Bulletin T.CXXIII de l’Académie Serbe des Sciences et des Arts - 2002 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 27

MATRIX TRANSFORMATIONS BETWEEN THE SEQUENCE SPACE BV^{P} AND CERTAIN BK SPACES

E. MALKOWSKY, V. RAKOČEVIĆ, SNEŽANA ŽIVKOVIĆ

(Presented at the 4th Meeting, held on May 31, 2002)

A b s t r a c t. In this paper, we characterize matrix transformations between the sequence space bv^{p} (1 < p < ∞) and certain BK spaces. Furthermore, we apply the Hausdorff measure of noncompactness to give necessary and sufficient conditions for a linear operator between these spaces to be compact.

AMS Mathematics Subject Classification (2000): 40H05, 46A45 Key Words: Matrix transformations, measure of noncompactness

1. Introduction

We write ω for the set of all complex sequences x = (x_{k})_{k=0}^{∞}. Let ϕ, ℓ_{∞}, c and c_{0} denote the set of all finite, bounded, convergent and null sequences, and cs be the set of all convergent series. We write ℓ_{p} = {x ∈ ω : ∑_{k=0}^{∞} |x_{k}|^{p} < ∞} for 1 ≤ p < ∞, and bv = {x ∈ ω : ∑_{k=0}^{∞} |x_{k} − x_{k−1} < ∞} for the set of all sequences of bounded variation and extend this definition to reals p ≥ 1 by putting

bv^{p} = {x ∈ ω : ∑_{k=0}^{∞} |x_{k} − x_{k−1}|^{p} < ∞}

so that bv^{1} = bv. The sets bv^{p} also arise from the sets ℓ_{p} as the matrix domains of the difference operator in ℓ_{p}, that is a sequence x is in bv^{p}, if and only if the sequence (x_{k} − x_{k−1})_{k=0}^{∞} is in ℓ_{p}. It is this concept rather than the first one that plays an important role in our studies.

In this paper, we determine the β–duals of the sets bv^{p}, characterize some matrix transformations and apply the Hausdorff measure of noncompactness to give necessary and sufficient conditions for the entries of an infinite matrix to be a compact operator between the spaces bv^{p} for 1 < p < ∞ and certain BK spaces.

In this section, we give some notations and recall some definitions and well–known results.

By e and e^{(n)} (n ∈ IN_{0}), we denote the sequences such that e_{k} = 1 for k = 0, 1, …, and e_{n}^{(n)} = 1 and e_{k}^{(n)} = 0 (k ≠ n). For any sequence x = (x_{k})_{k=0}^{∞}, let x^{[n]} = ∑_{k=0}^{n} x_{k}e^{(k)} be its n–section.

A sequence (b^{(n)})_{n=0}^{∞} in a linear metric space X is called Schauder basis if, for every x ∈ X there is a unique sequence (λ_{n})_{n=0}^{∞} of scalars such that x = ∑_{n=0}^{∞} λ_{n}b^{(n)}.

An F K space is a complete linear metric sequence space with the property that convergence implies coordinatewise convergence; a BK space is a normed F K space. An F K space X ⊃ ϕ is said to have AK if every sequence x = (x_{k})_{k=0}^{∞} ∈ X has a unique representation x = ∑_{k=0}^{∞} x_{k}e^{(k)}, that is x = lim_{n→∞} x^{[n]}.

Let x and y be sequences, X and Y be subsets of ω and A = (a_{nk})_{n,k=0}^{∞} be an infinite matrix of complex numbers. We write xy = (x_{k}y_{k})_{k=0}^{∞}, x^{−1} ∗ Y = {a ∈ ω : ax ∈ Y } and M(X, Y) = ⋂_{x∈X} x^{−1} ∗ Y = {a ∈ ω : ax ∈ Y for all x ∈ X} for the multiplier space of X and Y. In the special case of Y = cs, we write x^{β} = x^{−1} ∗ cs and X^{β} = M(X, cs) for the β–dual of X. By A_{n} = (a_{nk})_{k=0}^{∞} we denote the sequence in the n–th row of A, and we write A_{n}(x) = ∑_{k=0}^{∞} a_{nk}x_{k} (n = 0, 1, …) and A(x) = (A_{n}(x))_{n=0}^{∞}, provided A_{n} ∈ x^{β} for all n. The set X_{A} = {x ∈ ω : A(x) ∈ X} is called the matrix domain of A in X and (X, Y) denotes the class of all matrices that map X into Y, that is A ∈ (X, Y) if and only if X_{A} ⊂ Y, or equivalently A_{n} ∈ X^{β} for all n and A(x) ∈ Y for all x ∈ X.

Let X and Y be Banach spaces. Then B(X, Y) is the set of all continuous linear operators L : X ↦ Y, a Banach space with the operator norm defined by ∥L∥ = sup{∥L(x)∥ : ∥x∥ ≤ 1} (L ∈ B(X, Y)). If Y = ^{|}C then we write X^{∗} = B(X,^{|}C) for the space of continuous linear functionals on X with its norm defined by ∥f∥ = sup{|(x)| : ∥x∥ ≤ 1} (f ∈ X^{∗}). We recall that a linear operator L : X ↦ Y is called compact if D(L) = X for the domain of L and if, for every bounded sequence (x_{n}) in X, the sequence (L(x_{n})) has a convergent subsequence in Y. It is well known (cf. [10, Theorem 4.2.8, p. 87]) that if X and Y are BK spaces and A ∈ (X, Y) then L_{A} ∈ B(X, Y) where L_{A} is defined by L_{A}(x) = A(x) for all x ∈ X; we denote this by (X, Y) ⊂ B(X, Y).

Let 1 < p < ∞ and μ = (μ_{n})_{n=0}^{∞} be a non–decreasing sequence of positive reals tending to infinity. We define the matrices Σ and Δ by Σ_{nk} = 1 for 0 ≤ k ≤ n, Σ_{nk} = 0 for k > n, Δ_{n,n−1} = −1, Δ_{nn} = 1 and Δ_{nk} = 0 otherwise, and use the convention that any term with a negative subscript is equal to zero. So bv^{p} = (ℓ_{p})_{Δ}, as has been mentioned above.

Proposition 1.1. The space bv^{p} is a BK space with

∥x∥_{bv^{p}} = (∑_{k=0}^{∞} |x_{k} − x_{k−1}|^{p})^{1/p};

the sequence (b^{(k)})_{k=0}^{∞} with b^{(k)} = Σ(e^{(k)}), that is b_{j}^{(k)} = 0 for j < k and b_{j}^{(k)} = 1 for j ≥ k (k = 0, 1, …), is a Schauder basis of bv^{p}.

P r o o f. Since ℓ_{p} is a BK space with ∥x∥_{p} = (∑_{k=0}^{∞} |x_{k}|^{p})^{1/p}, bv^{p} is a BK space with ∥ · ∥_{bv^{p}} by [7, Theorem 3.3, p. 178]. Furthermore ℓ_{p} has AK. Hence the sequence (b^{(k)})_{k=0}^{∞} is a Schauder basis of bv^{p} by [5, Theorem 2.2]. □

2. The β–dual of the space bv^{p}

In this section, we give the β–dual of bv^{p} for p ≥ 1. If X ⊃ ϕ is a BK space and a ∈ ω then we write

∥a∥_{X}^{∗} = ∥a∥^{∗} = sup {|∑_{k=0}^{∞} a_{k}x_{k}| : ∥x∥ = 1} ,

provided the expression on the right is defined and finite which is the case whenever a ∈ X^{β} (cf. [10, Theorem 7.2.9, p.107]). Let 1 < p < ∞ and q = p/(p − 1). We write (n + 1)^{1/q} = ((n + 1)^{1/q})_{n=0}^{∞}.

Theorem 2.1. Let 1 < p < ∞. We define the matrix E by E_{nk} = 0 for 0 < k < n − 1 and E_{nk} = 1 for k ≥ n (n = 0, 1, …) and write M(bv^{p}) = ((n + 1)^{1/q})^{−1} ∗ ℓ_{∞}.

(a) Then

(bv^{p})^{β} = (ℓ_{q} ∩ M(bv^{p}))_{E}.

(b) Furthermore

∥a∥_{bv^{p}}^{∗} = ∥E(a)∥_{q} for all a ∈ (bv^{p})^{β}.  (2.1)

P r o o f. (a) By [5, Theorem 2.5], (bv^{p})^{β} = (ℓ_{p}^{β} ∩ M(bv^{p}, c))_{E} = (l_{q})_{E} ∩ (M(bv^{p}, c))_{E}. We are going to show

M(bv^{p}, c) ⊂ M(bv^{p}) ⊂ M(bv^{p}, c_{0}).  (2.2)

First we assume a ∈ M(bv^{p}, c). Then ax ∈ c for all x ∈ bv^{p}. Now x ∈ bv^{p} if and only if y = Δ(x) ∈ ℓ_{p}. Then x = Σ(y) and a_{n}x_{n} = ∑_{k=0}^{n} a_{n}y_{k} (n = 0, 1, …) for all y ∈ ℓ_{p}. We define the matrix C = (c_{nk})_{n,k=0}^{∞} by c_{nk} = a_{n} for 0 ≤ k ≤ n and c_{nk} = 0 for n > k (n = 0, 1, …). Then C ∈ (ℓ_{p}, c), and [10, Example 8.4.5B, p. 129] yields

sup_{n} ∑_{k=0}^{∞} |c_{nk}|^{q} = sup_{n} ∑_{k=0}^{n} |a_{n}|^{q} = sup_{n}(n + 1)|a_{n}|^{q} < ∞,  (2.3)

hence a(n + 1)^{1/q} ∈ ℓ_{∞}. This shows

M(bv^{p}, c) ⊂ M(bv^{p}).  (2.4)

Conversely, we assume a ∈ M(bv^{p}). Then there exists a constant K such that (n + 1)^{1/q}|a_{n}| ≤ K for all n, and so |a_{n}| ≤ K/(n + 1)^{1/q} → 0 (n → ∞), that is

a ∈ c_{0}.  (2.5)

Defining the matrix C as above, we see that (2.3) holds again, and by [10, Example 8.4.5D, p.129], conditions (2.3) and (2.5) yield C ∈ (ℓ_{p}, c_{0}), that is ax ∈ c_{0} for all x ∈ bv^{p}. Thus we have shown M(bv^{p}, c_{0}), and with (2.4), we obtain (2.2).

(b) Let a ∈ (bv^{p})^{β} be given. We observe that x ∈ bv^{p} if and only if y = Δ(x) ∈ ℓ_{p}. Abel’s summation by parts yields, with R = E(a),

∑_{k=0}^{n} a_{k}x_{k} = ∑_{k=0}^{n+1} R_{k}y_{k} − R_{n+1}x_{n+1} (n = 0, 1, …).

Since a ∈ (bv^{p})^{β} implies R ∈ M(bv^{p}, c_{0}) by Part (a), it follows that

∑_{k=0}^{∞} a_{k}x_{k} = ∑_{k=0}^{∞} R_{k}y_{k}.  (2.6)

Now ∥x∥_{bv^{p}} = ∥y∥_{p} implies ∥a∥_{bv^{p}}^{∗} = ∥R∥_{ℓ_{p}}^{∗} and (2.1) follows from the fact that ℓ_{p}^{∗} and ℓ_{q} are norm isomorphic. □

Remark 1. We observe that neither ℓ_{q} ⊂ M(bv^{p}) nor M(bv^{p}) ⊂ ℓ_{q}. If we define the sequences a and ã by

a_{k} = \{ \begin{matrix}\frac{1}{ν + 1} & (k = 2^{ν}) \\ 0 & (k ≠ 2^{ν}) \end{matrix} (ν = 0, 1, …) and ã_{k} = \frac{1}{(k + 1)^{1/q}} (k = 0, 1, …)

then a ∈ ℓ_{q} \ M(bv^{p}) and ã ∈ M(bv^{p}) \ ℓ_{q}, since

∑_{k=0}^{∞} |a_{k}|^{q} = ∑_{ν=0}^{∞} \frac{1}{(ν + 1)^{q}} < ∞ but |a_{2^{ν}}|(2^{ν} + 1)^{1/q} ≥ \frac{2^{ν/q}}{ν + 1} → ∞ (ν → ∞) and

ã_{k}(k + 1)^{1/q} = 1 for k = 0, 1, … but ∑_{k=0}^{∞} ã_{k} = ∑_{k=0}^{∞} \frac{1}{k + 1} = ∞.

3. Matrix Transformations on the spaces bv^{p}

In this section we characterize matrix transformations on the spaces bv^{p}. Throughout let 1 < p < ∞ and q = p/(p − 1). A subset X of ω is said to

be normal if x ∈ X and y ∈ ω with |y_{k}| ≤ |x_{k}| (k = 0, 1, …) together imply y ∈ X. We need the following general results.

Proposition 3.1.([5, Theorem 2.7 (a)]) Let X ⊃ ϕ be a normal F K space with AK and Y be a linear space. If M(X_{Δ}, c) = M(X_{Δ}, c_{0}) then A ∈ (X_{Δ}, Y) if and only if

R^{A} ∈ (X, Y) where r_{nk}^{A} = ∑_{j=k}^{∞} a_{nk} (n, k = 0, 1, …)  (3.1)

and

R_{n}^{A} ∈ (X_{Δ}, c) for all n.  (3.2)

Proposition 3.2.(cf. [7, Theorem 1.23, p. 155 ]) Let X ⊃ ϕ and Y be BK spaces.

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

∥A∥_{X}^{∗} = sup_{n} ∥A_{n}∥_{X}^{∗} < ∞.  (3.3)

Furthermore, if A ∈ (X, ℓ_{∞}) then ∥L_{A}∥ = ∥A∥_{X}^{∗}.

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

First we characterize the classes (bv^{p}, ℓ_{∞}), (bv^{p}, c_{0}) and (bv^{p}, c).

Theorem 3.1.We have

(a) A ∈ (bv^{p}, ℓ_{∞}) if and only if

∥A∥_{(bv^{p},ℓ_{∞})} = sup_{n} (∑_{k=0}^{∞} |∑_{j=k}^{∞} a_{nj}|^{q})^{1/q} < ∞  (3.4)

and

sup_{k} (k^{1/q} |∑_{j=k}^{∞} a_{nj}|) < ∞ for all n;  (3.5)

(b) A ∈ (bv^{p}, c_{0}) if and only if conditions (3.4) and (3.5) hold and

lim_{n→∞} ∑_{j=k}^{∞} a_{nj} = 0 for each k;  (3.6)

(c) A ∈ (bv^{p}, c) if and only if conditions (3.4) and (3.5) hold and

lim_{n→∞} ∑_{j=k}^{∞} a_{nj} = α_{k} for each k.  (3.7)

(d) Let Y denote any of the spaces ℓ_{∞}, c_{0} or c. If A ∈ (bv^{p}, Y) then ∥L_{A}∥ = ∥A∥_{(bv^{p},ℓ_{∞})}.

P r o o f. (a) By Theorem 2.1, M(bv^{p}, c) = M(bv^{p}, c_{0}), so Proposition 3.1. yields that A ∈ (bv^{p}, ℓ_{∞}) if and only if R ∈ (ℓ_{p}, ℓ_{∞}) and R_{n} ∈ M(bv^{p}, c) for all n where r_{nk} = ∑_{j=k}^{∞} a_{nj} for all n and k. Now M(bv^{p}, c) = ((k^{1/q})_{k=0}^{∞})^{−1} ∗ ℓ_{∞}, and this is condition (3.5). Furthermore, by [10, Example 8.4.5D, p.

129], R ∈ (ℓ_{p}, ℓ_{∞}) if and only if sup_{n} ∑_{k=0}^{∞} |r_{nk}|^{q} < ∞, and this is condition (3.4).

(b) Since (b^{(k)})_{k=0}^{∞} with b^{(k)} = Σ(e^{(k)}) for all k is a Schauder basis of bv^{p} and b_{j}^{(k)} = 0 for j < k and b_{j}^{(k)} = 1 for j ≥ k (k = 0, 1, …) by Proposition 1.1, we have

A_{n}(b^{(k)}) = ∑_{j=0}^{∞} a_{nj}b_{j}^{(k)} = ∑_{j=k}^{∞} a_{nj} for each k.

Now Part (b) follows from Part (a) and Proposition 3.2.

(c) Part (c) is proved in exactly the same way as Part (b).

(d) If A ∈ (bv^{p}, ℓ_{∞}) then ∥A∥_{bv^{p}}^{∗} = ∥L_{A}∥ by Proposition 3.2. Since ∥A∥_{bv^{p}}^{∗} = sup_{n} ∥A_{n}∥_{bv^{p}}^{∗} for all n, the conclusion follows from (2.1) in Theorem 2.1. Since (bv^{p}, c_{0}) ⊂ (bv^{p}, c) ⊂ (bv^{p}, ℓ_{∞}), the assertion also follows for Y = c_{0} or Y = c by what we have just shown and Parts (b) and (c). □

Now we characterize the classes (bv^{p}, ℓ_{1}) and (bv^{p}, bv). We need the following result.

Proposition 3.3.Let X ⊃ ϕ be a BK space. Then A ∈ (X, ℓ_{1}) if and only if

∥A∥_{(X,ℓ_{1})} = sup_{{N⊂IN_{0} \atop N finite}} ∥ ∑_{n∈N} A_{n}∥ < ∞ (cf. [4, Satz 1]).

Furthermore, if A ∈ (X, ℓ_{1}) then

∥A∥_{(X,ℓ_{1})} ≤ ∥L_{A}∥ = 4 · ∥A∥_{(X,ℓ_{1})}.  (3.8)

P r o o f. We have to show (3.8). Let A ∈ (X, ℓ_{1}) and m ∈ IN_{0} be given. Then, for all N ⊂ {0, …, m} and for all x ∈ X with ∥x∥ = 1,

| ∑_{n∈N} A_{n}(x)| ≤ ∑_{n=0}^{m} |A_{n}(x)| ≤ ∥L_{A}∥,

and this implies

∥A∥_{(X,ℓ_{1})} ≤ ∥L_{A}∥.  (3.9)

Furthermore, given ε > 0, there is x ∈ X with ∥x∥ = 1 such that

∥A(x)∥_{1} = ∑_{n=0}^{∞} |A_{n}(x)| ≥ ∥L_{A}∥ − \frac{ε}{2},

and there is an integer m(x) such that

∑_{n=0}^{m(x)} |A_{n}(x)| ≥ ∥A(x)∥_{1} − \frac{ε}{2}.

Consequently ∑_{n=0}^{m(x)} |A_{n}(x)| ≥ ∥L_{A}∥ − ε. By [7, Lemma 3.9, p. 181],

4 · max_{N⊂{0,...,m(x)}} | ∑_{n∈N} A_{n}(x)| ≥ ∑_{n=0}^{m(x)} |A_{n}(x)| ≥ ∥L_{A}∥ − ε,

and so 4 · ∥A∥_{(X,ℓ_{1})} ≥ ∥L_{A}∥ − ε. Since ε > 0 was arbitrary, we have 4 · ∥A∥_{(X,ℓ_{1})} ≥ ∥L_{A}∥, and together with (3.9) this yields (3.8) □

A matrix T is called a triangle if t_{nk} = 0 (k > n) and t_{nn} ≠ 0 for all n.

Proposition 3.4.([7, Theorem 3.8, p. 180]) Let T be a triangle. Then, for arbitrary subsets X and Y of ω, A ∈ (X, Y_{T}) if and only if B = T A ∈ (X, Y). Furthermore, if X and Y are BK spaces and A ∈ (X, Y_{T}) then

∥L_{A}∥ = ∥L_{B}∥.  (3.10)

Theorem 3.2. We have

(a) A ∈ (bv^{p}, ℓ_{1}) if and only if condition (3.5) holds and

∥A∥_{(bv^{p},ℓ_{1})} = sup_{{N⊂IN_{0} \atop N finite}} (∑_{k=0}^{∞} | ∑_{n∈N} (∑_{j=k}^{∞} a_{nk})|^{q})^{1/q} < ∞.  (3.11)

Furthermore, if A ∈ (bv^{p}, ℓ_{1}) then

∥A∥_{(bv^{p},ℓ_{1})} ≤ ∥L_{A}∥ ≤ 4 · ∥A∥_{(bv^{p},ℓ_{1})}.  (3.12)

(b) A ∈ (bv^{p}, bv) if and only if condition (3.5) holds and

∥A∥_{(bv^{p},bv)} = sup_{{N⊂IN_{0} \atop N finite}} (∑_{k=0}^{∞} | ∑_{n∈N} (∑_{j=k}^{∞}(a_{nk} − a_{n−1,k}))|^{q})^{1/q} < ∞.  (3.13)

Furthermore, if A ∈ (bv^{p}, bv) then

∥A∥_{(bv^{p},bv)} ≤ ∥L_{A}∥ = 4 · ∥A∥_{(bv^{p},bv)}.  (3.14)

P r o o f. (a) Part (a) follows from Proposition 3.3. and Theorem 2.1.

(b) Part (b) follows from Part (a) and Proposition 3.4. □

4. Measure of noncompactness and transformations

If X and Y are metric spaces, then f : X ↦ Y is a compact map if f(Q) is relatively compact (i.e., if the closure of f(Q) is a compact subset of Y) subset of Y for each bounded subset Q of X. In this section, among other things, we investigate when in some special cases the operator L_{A} is compact. Our investigations use the measure of noncompactness. Recall that if Q is a bounded subset of a metric space X, then the Hausdorff measure of noncompactness of Q, is denoted by χ(Q), and

χ(Q) = inf{ϵ > 0 : Q has a finite ϵ − net in X}.

The function χ is called the Hausdorff measure of noncompactness, and for its properties see ([1, 2, 8]). Denote by \overline{Q} the closure of Q. For the convenience of the reader, let us mention that: If Q, Q_{1} and Q_{2} are bounded subsets of a metric space (X, d), then

χ(Q) = 0 ⟺ Q is a totally bounded set,
χ(Q) = χ(\overline{Q}),
Q_{1} ⊂ Q_{2} ⟹ χ(Q_{1}) ≤ χ(Q_{2}),
χ(Q_{1} ∪ Q_{2}) = max{χ(Q_{1}), χ(Q_{2})},
χ(Q_{1} ∩ Q_{2}) ≤ min{χ(Q_{1}), χ(Q_{2})}.

If our space X is a normed space, then the function χ(Q) has some additional properties connected with the linear structure. We have e.g.

\begin{aligned}χ(Q_{1} + Q_{2}) & ≤ χ(Q_{1}) + χ(Q_{2}), \\ χ(λQ) & = |λ|χ(Q) for each λ ∈ ℂ. \end{aligned}

If X and Y are normed spaces, and A ∈ B(X, Y), then the Hausdorff measure of noncompactness of A, denoted by ∥A∥_{χ}, is defined by ∥A∥_{χ} = χ(AK), where K = {x ∈ X : ∥x∥ ≤ 1} is the unit ball in X. Furthermore, A is compact if and only if ∥A∥_{χ} = 0, and ∥A∥_{χ} ≤ ∥A∥.

Recall the following well known result (see e.g. [2, Theorem 6.1.1] or [1, 1.8.1]): Let X be a Banach space with a Schauder basis {e_{1}, e_{2}, …}, Q a bounded subset of X, and P_{n} : X ↦ X the projector onto the linear span of {e_{1}, e_{2}, …, e_{n}}. Then

\frac{1}{a} lim sup_{n→∞} (sup_{x∈Q} ∥(I − P_{n})x∥) ≤ χ(Q) ≤

≤ inf_{n} sup_{x∈Q} ∥(I − P_{n})x∥ ≤ lim sup_{n→∞} (sup_{x∈Q} ∥(I − P_{n})x∥) ,

(4.1)

where a = lim sup_{n→∞} ∥I − P_{n}∥.

Theorem 4.1.Let A be an infinite matrix, 1 < p < ∞, q = p/(p − 1) and for any integers n, r, n > r, set

∥A∥_{(bv^{p},ℓ_{∞})}^{(r)} = sup_{n>r} (∑_{k=0}^{∞} |∑_{j=k}^{∞} a_{nj}|^{q})^{1/q} .

(a) If A ∈ (bv^{p}, c_{0}), then

∥L_{A}∥_{χ} = lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{∞})}^{(r)}.  (4.2)

(b) If A ∈ (bv^{p}, c), then

\frac{1}{2} · lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{∞})}^{(r)} ≤ ∥L_{A}∥_{χ} ≤ lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{∞})}^{(r)}.  (4.3)

(c) If A ∈ (bv^{p}, ℓ_{∞}), then

0 ≤ ∥L_{A}∥_{χ} ≤ lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{∞})}^{(r)}.  (4.4)

P r o o f. Let us remark that the limits in (4.2), (4.3) and (4.4) exist. Set K = {x ∈ bv^{p} : ∥x∥ ≤ 1}. In the case (a) by inequality (4.1) we have

∥L_{A}∥_{χ} = χ(AK) = lim_{r→∞} [sup_{x∈K} ∥(I − P_{r})Ax∥] ,  (4.5)

where P_{r} : c_{0} ↦ c_{0}, r = 0, 1, …, is the projector on the first r+1 coordinates, i.e., P_{r}(x) = (x_{0}, …, x_{r}, 0, 0, …), for x = (x_{k}) ∈ c_{0}; (let us remark that ∥I − P_{r}∥ = 1, r = 1, 2, . . .. Let A_{(r)} = (ã_{nk}) be infinite matrix defined by ã_{nk} = 0 if 0 ≤ n ≤ r and ã_{nk} = a_{nk} if r < n. Now, by Theorem 4.1 (d) we have

sup_{x∈K} ∥(I − P_{r})Ax∥ = ∥L_{A_{(r)}}∥ = ∥A_{(r)}∥_{(bv^{p},ℓ_{∞})} = ∥A∥_{(bv^{p},ℓ_{∞})}^{(r)}.  (4.6)

Clearly, by (4.5) and (4.6) we get (4.2).

(b) Let us remark that every sequence x = (x_{k})_{k=0}^{∞} ∈ c has a unique representation x = le+∑_{k=0}^{∞}(x_{k}−l)e^{(k)} where l ∈ ^{|}C is such that x−le ∈ c_{0}. Let us define P_{r} : c ↦ c by P_{r}(x) = le + ∑_{k=0}^{m}(x_{k} − l)e^{(k)}, r = 0, 1, . . .. It is known that ∥I − P_{r}∥ = 2, r = 0, 1, . . .. Now the proof of (b) is similar as in the case (a), and we omit it (it should be borne in mind that now a in (4.1) is 2). Let us prove (4.4). Now define P_{r} : ℓ_{∞} ↦ ℓ_{∞}, by P_{r}(x) = (x_{0}, x_{1}, …, x_{r}, 0, …), x = (x_{k}) ∈ ℓ_{∞}, r = 0, 1, . . .. It is clear that

AK ⊂ P_{r}(AK) + (I − P_{r})(AK).

Now, by the elementary properties of the function χ we have

\begin{aligned}χ(AK) & ≤ χ(P_{r}(AK)) + χ((I − P_{r})(AK)) = χ((I − P_{r})(AK)) \\ & ≤ sup_{x∈K} ∥(I − P_{r})Ax∥ = ∥L_{A_{(r)}}∥. \end{aligned} (4.7)

By (4.7) and Theorem 4.1 (d) we get (4.4). □

Now as a corollary of the above theorem we have

Corollary 4.1.If either A ∈ (bv^{p}, c_{0}) or A ∈ (bv^{p}, c), then

L_{A} is compact if and only if lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{∞})}^{(r)} = 0.  (4.8)

If A ∈ (bv^{p}, ℓ_{∞}), then

L_{A} is compact if lim_{r→∞} ∥A∥_{(bv^{p},l_{∞})}^{(r)} = 0.  (4.9)

The following example will show that it is possible for L_{A} in (4.9) to be compact in the case lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{∞})}^{(r)} > 0, and hence in general in (4.9) we have just ” if”.

Example 4.1. Let the matrix A be defined by A_{n} = e^{(0)} (n = 0, 1, …).

Then sup_{n} (∑_{k=0}^{∞} |∑_{j=k}^{∞} a_{nj}|^{q})^{1/q} = 1 < ∞ and sup_{k} (k^{1/q} |∑_{j=k}^{∞} a_{nj}|) = 0 < ∞ for all n. By Theorem 4.1 (a) it follows A ∈ (bv^{p}, ℓ_{∞}). Further,

∥A∥_{(bv^{p},ℓ_{∞})}^{(r)} = sup_{n>r} (∑_{k=0}^{∞} |∑_{j=k}^{∞} a_{nj}|^{q})^{1/q} = sup_{n>r} |∑_{j=0}^{∞} a_{nj}| = 1 for all r,

whence

lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{∞})}^{(r)} = 1 > 0.

Since L_{A}(x) = x_{0}e for all x ∈ bv^{p}, L_{A} is a compact operator.

Theorem 4.2. Let A be an infinite matrix, 1 < p < ∞, q = p/(p − 1) and for any integer r, set

∥A∥_{(bv^{p},ℓ_{1})}^{(r)} = sup_{{N⊂IN_{0}\\{0,1,...,r\} \atop N finite}} (∑_{k=0}^{∞} | ∑_{n∈N} (∑_{j=k}^{∞} a_{nj})|^{q})^{1/q} .

If A ∈ (bv^{p}, ℓ_{1}), then

lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{1})}^{(r)} ≤ ∥L_{A}∥_{χ} ≤ 4 lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{1})}^{(r)}.  (4.10)

P r o o f. Every sequence x = (x_{k})_{k=0}^{∞} ∈ ℓ_{1} has a unique representation

x = ∑_{k=0}^{∞} x_{k}e^{(k)}.

We define P_{r} : ℓ_{1} ↦ ℓ_{1} by P_{r}(x) = (x_{0}, x_{1}, …, x_{r}, 0, 0, …), r = 0, 1, . . .. Since ∥I − P_{r}∥ = 1, r = 0, 1, …, by Theorem 4.2 (a) and (4.1) we get (4.10) (the proof is similar as in the case (4.2)). □

Corollary 4.2.Let A be as in Theorem 5.2. If A ∈ (bv^{p}, ℓ_{1}), then

L_{A} is compact if end only if lim_{r→∞} ∥A∥_{(bv^{p},ℓ_{1})}^{(r)} = 0.

Theorem 4.3.Let A be an infinite matrix, 1 < p < ∞, q = p/(p − 1) and for any integer r, set

∥A∥_{(bv^{p},bv)}^{(r)} = sup_{{N⊂IN_{0}\\{0,1,...,r\} \atop N finite}} (∑_{k=0}^{∞} | ∑_{n∈N} (∑_{j=k}^{∞}(a_{nj} − a_{n−1,j}))|^{q})^{1/q} .

If A ∈ (bv^{p}, bv), then

lim_{r→∞} ∥A∥_{(bv^{p},bv)}^{(r)} ≤ ∥L_{A}∥_{χ} ≤ 4 lim_{r→∞} ∥A∥_{(bv^{p},bv)}^{(r)}.  (4.11)

P r o o f. Let b^{(k)} k = 0, 1, …, be as in Proposition 2.1. (b^{(k)})_{k=0}^{∞} is Schauder basis of bv and it holds

x = ∑_{k=0}^{∞}(x_{k} − x_{k−1})b^{(k)}, x ∈ bv.

Now let us define P_{r} : bv ↦ bv by

P_{r}(x) = ∑_{k=0}^{r}(x_{k} − x_{k−1})b^{(k)}, r = 0, 1, ….

Therefore (I − P_{r})(x) = (0, …, 0, x_{r+1} − x_{r}, x_{r+2} − x_{r}, …). By

∥(I − P_{r})(x)∥_{bv} = = |x_{r+1} − x_{r}| + |x_{r+2} − x_{r} − (x_{r+1} − x_{r})| + |x_{r+3} − x_{r} − (x_{r+2} − x_{r})| + … = |x_{r+1} − x_{r}| + |x_{r+2} − x_{r+1}| + |x_{r+3} − x_{r+2}| + … ≤ ∥x∥_{bv},

(4.12)

we get ∥I − P_{r}∥ ≤ 1. Since I − P_{r} is a projector, we have ∥I − P_{r}∥ ≥ 1. Therefore ∥I − P_{r}∥ = 1. Now, by Theorem 4.2 (b) and (4.1) we get (4.11). □

Now as a corollary of the above theorem we have

Corollary 4.3. Let A be as in Theorem 5.3. If A ∈ (bv^{p}, bv), then

L_{A} is compact if and only if lim_{r→∞} ∥A∥_{(bv^{p},bv)}^{(r)} = 0.

REFERENCES

[1] R. R. A k h m e r o v, M. I. K a m e n s k i i, A. S. P o t a p o v, A. E. R o d k i n a, B. N. S a d o v s k i i, Measures of noncompactness and condensing operators, Operator Theory: Advances and Applications, 55, Birkhäuser Verlag, Basel, 1992.

[2] J. B a n á s, K. G o e b l, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker, New York and Basel, 1980.

[3] A. M. J a r r a h, E. M a l k o w s k y, BK spaces, bases and linear operators, Rend. Circ. Mat. Palermo II, 52, 1998, (177–191)

[4] E. M a l k o w s k y, Klassen von Matrix Abbildungen in paranormierten FR–Räumen, Analysis 7, 1987, (275–292)

[5] , Linear operators between some matrix domains, Rend. Circ. Mat. Palermo, to appear

[6] , V. R a k o č e v i ć, The measure of noncompactness of linear operators between certain sequence spaces, Acta Sci. Math (Szeged), 64, 1998, (151–170)

[7]  ,  , An introduction into the theory of sequence spaces and measures

of noncompactness, Zbornik radova 9(17), Matematički Institut SANU, Belgrade, 2000, (143–234)

[8] V. R a k o č e v i ć, Funkcionalna analiza, Naučna knjiga, Beograd, 1994.

[9] A. W i l a n s k y, Functional Analysis, Blaisdell Publishing Company, 1964

[10] , Summability through Functional Analysis, North–Holland Mathematics Studies, 85, 1984

E. Malkowsky Department of Mathematics University of Giessen Arndtstrasse 2 D–35392 Giessen Germany V. Rakočević, Snežana Živković Department of Mathematics Faculty of Science and Mathematics University of Niš Višegradska 33 18000 Niš Yugoslavia