q)_{∞}, (\overline{N}, p)_{∞}) etc. are obtained by replacing the entries of A above by the entries of C = T A where t_{nl} = p_{l}/P_{l} (0 ⩽ l ⩽ n) and t_{nl} = 0 (l > n) for all n.

3.2. Spaces of sequences of m^{th} order differences. Let m be a positive integer. We define the operators Δ^{(m)}, Σ^{(m)} : ω → ω by

(Δ^{(1)}x)_{k} = Δ^{(1)}x_{k} = x_{k} − x_{k−1}, (Σ^{(1)}x)_{k} = Σ^{(1)}x_{k} = ∑_{j=0}^{k} x_{j} (k = 0, 1,…),
Δ^{(m)} = Δ^{(1)} ^{∘} Δ^{(m−1)},  Σ^{(m)} = Σ^{(1)} ^{∘} Σ^{(m−1)} (m ⩾ 2).

We shall write X(Δ^{(m)}) = X_{Δ^{(m)}} = {x ∈ ω : Δ^{(m)}x ∈ X} for X ∈ {l_{∞}, c, c_{0}}.

Theorem 3.6. ([10, Proposition 1, Theorem 1]) (a) The sets l_{∞}(Δ^{(m)}), c(Δ^{(m)}) and c_{0}(Δ^{(m)}) are BK spaces,

∥x∥_{Δ^{(m)}} = sup_{k} |(Δ^{(m)}x)_{k}| = sup_{k}| ∑_{j=0}^{m}(−1)^{j}\binom{m}{j}x_{k−j}|.

(b) We define the sequences b^{k}(m) by

b_{n}^{(−1)}(m) = \binom{m + n}{n}, and, for k ⩾ 0, b_{n}^{(k)}(m) = \{ \begin{matrix}0, & (n ⩽ k − 1), \\ \binom{m+n−k−1}{n−k}, & (n ⩾ k). \end{matrix}

Then every sequence x = (x_{k})_{k=0}^{∞} ∈ c_{0}(Δ^{(m)}) has a unique representation

x = ∑_{k=0}^{∞} λ_{k}(m)b^{(k)}(m) where λ_{k}(m) = (Δ^{(m)}x)_{k} (k = 0, 1,…),

Matrix transformations in sequence spaces 193

and every sequence x = (x_{k})_{k=0}^{∞} ∈ c(Δ^{(m)}) has a unique representation

x = lb^{(−1)}(m) + ∑_{k=0}^{∞}(λ_{k}(m) − l)b^{(k)}(m) where l = lim_{k→∞}(Δ^{(m)}x)_{k}.

Given any sequence a we define the sequence R^{(m)}(a) by

R_{k}^{(1)}(a) = ∑_{j=k}^{∞} a_{j} (k = 0, 1,…), R^{(m)}(a) = R^{(1)}(R^{(m−1)}(a)) (m ⩾ 2)

provided the series converge. Further we write

R^{(m)}(X) = { x ∈ ω : R^{(m)}(x) ∈ X } for any X ⊂ ω.

Theorem 3.7. ([10, Theorem 3, Lemma 4]) We put

M_{∞}^{β}(m) = (((k^{m}))^{−1} ∗ cs) ∩ R^{(m)}(l_{1}),
M_{0}^{β}(m) = ( ⋂_{v∈c_{0}^{+}}(Σ^{(m)}v)^{−1} ∗ cs) ∩ R^{(m)}(l_{1}).

Then

(c(Δ^{(m)}))^{β} = (l_{∞}(Δ^{(m)}))^{β} = M_{∞}^{β}(m), (c_{0}(Δ^{(m)}))^{β} = M_{0}^{β}(m),

(l_{∞}(Δ^{(m)}))^{β} ≠ (c_{0}(Δ^{(m)}))^{β};

∥a∥^{∗} = ∑_{k=0}^{∞} |R_{k}^{(m)}| on (c_{0}(Δ^{(m)}))^{β}, (c(Δ^{(m)}))^{β} and (l_{∞}(Δ^{(m)}))^{β}.

Since obviously A_{n}(b^{(−1)}(m)) = ∑_{j=0}^{∞} \binom{m + j}{j}a_{nj} and, for k ⩾ 0,

A_{n}(b^{(k)}(m)) = ∑_{j=k}^{∞} \binom{m − 1 + j − k}{j − k}a_{nj} for all n,

we conclude from Theorems 3.6, 2.3 and 3.7:

Theorem 3.8. We consider the conditions

M(l_{∞}(Δ^{(m)}), l_{∞}) = sup_{n} ∥R^{(n)}(A_{n})∥_{1} < ∞,  (3.8)
(3.9) A_{n} ∈ (k^{(m)})^{−1} ∗ cs  (3.10) A_{n} ∈ ⋂_{v∈c_{0}^{+}}(Σ^{(m)}v)^{−1} ∗ cs
(3.11) lim_{n→∞} A_{n}(b^{(k)}(m)) = 0, k ⩾ 0  (3.12) lim_{n→∞} A_{n}(b^{(k)}(m)) = l_{k}, k ⩾ 0
(3.13) lim_{n→∞} A_{n}(b^{(k)}(−1)) = 0  (3.14) lim_{n→∞} A_{n}(b^{(k)}(−1)) = l_{−1}.

Obviuosly (l_{∞}(Δ^{(m)}), l_{∞}) = (c(Δ^{(m)}), l_{∞}), and

A ∈ (l_{∞}(Δ^{(m)}), l_{∞}) if and only if (3.8) and (3.9);
A ∈ (c_{0}(Δ^{(m)}), l_{∞}) if and only if (3.8) and (3.10);
A ∈ (c_{0}(Δ^{(m)}), c_{0}) if and only if (3.8), (3.10) and (3.11);
A ∈ (c_{0}(Δ^{(m)}), c) if and only if (3.8), (3.10) and (3.12);
A ∈ (c(Δ^{(m)}), c_{0}) if and only if (3.8), (3.9), (3.11) and (3.13);
A ∈ (c(Δ^{(m)}), c)  if and only if (3.8), (3.9), (3.12) and (3.14).

3.3. Spaces of sequences that are Λ-strongly convergent or bounded. Let μ = (μ_{n})_{n=0}^{∞} be a nondecreasing sequence of positive reals tending to infinity. If (n(ν))_{ν=0}^{∞} is a sequence such that 0 = n(0) < n(1) < n(2) < ⋯, then we shall write K^{⟨ν⟩} = {k ∈ Z : n(ν) ⩽ k ⩽ n(ν + 1) − 1}, and Σ_{ν} and max_{ν} for the sum and maximum taken over all k in K^{⟨ν⟩}. We define the matrices B = (b_{nk})_{n,k=0}^{∞}, B̃ = (b̃_{νk})_{ν,k=0}^{∞} and Δ(μ) by

b_{nk} = \{ \begin{matrix}\frac{1}{λ_{n}} & (0 ⩽ k ⩽ n) \\ 0 & (k < n) \end{matrix} and b̃_{νk} = \{ \begin{matrix}\frac{1}{λ_{n(ν+1)}} & (k ∈ K^{⟨ν⟩}) \\ 0 & (k ∉ K^{⟨ν⟩}). \end{matrix}

Δ_{nk}(μ) = \{ \begin{matrix}−μ_{n−1} & (k = n − 1) \\ μ_{n} & (k = n) (n = 0, 1,…) where μ_{−1} = 0 \\ 0 & (otherwise). \end{matrix}

The following sets were defined in [12]:

c_{0}(μ) = ((c_{0})_{[B]})_{Δ(μ)},  c̃_{0}(μ) = ((c_{0})_{[B̃]})_{Δ(μ)},
c(μ) = {x ∈ ω : x − le ∈ c_{0}(μ)}, c̃(μ) = {x ∈ ω : x − le ∈ c̃_{0}(μ)},
c_{∞}(μ) = ((l_{∞})_{[B]})_{Δ(μ)},  c̃_{∞}(μ) = ((l_{∞})_{[B̃]})_{Δ(μ)}.

Theorem 3.9. ([8, Theorem 2(c)]) The spaces c_{0}(μ), c(μ) and c_{∞}(μ) are BK spaces with

∥x∥′ = ∥B(|Δ(μ)(x)|)∥_{∞} = sup_{n⩾0}(\frac{1}{μ_{n}} ∑_{k=0}^{n} |μ_{k}x_{k} − μ_{k−1}x_{k−1}|);

c_{0}(μ) has AK; every sequence x = (x_{k})_{k=0}^{∞} ∈ c_{μ} has a unique representation

x = le + ∑_{k=1}^{∞}(x_{k} − l)e^{(k)} where l ∈ C is such that x − le ∈ c_{0}(μ).

A sequence Λ = (λ_{n})_{n=0}^{∞} of positive reals is called exponentially bounded if there is an integer m ⩾ 2 such that for all integers ν there is at least one λ_{n} in the interval [m^{ν}, m^{ν+1}). It is known (cf. [7, Lemma1]) that a nondecreasing sequence Λ = (λ_{n})_{n=0}^{∞} of positive reals is exponentially bounded if and only if the following condition holds:

There are reals s ⩽ t in the open unit interval such that for some

subsequence (λ_{n(ν+1)})_{ν=0}^{∞}, s ⩽ \frac{λ_{n(ν)}}{λ_{n(ν+1)}} ⩽ t for all ν = 0, 1,… (E)

A subsequence (λ_{n(ν+1)})_{ν=0}^{∞} of an exponentially bounded sequence Λ = (λ_{n})_{n=0}^{∞} that satisfies condition (E) will be called an associated subsequence. From now on, let Λ = (λ_{n})_{n=0}^{∞} always be a nondecreasing exponentially bounded sequence of positive reals and (λ_{n(ν+1)})_{ν=0}^{∞} an associated subsequence.

Theorem 3.10. ([8, Theorem 2]) We have c_{0}(Λ) = c̃_{0}(Λ), c(Λ) = c̃(Λ) and c_{∞}(Λ) = c̃_{∞}(Λ). The norms ∥x∥′ and

∥x∥ = ∥B̃(|Δ(Λ)(x)|)∥_{∞} = sup_{ν⩾0}(\frac{1}{λ_{n(ν+1)}} Σ_{ν}|λ_{k}x_{k} − λ_{k−1}x_{k−1}|)

are equivalent on c_{0}(Λ), c(Λ) and c_{∞}(Λ). Thus each of the spaces c_{0}(Λ), c(Λ) and c_{∞}(Λ) is a BK space with ∥ · ∥ (cf. [18, Corollary 4.2.4, p. 56).

Theorem 3.11. ([9, Lemma2]) We put

C(Λ) = \{a ∈ ω : ∑_{ν=0}^{∞} λ_{n(ν+1)} max_{ν} | ∑_{k=n}^{∞} \frac{a_{k}}{λ_{k}} | < ∞\},
∥a∥_{C(Λ)} = ∑_{ν=0}^{∞} λ_{n(ν+1)} max_{ν} | ∑_{k=n}^{∞} \frac{a_{k}}{λ_{k}} | foa all a ∈ C(Λ).

Then (c_{0}(Λ))^{β} = (c(Λ))^{β} = (c_{∞}(Λ))^{β} = C(Λ) and ∥a∥^{∗} = ∥a∥_{C(Λ)} on C(Λ).

As in the previous sections, it is now easy to characterize the classes (X, Y) where X = c_{∞}(Λ), c(Λ), c_{0}(Λ) and Y = l_{∞}, c, c_{0}, c_{∞}(μ), c(μ), c_{0}(μ). For instance, if we put Δ_{n}(μ_{n}a_{nj}) = μ_{n}a_{nj} − μ_{n−1}a_{n−1,j} and

M(c_{∞}(Λ), c_{∞}(μ)) = sup_{m}(max_{N_{m}}( ∑_{ν=0}^{∞} λ_{k(ν+1)} max_{ν}| ∑_{j=k}^{∞} \frac{1}{λ_{j}} (\frac{1}{μ_{m}} ∑_{n∈N_{m}} Δ_{n}(μ_{n}a_{nj}))|)),

then A ∈ (c_{∞}(Λ), c_{∞}(μ)) if and only if M(c_{∞}(Λ), c_{∞}(μ)) < ∞ and A ∈ (c(Λ), c(μ)) if and only if M(c_{∞}(Λ), c_{∞}(μ)) < ∞,

lim_{m→∞}(\frac{1}{μ_{m}} ∑_{n=0}^{m} |Δ_{n}(μ_{n}(a_{nk} − l_{k}))|) = 0 for all k,

lim_{m→∞}(\frac{1}{μ_{m}} ∑_{n=0}^{m}|Δ_{n}(μ_{n}( ∑_{k=0}^{∞} a_{nk} − l_{k}))|) = 0.

REFERENCES

[1] A. M. Aljarrah, E. Malkowsky, BK spaces, bases and linear operators, to appear in Proc. Conference in Acquafredda di Maratea, Italy, 1996.

[2] G. H. Hardy, Divergent Series, Oxford University Press, 1973.

[3] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull. 24, 2 (1981), 169–176.

[4] I. J. Maddox, On Kuttner’s theorem, J. London Math. Soc. 43 (1968), 285–290.

[5] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, 1971.

[6] E. Malkowsky, Matrix transformations between spaces of absolutely and strongly summable sequences, Habilitationsschrift, Giessen 1988.

[7] E. Malkowsky, A note on the Köthe-Toeplitz duals of generalized sets of bounded and convergent difference sequences, J. Analysis, 4 (1996), 81–91.

[8] E. Malkowsky, The continuous duals of the spaces c_{0}(Λ) and c(Λ) for exponentially bounded sequences Λ, Acta Sci. Math. Szeged, 61 (1995), 241–250.

[9] E. Malkowsky, Linear Operators in certain BK spaces, Bolyai Society Mathematical Studies, 5 (1996), 259–273.

[10] E. Malkowsky, S. D. Parashar, Matrix transformations in spaces of bounded and convergent difference sequences of order m, to appear in Analysis

[11] E. Malkowsky, V. Rakočević, The measure of noncompactness of linear operators between certain sequence spaces, to appear in Acta Sci. Math. Szeged.

[12] F. Mòricz, On Λ-strong convergence of numerical sequences and Fourier series, Acta Math. Hung. 54, 3–4 (1989), 319–327.

[13] A. Peyerimhoff, Lectures on Summability, Lecture Notes in Mathematics 107, Springer Verlag.

[14] W. Ruckle, Sequence Spaces, Pitman, 1981.

[15] M. Stieglitz, H. Tietz, Matrixtransformationen in Folgenräumen. Eine Ergebnisübersicht, Math. Z. 154 (1977), 1–16.

[16] A. Wilansky, Functional Analysis, Blaisdell Publishing Company, 1964.

[17] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw Hill 1978.

[18] A. Wilansky, Summability through Functional Analysis, North-Holland Mathematics Studies 85, 1984.

[19] K. Zeller, W. Beekmann, Theorie der Limitierungsverfahren, Springer-Verlag, 1968.

(received 01.08.1997.)

Mathematisches Institut, Universität Giessen, Arndtstrasse 6, D-35392 Giessen, Germany E-mail: ema@thor.informatik.uni-giessen.de