IJMMS 2003:28, 1783–1801 PII. S0161171203204324 http://ijmms.hindawi.com © Hindawi Publishing Corp.
ON SOME BK SPACES
BRUNO DE MALAFOSSE
Received 25 April 2002
◦ (c) We characterize the spaces sα(∆), sα(∆),andsα (∆) and we deal with some sets generalizing the well-known sets w0(λ), w∞(λ), w(λ), c0(λ), c∞(λ),andc(λ).
2000 Mathematics Subject Classification: 46A45, 40C05.
1. Notations and preliminary results. For a given infinite matrix A =
(anm)n,m≥1, the operators An, for any integer n ≥ 1, are defined by
∞ An(X) = anmxm, (1.1) m=1 where X = (xn)n≥1 is the series intervening in the second member being con- vergent. So, we are led to the study of the infinite linear system
An(X) = bn,n= 1,2,..., (1.2)
where B=(b n)n≥1 is a one-column matrix and X the unknown, see [2, 3, 5, 6, 7, 9].
Equation (1.2) can be written in the form AX = B, where AX = (An(X))n≥1.In this paper, we will also consider A as an operator from a sequence space into another sequence space. A Banach space E of complex sequences with the norm E is a BK space if P X → P X E each projection n : n is continuous. A BK space is said to have AK B = (b ) B = ∞ b e (see [8]) if for every n n≥1, n=1 m m, that is, ∞ bmem → 0 (n →∞). (1.3) m=N+1 E
We shall write s, c, c0,andl∞ for the sets of all complex, convergent sequences, sequences convergent to zero, and bounded sequences, respectively. We shall write cs and l1 for the sets of convergent and absolutely convergent series, respectively. We will use the set U +∗ = u ∈ s | u > ∀n . n n≥1 n 0 (1.4) 1784 BRUNO DE MALAFOSSE
Using Wilansky’s notations [12], we define, for any sequence α = (αn)n≥1 ∈ U +∗ and for any set of sequences E, the set xn α∗E = (xn)n≥1 ∈ s | ∈ E . (1.5) αn n
Writing s◦ E = c , α if 0 (c) α∗E = sα if E = c, (1.6) sα if E = l∞, we have for instance α∗c = s◦ = x ∈ s | x = o α n →∞ . 0 α n n≥1 n n (1.7)
Each of the spaces α∗E, where E ∈{c0,c,l∞}, is a BK space normed by xn Xsα = sup , (1.8) n≥1 αn
◦ and sα has AK. +∗ Now, let α = (αn)n≥1 and β = (βn)n≥1 ∈ U . We shall write Sα,β for the set of infinite matrices A = (anm)n,m≥1 such that
∞ a α ∈ l1 ∀n ≥ , a α = O β (n →∞). nm m m≥1 1 nm m n (1.9) m=1
The set Sα,β is a Banach space with the norm ∞ α A = a m . Sα,β sup nm (1.10) n≥ βn 1 m=1
Let E and F be any subsets of s.WhenA maps E into F,wewriteA ∈ (E,F), X ∈ E AX ∈ F AX ∈ F n ≥ see [10]. So, for every , ( will mean that for each 1, y = ∞ a x (y ) ∈ F the series defined by n m=1 nm m is convergent and n n≥1 ). It has been proved in [8] that A ∈ (sα,sβ) if and only if A ∈ Sα,β. So, we can write (sα,sβ) = Sα,β. When sα = sβ, we obtain the unital Banach algebra Sα,β = Sα,(see[2, 3, 9]) normed by ASα =ASα,α .
We also have A ∈ (sα,sα) if and only if A ∈ Sα.IfI −ASα < 1, we say that A ∈ Γα. The set Sα being a unital algebra, we have the useful result: if A ∈ Γα, A is bijective from sα into itself. ON SOME BK SPACES 1785
n ◦ (c) ◦ If α = (r )n≥1,thenΓα, Sα, sα, sα,andsα are replaced by Γr , Sr , sr , sr ,and s(c) r = s = l s◦ = c r , respectively, (see [2, 3, 5, 6, 7, 8, 9]). When 1, we obtain 1 ∞, 1 0, s(c) = c e = ( , ,...) S = S and 1 , and putting 1 1 , we have 1 e. It is well known, see [10], that s1,s1 = c0,s1 = c,s1 = S1. (1.11)
We write en = (0,...,1,...) (where 1 is in the nth position). For any subset E of s,weput
AE ={Y ∈ s |∃X ∈ EY= AX}. (1.12)
If F is a subset of s,wedenote
F(A)= FA ={X ∈ s | Y = AX ∈ F}. (1.13)
We can see that F(A)= A−1F.
◦ (c) 2. Sets sα(∆), sα(∆),andsα (∆). In this section, we will give necessary and ◦ (c) sufficient conditions permitting us to write the sets sα(∆), sα(∆), and sα (∆) ◦ (c) by means of the spaces sξ , sξ ,orsξ . For this, we need to study the sequence C(α)α.
2.1. Properties of the sequence C(α)α. Here, we will deal with the opera- tors represented by C(λ) and ∆(λ),see[2, 5, 7, 8, 9]. Let U = u ∈ s | u ≠ ∀n . n n≥1 n 0 (2.1)
We define C(λ) = (cnm)n,m≥1,forλ = (λn)n≥1 ∈ U,by 1 if m ≤ n, λ cnm = n (2.2) 0 otherwise.
It can be proved that the matrix ∆(λ) = (cnm)n,m≥1,with λ m = n, n if
c = −λn− if m = n−1,n≥ 2, (2.3) nm 1 0 otherwise, is the inverse of C(λ),see[8]. If λ = e, we get the well-known operator of first difference represented by ∆(e) = ∆ and it is usually written Σ = C(e). Note that −1 ∆ = Σ , and ∆ and Σ belong to any given space SR with R>1. 1786 BRUNO DE MALAFOSSE
We use the following sets:
n C = α ∈ U +∗ | C(α)α = 1 α ∈ c , α k n k= 1 n≥1 +∗ C1 = α ∈ U | C(α)α ∈ s1 = l∞ , (2.4) α Γ = α ∈ U +∗ | n−1 < . lim 1 n→∞ αn
Note that ∆ ∈ Γα implies α ∈ Γ . It can be easily seen that α ∈ Γ if and only if there is an integer q ≥ 1 such that αn−1 γq(α) = sup < 1. (2.5) n≥q+1 αn
See [7]. In order to express the following results, we will denote by [C(α)α]n (in- stead of [C(α)]n(α))thenth coordinate of C(α)α. We get the following propo- sition.
Proposition 2.1. Let α ∈ U +∗.Then
(i) αn−1/αn → 0 if and only if [C(α)α]n → 1; (ii) (a) α ∈ C implies that (αn−1/αn)n≥1 ∈ c,
(b) [C(α)α]n → l implies that αn−1/αn → 1−1/l; (iii) if α ∈ C1, there are K>0 and γ>1 such that
n αn ≥ Kγ ∀n; (2.6)
(iv) the condition α ∈ Γ implies that α ∈ C1 and there exists a real b>0 such that
1 n C(α)α ≤ +bχ for n ≥ q +1,χ= γq(α) ∈ ]0,1[. (2.7) n 1−χ
Proof. (i) Assume that αn−1/αn → 0. Then there is an integer N such that
αn− 1 n ≥ N +1 ⇒ 1 ≤ . (2.8) αn 2
n So, there exists a real K>0 such that αn ≥ K2 for all n and
n−k αk αk αn− 1 = ··· 1 ≤ for N ≤ k ≤ n−1. (2.9) αn αk+1 αn 2 ON SOME BK SPACES 1787
Then n −1 N −1 n −1 1 1 αk αk = αk + αn αn αn k=1 k=1 k=N (2.10) N −1 n −1 n−k 1 1 ≤ αk + ; K2n 2 k=1 k=N and since
n− n−k n−N 1 1 1 = 1− → 1 (n →∞), (2.11) k=N 2 2 we deduce that n −1 1 αk = O(1), C(α)α n ∈ l∞. (2.12) αn k=1
Using the identity
α1 +···+αn−1 αn−1 C(α)α n = +1 αn−1 αn (2.13) α = C(α)α n−1 + , n−1 1 αn we get [C(α)α]n → 1. This proves the necessity. Conversely, if [C(α)α]n → 1, then αn− C(α)α n −1 1 = → 0. (2.14) α C(α)α n n−1
Assertion (ii) is a direct consequence of identity (2.14). Σ = n α M> (iii) We put n k=1 k.Thenforareal 1,
Σn C(α)α n = ≤ M ∀n. (2.15) Σn −Σn−1
n−1 So, Σn ≥ (M/(M −1))Σn−1 and Σn ≥ (M/(M −1)) α1 for all n. Therefore, from n−1 α1 M Σn ≤ C(α)α n = ≤ M, (2.16) αn M −1 αn
n 2 we conclude that αn ≥ Kγ for all n,withK = (M −1)α1/M and γ = M/(M− 1)>1. 1788 BRUNO DE MALAFOSSE
(iv) If α ∈ Γ , then there is an integer q ≥ 1 for which
αk−1 k ≥ q +1 ⇒ ≤ χ<1withχ = γq(α). (2.17) αk
So, there is a real M > 0 for which
M αn ≥ ∀n ≥ q +1. (2.18) χn σ = /α ( q α ) d = [C(α)α] −σ Writing nq 1 n k=1 k and n n nq,weget n n −1 n −j 1 αn−k dn = αk = 1+ αn αn−k+ k=q+1 j=q+1 k=1 1 (2.19) n ≤ χn−j ≤ 1 . 1−χ j=q+1
And using (2.18), we get q 1 n σnq ≤ χ αk . (2.20) M k=1
So n C(α)α n ≤ a+bχ (2.21) a = /( −χ) b = ( /M )( q α ) with 1 1 and 1 k=1 k . Remark 2.2. Note that α ∈ C1 does not imply that α ∈ Γ .
2.2. New properties of the operator represented by ∆. Throughout this paper, we will denote by Dξ the infinite diagonal matrix (ξnδnm)n,m≥1 for any given sequence ξ = (ξn)n≥1. Now, we require some lemmas.
◦ ◦ Lemma 2.3. The condition ∆ ∈ (sα,sα) is equivalent to ∆α = D1/α∆Dα ∈ (c) (c) (c0,c0) and ∆ ∈ (sα ,sα ) implies ∆α = D1/α∆Dα ∈ (c,c).
◦ Proof. First, Dα is bijective from c0 into sα. In fact, the equation DαX = B, ◦ for every B = (bn)n ∈ sα, admits a unique solution X = D1/αB = (bn/αn)n ∈ c0. ◦ ◦ Suppose now that ∆ ∈ (sα,sα).ThenforeveryX ∈ c0, we get successively X = ◦ ◦ DαX ∈ sα, ∆X ∈ sα,and∆α = D1/α∆Dα ∈ (c0,c0). Conversely, assume that ◦ ◦ ∆α ∈ (c0,c0) and let X ∈ sα.ThenX = DαX with X ∈ c0.So,∆X ∈ Dαc0 = sα ◦ ◦ (c) (c) and ∆ ∈ (sα,sα). By a similar reasoning, we get ∆ ∈ (sα ,sα ) ⇒ ∆α ∈ (c,c).
We need to recall here the following well-known results given in [12]. ON SOME BK SPACES 1789
Lemma 2.4. The condition A ∈ (c,c) is equivalent to the following conditions:
(i) A ∈ S1; (a ) ∈ c m ≥ (ii) nm n≥1 for each 1; ( ∞ a ) ∈ c (iii) m=1 nm n≥1 .
If for any given sequence X = (xn)n ∈ c,withlimn xn = l, An(X) is conver- gent for all n and limn An(X) = l, it is written that
limX = A−limX, (2.22) and A is called a Toeplitz matrix. We also have the next result.
Lemma 2.5. The operator A ∈ (c,c) is a Toeplitz matrix if and only if
(i) A ∈ S1; a = m ≥ (ii) limn nm 0 for each 1; ( ∞ a ) = (iii) limn m=1 nm 1. Now, we can assert the following theorem.
Theorem 2.6. We have successively (i) sα(∆) = sα if and only if α ∈ C1; ◦ ◦ (ii) sα(∆) = sα if and only if α ∈ C1; (c) (c) (iii) sα (∆) = sα if and only if α ∈ C;
(iv) ∆α = D1/α∆Dα is bijective from c into itself with limX = ∆α −limX if and only if
αn− 1 → 0. (2.23) αn
Proof. (i) We have sα(∆) = sα if and only if ∆, Σ ∈ (sα,sα). This means that ∆, Σ ∈ Sα, that is,
αn−1 ∆Sα = sup 1+ < ∞, ΣSα = sup C(α)α n < ∞. (2.24) n≥1 αn n≥1
Since 0 <αn−1/αn ≤ [C(α)α]n, we deduce that ∆, Σ ∈ Sα if and only if ΣSα < ∞, that is, α ∈ C1. ◦ ◦ (ii) From Lemma 2.3,ifsα(∆) = sα,then∆α = D1/α∆Dα ∈ (c0,c0).So,∆α ∈
(c0,l∞) = S1 and since ∆α = (dnm)n,m≥1 with
m = n, 1if α d = − n−1 m = n− , nm if 1 (2.25) αn 0 otherwise, 1790 BRUNO DE MALAFOSSE
◦ ◦ we deduce that αn−1/αn = O(1), n →∞. Further, sα(∆) = sα implies Σα =
D1/αΣDα ∈ (c0,c0) and Σα ∈ (c0,l∞) = S1.SinceΣα = (σnm)n,m≥1 with αm if m ≤ n, α σnm = n (2.26) 0ifm>n, we deduce that n 1 sup αk < ∞, (2.27) n≥ αn 1 k=1
◦ ◦ that is, α ∈ C1. Conversely, assume that α ∈ C1. First, ∆ ∈ (sα,sα). Indeed, from the inequality
αn−1 ≤ sup C(α)α n < ∞, (2.28) αn n≥1
◦ we deduce that for every X ∈ sα, xn/αn = o(1),
xn −xn− xn xn− αn− 1 = − 1 1 = o(1) (2.29) αn αn αn−1 αn
◦ ◦ and ∆X ∈ sα. Further, take B = (bn)n≥1 ∈ sα. Then there exists ν = (νn)n≥1 ∈ c0 such that bn = αnνn. We must prove that the equation ∆X = B admits a unique ◦ solution in the space sα. First, we obtain n X = ΣB = αkνk . (2.30) k= 1 n≥1
◦ In order to show that X = (xn)n≥1 ∈ sα, we will consider any given ε>0. From Proposition 2.1(iii), the condition α ∈ C1 implies that αn →∞. So, there exists an integer N such that N 1 ε Sn = αkνk ≤ for n ≥ N, αn 2 k=1 (2.31) ε sup νk ≤ . C(α)α n≥N+1 2supn≥1 n R = /α | n α ν | Writing n 1 n k=N+1 k k , we conclude that ε Rn ≤ sup νk C(α)α n ≤ . (2.32) N+1≤k≤n 2 ON SOME BK SPACES 1791
Finally, we obtain N n xn 1 1 = αkνk + αkνk αn αn αn k=1 k=N+1 (2.33)
≤ Sn +Rn ≤ ε for n ≥ N,
0 and X ∈ sα. (c) (c) (iii) As above, sα (∆) = sα if and only if ∆α, Σα ∈ (c,c); and from Lemma 2.4, ∆ ∈ (c,c) (α /α ) ∈ c ∆ ∈ S we have α if and only if n−1 n n . In fact, we have α 1 and n d = +α /α n →∞ Σ ∈ (c,c) m=1 nm 1 n−1 n tends to a limit as . Afterwards, α is equivalent to (a) Σα ∈ S1,thatis,α ∈ C1; (b) limn(αm/αn) = 0 for all m ≥ 1; (c) α ∈ C. From Proposition 2.1(iii), (c) implies that αn tends to infinity, so (c) implies (a) and (b). Finally, from Proposition 2.1(ii), we conclude that α ∈ C implies
(αn−1/αn)n ∈ c. This completes the proof of (iii). (iv) From Lemma 2.5, it can be easily verified that ∆α ∈ (c,c) and limX =
∆α −limX if and only if αn−1/αn → 0. We conclude, using (iii), since αn−1/αn = o(1) implies that α ∈ C.
Remark 2.7. In Theorem 2.6(iv), we see that Σα ∈ (c,c) and limX = Σα − limX if and only if αn−1/αn → 0. In fact, we must have for each m ≥ 1, σnm = αm/αn = o(1)(n→∞) and n n −1 αm lim σnm = lim 1+ = 1, (2.34) n n αn m=1 m=1 and from Proposition 2.1(i), the previous property is satisfied if and only if
αn−1/αn → 0.
Remark 2.8. It can be seen that the condition (αn−1/αn)n ∈ c does not imply that α ∈ C1. It is enough to consider C(e)e = (n)n ∉ c0.
The next corollary is a direct consequence of the previous results.
Corollary 2.9. Consider the following properties: (i) α ∈ C; (c) (c) (ii) sα (∆) = sα ; (iii) α ∈ Γ ; (iv) α ∈ C1; (v) sα(∆) = sα; ◦ ◦ (vi) sα(∆) = sα. Then (i) (ii)⇒(iii)⇒(iv) (v) (vi).
We obtain from the precedent the following corollary. 1792 BRUNO DE MALAFOSSE
◦ ◦ Corollary 2.10. (i) (sα −sα)(∆) = sα −sα if and only if α ∈ C1, (c) ◦ (c) ◦ (ii) (sα −sα)(∆) = sα −sα if and only if α ∈ C, (c) (c) (iii) α ∈ C implies (sα −sα )(∆) = sα −sα .
◦ ◦ Proof. (i) If ∆ is bijective from sα −sα into itself, then for every B ∈ sα −sα, ◦ ◦ we have X = ΣB ∈ sα −sα.Sinceα ∈ sα −sα, we conclude that Σα ∈ sα, that is, C(α)α ∈ l∞. Conversely, from Theorem 2.6(i) and (ii), it can be easily seen that ◦ ◦ ∆ is bijective from sα to sα and from sα to sα,sinceα ∈ C1.So,∆ is bijective ◦ ◦ from sα −sα to sα −sα. (c) ◦ (ii) Suppose that ∆ is bijective from sα −sα into itself. Reasoning as above, (c) ◦ (c) we have α ∈ sα −sα and Σα ∈ sα ,soD1/αΣα = C(α)α ∈ c. Conversely, using (c) (c) ◦ Theorem 2.6(i) and (iii), we see that ∆ is bijective from sα to sα and from sα ◦ (c) ◦ (c) ◦ to sα since α ∈ C and C ⊂ C1.So,(sα −sα)(∆) = sα −sα. (c) Similarly, (iii) comes from the fact that ∆ is bijective from sα into itself and from sα into itself, since α ∈ C.
Remark 2.11. Assume that limn→∞[C(α)α]n = l.Then
xn xn −xn− L → L implies 1 → . (2.35) αn αn l
Indeed, from Proposition 2.1(ii) (b), αn−1/αn → 1−(1/l) and
xn −xn− xn xn− αn− 1 L 1 = − 1 1 → L−L 1− = . (2.36) αn αn αn−1 αn l l
h h 3. Generalization to the sets sr (∆ ) and sα(∆ ) for h real. In this section, we consider the operator ∆h, where h is a real, and give among other things a s (∆h) = s necessary and sufficient condition to have α α. f(z)= ∞ a zk First, recall that we can associate to any power series k=0 k ,de- |z| a a a · 0 1 2 a a · 0 1 ϕ(f) = (3.1) 0 a0 · · (see [3, 4, 5]). Practically, we will write ϕ[f(z)] instead of ϕ(f). We have the following lemma. Lemma 3.1. (i) The map ϕ : f → A is an isomorphism from the algebra of the power series defined in |z| Now, for h ∈ R −N, we define (see [13]) −h+k−1 −h(−h+1)···(−h+k−1) = if k>0, k k! (3.3) −h+k−1 = 1ifk = 0, k and putting ∆+ = ∆t , we get for any h ∈ R, ∞ h −h+k−1 ∆+ = ϕ (1−z)h = ϕ zk for |z| < 1. (3.4) k k=0 h Then if ∆ = (τnm)n,m, −h+n−m−1 if m ≤ n, τ = n−m nm (3.5) 0ifm>n. Using the isomorphism ϕ, we get the following proposition. Proposition 3.2 (see [5]). (i) The operator represented by ∆ is bijective from + sr into itself for every r>1, and ∆ is bijective from sr into itself for all r , 0 (vi) For every integer h ≥ 1, h s ⊂ s ∆ ⊂ s h ⊂ sr . (3.7) 1 1 (n )n≥1 r>1 (vii) If h>0 and h ∉ N, then q is the greatest integer strictly less than (h+1). For all r>1, + h Ker ∆ sr = span V1,V2,...,Vq , (3.8) where V = et,V= A1,A1,... t, 1 2 1 2 t (3.9) V = ,A2,A2,... t,..., V = , ,...,Aq−1,Aq−1,...,Aq−1,... 3 0 2 3 q 0 0 q−1 q n ; j Ai = i!/(i−j)!,with0 ≤ j ≤ i, being the number of permutations of i things taken j at a time. We give here an extension of the previous results, where sr is replaced by sα. h Proposition 3.3. Let h be a real greater than 0. The condition sα(∆ ) = sα is equivalent to n −1 1 h+n−k−1 γn(h) = αk = O(1)(n →∞). (3.10) αn n−k k=1 h h Proof. The operator ∆ is bijective from sα into itself if and only if ∆ , h h Σ ∈ (sα,sα). We have ∆ ∈ (sα,sα) if and only if h D1/α∆ Dα ∈ S1, (3.11) h and using (3.5), we deduce that ∆ ∈ (sα,sα) if and only if n −h+n−k− 1 1 α = O( ). k 1 (3.12) αn n−k k=1 Further, (Σt)h = ϕ[(1−z)−h], where ∞ h+n−1 ϕ(z) = 1+ zn with |z| < 1. (3.13) n n=1 ON SOME BK SPACES 1795 h So, D1/αΣ Dα ∈ S1 if and only if (3.10) holds. Finally, since h>0, we have −h+n−k−1 h+n−k−1 ≤ for k = 1,2,...,n−1, (3.14) n−k n−k and we conclude since (3.10)implies(3.12). We deduce immediately the next result. Corollary 3.4. Let h be an integer greater than or equal to 1. The following properties are equivalent: (i) α ∈ C1; (ii) sα(∆) = sα; h (iii) sα(∆ ) = sα; h−1 (iv) C(α)(Σ α) ∈ l∞. h Proof. From the proof of Proposition 3.3, sα(∆ ) = sα is equivalent to h h−1 h−1 D1/αΣ Dα = C(α)Σ Dα ∈ S1,thatis,C(α)(Σ α) ∈ l∞. So, (iii) and (iv) are equivalent. It remains to prove that (ii) (iii). If sα(∆) = sα, ∆ and consequently h ∆ are bijective from sα into itself and condition (iii) holds. Conversely, assume h that sα(∆ ) = sα holds. Then (3.10) holds, and since h+n−k−1 ≥ 1fork = 1,2,...,n−1, (3.15) n−k we deduce that C(α)α n ≤ γn(h) = O(1), n →∞. (3.16) So, (i) holds and (ii) is satisfied. 4. Generalization of well-known sets. In this section, we see that under ◦ ∗ ◦ some conditions, the spaces w α(λ), wα(λ), wα (λ), c α(λ,µ), cα(λ,µ), and ∗ ◦ cα (λ,µ) can be written by means of the sets sξ or sξ . ◦ ∗ 4.1. Sets w α(λ), wα(λ),andwα (λ). We recall some definitions and prop- erties of some spaces. For every sequence X = (xn)n, we define |X|=(|xn|)n and w α(λ) = X ∈ s | C(λ) |X| ∈ sα , ◦ ◦ wα(λ) = X ∈ s | C(λ) |X| ∈ sα , (4.1) ∗ t ◦ wα (λ) = X ∈ s | X −le ∈ wα(λ) for some l ∈ C . 1796 BRUNO DE MALAFOSSE For instance, we see that n w (λ) = X = x ∈ s | 1 x < ∞ . α n n sup k (4.2) n≥ λn αn 1 k=1 If there exist A, B>0 such that A<αn (see [1]). We have the next result. Theorem 4.1. Let α and λ be any sequences of U +∗. (i) Consider the following properties: (a) αn−1λn−1/αnλn → 0; (c) (c) (b) sα (C(λ)) = sαλ ; (c) αλ ∈ C1; (d) w α(λ) = sαλ; ◦ ◦ (e) wα(λ) = sαλ; ∗ ◦ (f) wα (λ) = sαλ. Then (a)⇒(b), (c) (d), and (c)⇒(e) and (f). ◦ ∗ (ii) If αλ ∈ C1, w α(λ), wα(λ), and wα (λ) are BK spaces with respect to the norm x X = n , sαλ sup (4.4) n≥1 αnλn ◦ ∗ and wα(λ) = wα (λ) has AK. Proof. (i) First, we prove that (a)⇒(b). We have (c) (c) (c) (c) sα C(λ) = ∆(λ)sα = ∆Dλsα = ∆sαλ , (4.5) and from Proposition 2.1(i) and Theorem 2.6(iii), we get successively αλ ∈ C, (c) (c) ∆sαλ = sαλ , and (b) holds. (c) (d). Assume that (c) holds. Then w α(λ) = X ||X|∈∆(λ)sα . (4.6) ON SOME BK SPACES 1797 Since ∆(λ) = ∆Dλ,weget∆(λ)sα = ∆sαλ. Now, using (c), we see that ∆ is bijective from sαλ into itself and wα(λ) = sαλ. Conversely, assume that wα(λ) = sαλ.Thenαλ ∈ sαλ implies that C(λ)(αλ) ∈ sα, and since D1/αC(λ)(αλ) ∈ s1 = l∞, we conclude that C(αλ)(αλ) ∈ l∞. The proof of (c)⇒(e) follows on the same ◦ lines of the proof of (c)⇒(d) replacing sαλ by sαλ. ∗ We prove that (c) implies (f). Take X ∈ wα (λ). There is a complex number l such that t ◦ C(λ) X −le ∈ sα. (4.7) So t ◦ ◦ X −le ∈ ∆(λ)sα = ∆sαλ, (4.8) ◦ ◦ and from Theorem 2.6(ii), ∆sαλ = sαλ. Now, since (c) holds, we deduce from t ◦ ∗ Proposition 2.1(iii) that αnλn →∞and le ∈ sαλ. We conclude that X ∈ wα (λ) t ◦ ◦ if and only if X ∈ le +sαλ = sαλ. Assertion (ii) is a direct consequence of (i). ◦ ∗ +∗ 4.2. Sets c α(λ,µ), cα(λ,µ),andcα (λ,µ). Let α = (αn)n ∈ U be a given sequence, we consider now for λ ∈ U, µ ∈ s the space c α(λ,µ) = wα(λ) ∆(µ) = X ∈ s | ∆(µ)X ∈ wα(λ) . (4.9) It is easy to see that c α(λ,µ) = X ∈ s | C(λ) ∆(µ)X ∈ sα , (4.10) that is, n c (λ,µ) = X = x ∈ s | 1 µ x −µ x < ∞ , α n n sup k k k−1 k−1 n≥ λn αn 2 k=2 (4.11) see [1]. Similarly, we define the following sets: ◦ ◦ cα(λ,µ) = X ∈ s | C(λ) ∆(µ)X ∈ sα , (4.12) ∗ t ◦ cα (λ,µ) = X ∈ s | X −le ∈ cα(λ,µ) for some l ∈ C . Recall that if λ = µ, it is written that c0(λ) = (w0(λ))∆(λ), t c(λ) = X ∈ s | X −le ∈ c0(λ) for some l ∈ C , (4.13) 1798 BRUNO DE MALAFOSSE and c∞(λ) = (w∞(λ))∆(λ),see[11]. It can be easily seen that ◦ ∗ c0(λ) = ce (λ,λ), c∞(λ) = c e(λ,λ), c(λ) = ce (λ,λ). (4.14) These sets of sequences are called strongly convergent to 0, strongly conver- gent, and strongly bounded. If λ ∈ U +∗ is a sequence strictly increasing to infinity, c(λ) is a Banach space with respect to n 1 Xc∞(λ) = sup λkxk −λk−1xk−1 (4.15) n≥ λn 1 k=1 with the convention x0 = 0. Each of the spaces c0(λ), c(λ), and c∞(λ) is a BK space, relatively to the previous norm (see [1]). The set c0(λ) has AK and every X ∈ c(λ) has a unique representation given by ∞ t t X = le + xk −l ek, (4.16) k=1 t where X −le ∈ c0. The scalar l is called the strong c(λ)-limit of the sequence X. We obtain the next result. Theorem 4.2. Let α, λ, and µ be sequences of U +∗. (i) Consider the following properties: (a) αλ ∈ C1; (b) c α(λ,µ) = sα(λ/µ); ◦ ◦ (c) cα(λ,µ) = sα(λ/µ); ∗ t ◦ (d) cα (λ,µ) ={X ∈ s | X −le ∈ sα(λ/µ) for some l ∈ C}. Then (a) (b) and (a)⇒(c) and (d). ◦ ∗ (ii) If αλ ∈ C1, then c α(λ,µ), cα(λ,µ), and cα (λ,µ) are BK spaces with respect to the norm x X = µ n . sα(λ/µ) sup n (4.17) n≥1 αnλn ◦ ∗ The set cα(λ,µ) has AK and every X ∈ cα (λ,µ) has a unique representation t ◦ given by (4.16), where X −le ∈ sα(λ/µ). Proof. We show that (a)⇒(b). Take X ∈ cα(λ,µ). We have ∆(µ)X ∈ wα(λ), which is equivalent to X ∈ C(µ)sαλ = D1/µΣsαλ, (4.18) and using Theorem 2.6(i), ∆ and consequently Σ are bijective from sαλ into itself. So, Σsαλ = sαλ and X ∈ D1/µΣsαλ = sα(λ/µ). We conclude that (b) holds. n We prove that (b) implies (a). First, put α λ,µ = ((−1) (λn/µn)αn)n≥1. We have ON SOME BK SPACES 1799 α λ,µ ∈ sα(λ/µ) = c α(λ,µ) = sα(λ/µ), and since ∆(µ) = ∆Dµ and Dµα λ,µ = n ((−1) λnαn)n≥1,weget|∆(µ)α λ,µ|=(ξn)n≥1,with λ α if n = 1, ξ = 1 1 n (4.19) λn−1αn−1 +λnαn if n ≥ 2. From (b), we deduce that Σ|∆(µ)α λ,µ|∈sαλ. This means that n 1 Cn = λ1α1 + λk−1αk−1 +λkαk = O(1), n →∞. (4.20) αnλn k=2 From the inequality C(αλ)(αλ) n ≤ Cn, (4.21) we obtain (a). The proof of (a)⇒(c) follows on the same lines of the proof of ◦ (a)⇒(b) with sα replaced by sα. ∗ We show that (a) implies (d). Take X ∈ cα (λ,µ). There exists l ∈ C such that t ◦ ∆(µ) X −le ∈ wα(λ), (4.22) ◦ ◦ and from (c)⇒(e) in Theorem 4.1, we have wα(λ) = sαλ.So t ◦ ◦ X −le ∈ C(µ)sαλ = D1/µΣsαλ, (4.23) ◦ ◦ ◦ ◦ and from Theorem 2.6(ii), Σsαλ = sαλ,andD1/µΣsαλ = sα(λ/µ), we conclude that ∗ t ◦ X ∈ cα (λ,µ) if and only if X ∈ le +sα(λ/µ) for some l ∈ C. Assertion (ii) is a direct consequence of (i) and of the fact that for every ∗ X ∈ cα (λ), we have N t t xn −l X −le − xk −l e = sup µn = o(1), N →∞. k α λ k= n≥N+1 n n 1 sα(λ/µ) (4.24) We deduce immediately the following corollary. Corollary 4.3. Assume that α, λ, µ ∈ U +∗. (i) If αλ ∈ C1 and µ ∈ l∞, then ∗ ◦ cα (λ,µ) = sα(λ/µ). (4.25) 1800 BRUNO DE MALAFOSSE (ii) Then ◦ λ ∈ Γ ⇒ λ ∈ C1 ⇒ c0(λ) = sλ,c∞(λ) = sλ. (4.26) Proof. (i) Since µ ∈ l∞, we deduce, using Proposition 2.1(iii), that there are K>0 and γ>1 such that αnλn ≥ Kγn ∀n. (4.27) µn t ◦ So, le ∈ sα(λ/µ) and (4.25) holds. (ii) comes from Theorem 4.2 since Γ ⊂ C1. Example 4.4. We denote by e the base of the natural system of logarithms. From the well-known Stirling formula, we have nn+1/2 1 ∼ e n √ , (4.28) n! 2π n s n+( / ) = s λ = (n /n ) ∈ Γ so (n 1 2 /n!)n e . Further, ! n since λn− 1 1 = e −(n−1)ln(1+1/(n−1)) → < 1. (4.29) λn e We conclude that nn nn ◦ √1 ◦ √1 ce , = se , ce , = se . (4.30) n! n n n n! n n n References [1] A. M. Al-Jarrah and E. Malkowsky, BK spaces, bases and linear operators, Rend. Circ. Mat. Palermo (2) Suppl. (1998), no. 52, 177–191. [2] B. de Malafosse, Some properties of the Cesàro operator in the space sr , Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. Statist. 48 (1999), no. 1-2, 53–71. [3] , Bases in sequence spaces and expansion of a function in a series of power series, Mat. Vesnik 52 (2000), no. 3-4, 99–112. [4] , Application of the sum of operators in the commutative case to the infinite matrix theory, Soochow J. Math. 27 (2001), no. 4, 405–421. [5] , Properties of some sets of sequences and application to the spaces of bounded difference sequences of order µ, Hokkaido Math. J. 31 (2002), no. 2, 283–299. [6] , Recent results in the infinite matrix theory, and application to Hill equa- tion, Demonstratio Math. 35 (2002), no. 1, 11–26. [7] , Variation of an element in the matrix of the first difference operator and matrix transformations, Novi Sad J. Math. 32 (2002), no. 1, 141–158. [8] B. de Malafosse and E. Malkowsky, Sequence spaces and inverse of an infinite matrix, Rend. Circ. Mat. Palermo (2) 51 (2002), no. 2, 277–294. [9] R. Labbas and B. de Malafosse, On some Banach algebra of infinite matrices and applications, Demonstratio Math. 31 (1998), no. 1, 153–168. ON SOME BK SPACES 1801 [10] I. J. Maddox, Infinite Matrices of Operators, Lecture Notes in Mathematics, vol. 786, Springer-Verlag, Berlin, 1980. [11] F. Móricz, On Λ-strong convergence of numerical sequences and Fourier series, Acta Math. Hungar. 54 (1989), no. 3-4, 319–327. [12] A. Wilansky, Summability Through Functional Analysis, North-Holland Mathe- matics Studies, vol. 85, North-Holland Publishing, Amsterdam, 1984. [13] K. Zeller and W. Beekmann, Theorie der Limitierungsverfahren, Zweite, erweiterte und verbesserte Auflage. Ergebnisse der Mathematik und ihrer Grenzgebi- ete, vol. 15, Springer-Verlag, Berlin, 1970 (German). Bruno de Malafosse: Laboratoire de Mathematiques Appliquées Havrais, Institut Uni- versitaire de Technologie Le Havre, Université du Havre, BP 4006, 76610 Le Havre, France E-mail address: [email protected] Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 The Scientific International Journal of World Journal Differential Equations Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Mathematical Physics Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Journal of Journal of Mathematical Problems Abstract and Discrete Dynamics in Complex Analysis Mathematics in Engineering Applied Analysis Nature and Society Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 International Journal of Journal of Mathematics and Mathematical Discrete Mathematics Sciences Journal of International Journal of Journal of Function Spaces Stochastic Analysis Optimization Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014