BULLETIN of the Malaysian Mathematical Sciences Society
Bull. Malays. Math. Sci. Soc. (2) 32(2) (2009), 223–231
http://math.usm.my/bulletin
Triangles Which are Bounded Operators on Ak 1
E. Savas¸, 2 H. S ¸ evli and 3 B. E. Rhoades
1 Department
of Mathematics, Istanbul Commerce University Uskudar, Istanbul, Turkey 2 Department of Mathematics, Faculty of Arts & Sciences, Y¨ uz¨ unc¨ u Yıl University, Van, Turkey 3 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA 1
[email protected], 2
[email protected], 3
[email protected]
Abstract. A lower triangular infinite matrix is called a triangle if there are no zeros on the principal diagonal. The main result of this paper gives a minimal set of sufficient conditions for a triangle T : Ak → Ak for the sequence space Ak defined as follows:
(
Ak :=
{sn } :
∞ X
)
nk−1 | an |k < ∞ , an = sn − sn−1
.
n=1
2000 Mathematics Subject Classification: 40C05 Key words and phrases: Bounded operator, triangular matrices, Ak spaces, weighted mean methods.
1. Introduction and background P Let av denote a series with partial sums sn . For an infinite matrix T , tn , the n th term of the T -transform of {sn } is denoted by tn =
∞ X
tnv sv .
v=0
P P A series av is said to be absolutely T -summable if n |∆ tn−1 | < ∞, where ∆ is the forward difference operator defined by ∆ tn−1 = tn−1 − tn . Papers dealing with absolute summability date back at least as far as FeketeP[2]. A sequence {sn } is said to be of bounded variation (bv) if n |∆ sn | < ∞. Thus, to say that a series is absolutely summable by a matrix T is equivalent to saying that the T -transform the sequence is in bv. Necessary and sufficient conditions for a matrix T : bv → bv are known. (See, e.g. Stieglitz and Tietz [7]). Let σnα denote the nth terms of the transform of a Ces´aro matrix P (C, α) of a sequence {sn }. In 1957 Flett [3] made the following definition. A series an , with Received: September 8, 2008; Accepted: 27 November 2008.
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E. Savas¸, H. S¸evli and B. E. Rhoades
partial sums sn , is said to be absolutely (C, α) summable of order k ≥ 1, written P an is summable |C, α|k , if (1.1)
∞ X
k α − σnα < ∞. nk−1 σn−1
n=1
P He then proved the following inclusion theorem. If series an is summable |C, α|k , it is summable |C, β|r for each r ≥ k ≥ 1, α > −1, β > α + 1/k − 1/r. It P then follows that, if one chooses r = k, then a series an which is |C, α|k summable is also |C, β|k summable for k ≥ 1, β > α > −1. P Quite recently, Rhoades and Sava¸s [5] established sufficient conditions for a series an summable |A|k to imply that it is summable |B|k , where A and B are particular lower triangular matrices. P Let an be a series with partial sums sn . Define ( ) ∞ X k k−1 (1.2) Ak := {sn } : n | an | < ∞ , an = sn − sn−1 . n=1
If one sets α = 0 in the inclusion statement involving (C, α) and (C, β), then one obtains the fact that (C, β) ∈ B (Ak ) for each β > 0, where B (Ak ) denotes the algebra of all matrices that map Ak to Ak . In 1970, using the same definition as Flett, Das [1] defined a matrix T to be absolutely kth power conservative for k ≥ 1, if T ∈ B (Ak ); i.e., if {sn } is a sequence satisfying ∞ X k nk−1 | sn − sn−1 | < ∞, n=1
then
∞ X
k
nk−1 | tn − tn−1 | < ∞.
n=1
In that same paper he proved that every conservative Hausdorff matrix H ∈ B (Ak ), which contains as a special case the fact that (C, β) ∈ B (Ak ) for β > 0. In [6], it is shown that (C, α) ∈ B (Ak ) for each α > −1. Therefore being conservative is not a necessary condition for a matrix to map Ak to Ak . Given a matrix T one can find a matrix B such that the statement T ∈ B (Ak ) is equivalent to B ∈ B `k . Since necessary and sufficient conditions are not known for an arbitrary B ∈ B `k for k > 1, it is not reasonable to expect to find necessary and sufficient conditions for T ∈ B (Ak ). A lower triangular matrix with nonzero principal diagonal entries is called a triangle. In this paper we obtain a minimal set of sufficient conditions for a triangle T ∈ B (Ak ). As corollaries we obtain other A ∈ B (Ak ) results including that of [5]. We may associate with T two infinite matrices T¯ and Tˆ as follows: t¯nv =
n X
tnr
,
n, v = 0, 1, 2, ...
r=v
and tˆnv = t¯nv − t¯n−1,v
,
n = 1, 2, 3, ...,
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225
where tˆ00 = t¯00 = t00 . If T = (tnv ) is a lower triangular matrix, then T¯ = (t¯nv ) and Tˆ = tˆnv are also lower triangular matrices. If T is a triangle, then for each n ∈ N0 tˆnn = t¯nn = tnn 6= 0, and T¯ and Tˆ are triangles. 2. Main results Theorem 2.1. T = (tnv ) be a triangle satisfying (i) |tnn | = O (1) , n−1 P (ii) |tvv | tˆnv = O (|tnn |), v=0
and (iii)
∞ P
k−1
(n |tnn |)
tˆnv = O (v |tvv |)k−1 .
n=v+1
Then, T ∈ B (Ak ) , k ≥ 1. Proof. If yn denotes the nth term of the T -transform of a sequence {sn }, then n n v n n X X X X X yn = tnv sv = tnv ai = ai tnv v=0
=
n X
v=0
i=0
i=0
v=i
t¯ni ai ,
i=0
and Yn := yn − yn−1 =
n X
t¯nv av −
v=0
= =
n X v=0 n X
n−1 X
t¯n−1,v av
v=0
(t¯nv − t¯n−1,v ) av tˆnv av
v=0
=
n−1 X
tˆnv av + tˆnn an
v=0
= Tn1 + Tn2 . By Minkowski’s inequality it is sufficient to prove that ∞ X k nk−1 |Tnr | < ∞, r = 1, 2. n=1
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226
Using H¨older’s inequality, (ii), and (iii), we get ∞ X
n
k−1
k
|Tn1 | =
n=1
∞ X
n
n−1 k X ˆ tnv av v=0 )k n−1 X |tvv | tˆnv |av |
k−1
n=1
≤
∞ X
nk−1
n=1
≤
∞ X
n
n−1 X
k−1
n=1
|tvv |
v=0
tˆnv |av |k ×
1−k
|tvv |
(n−1 X
v=0
= O (1)
∞ X
v=0
nk−1
n=1
= O (1)
∞ X
n−1 X
1−k
|tvv |
= O (1)
tˆnv |av |k × { |tnn | }k−1
v=0 1−k
|tvv |
k
|av |
v=0 ∞ X
)k−1 |tvv | tˆnv
∞ X
k−1
(n |tnn |)
tˆnv
n=v+1 1−k
|tvv |
k
k−1
|av | v k−1 |tvv |
v=1
= O (1)
∞ X
k
v k−1 |av | = O (1).
v=1
Using (i), ∞ X
n
k−1
k
|Tn2 | =
∞ X
k
nk−1 |tnn an |
n=1
n=1
= O (1)
∞ X
k
nk−1 |an | = O (1) ,
n=1
since {sn } ∈ Ak . This completes the proof. Using an argument similar to that of [4] we now establish another set of sufficient conditions for a nonnegative triangle T with row sums one and decreasing columns to be in B (Ak ). We then show that the conditions of Theorem 2.2 imply those of Theorem 2.1. Theorem 2.2. Let T = (tnv ) be a nonnegative triangle satisfying (i) (ii) (iii)
ntnn 1, t¯n0 = 1 for n = 0, 1, 2, ... , tn−1,v ≥ tnv for n ≥ v + 1,
and (iv)
n−1 P
tvv tˆnv = O (tnn ).
v=0
Then, T ∈ B (Ak ) , k ≥ 1.
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227
Proof. By {xn } we denote the T -transform of {sn }. Then n X xn = tnv sv , v=0
and xn − xn−1 =
n X
tˆnv av
v=0
=
n−1 X
tˆnv av + tˆnn an
v=0
= Tn1 + Tn2 , say. Using H¨older’s inequality, (i) and (iv) m+1 X
n
k−1
k
|Tn1 | =
n=1
≤
n=1
n−1 k X tˆnv av
m+1 X
n−1 X
m+1 X
n
k−1
v=0
nk−1
n=1
v=0
= O (1)
m+1 X
)k−1 n−1 X 1−k k tˆnv |tvv | |av | × |tvv | tˆnv v=0
k−1
(ntnn )
n=1
= O (1)
m+1 X
n−1 X v=0
1−k
|tvv |
k
|av |
v=1
= O (1)
m+1 X
tˆnv |tvv |1−k |av |k m+1 X
k−1
(n |tnn |)
n=v+1 1−k
|tvv |
k
|av |
m+1 X
tˆnv .
n=v+1
v=1
Since T = (tnv ) is a positive matrix, from (ii) and (iii), tˆnv =
n X
tnr −
r=v
=1−
n−1 X
v−1 X
tnr − 1 +
r=0
=
v−1 X
tn−1,r
r=v v−1 X
tn−1,r
r=0
(tn−1,r − tnr ) ≥ 0,
r=0
and Tˆ = tˆnv is a positive matrix. Therefore, using (ii) m+1 X n=v+1
m+1 X tˆnv = (t¯nv − t¯n−1,v ) n=v+1
= t¯m+1,v − tvv = O (t¯m+1,v ) = O (t¯m+1,0 ) = O (1) .
tˆnv
E. Savas¸, H. S¸evli and B. E. Rhoades
228
Using (i) and since {sn } ∈ Ak , m+1 X
k
nk−1 |Tn1 | = O (1)
m+1 X
n=1
1−k
|tvv |
k
|av |
v=1
= O (1)
m+1 X
k
v k−1 |av | = O (1) , as m → ∞.
v=1
Using (i), m+1 X
k
nk−1 |Tn2 | =
n=1
m+1 X
k
nk−1 |tnn an |
n=1
= O (1)
m+1 X
k
nk−1 |an | = O (1) , as m → ∞,
n=1
since {sn } ∈ Ak . Hence the proof is complete. Corollary 2.1. The conditions of Theorem 2.2 imply these of Theorem 2.1. Proof. Conditions (i) and (iv) of Theorem 2.2 with tnv nonnegative imply conditions (i) and (ii) of Theorem 2.1, respectively. Since T = (tnv ) is a positive matrix, from (ii) and (iii) of Theorem 2.2, then ˆ T = tˆnv is a positive matrix. Using (i) and (ii) of Theorem 2, ∞ X
1 k−1
(v |tvv |)
k−1
(n |tnn |)
∞ X tˆnv = O (1) tˆnv = O (1),
n=v+1
n=v+1
and condition (iii) of Theorem 2.1 is satisfied. Setting A = I, the identity matrix, in Theorem 1 of [5] gives the following result. Corollary 2.2. Let T be a triangle satisfying (i) |tnn | = O (1) , n−1 P (ii) ∆v tˆnv = O (|tnn |), v=0 ∞ P k−1 k (n |tnn |) (iii) ∆v tˆnv = O v k−1 |tvv | , (iv)
n=v+1 n−1 P
|tvv | tˆn,v+1 = O (|tnn |),
v=0
and (v)
∞ P
k−1
(n |tnn |)
tˆn,v+1 = O (v |tvv |)k−1 .
n=v+1
Then, T ∈ B (Ak ) , (k ≥ 1) . Proof. Condition (i) of Corollary 2.2 is condition (i) of Theorem 2.1. Conditions (ii)–(iii) of Theorem 2.1 can be obtained from conditions (i)–(v) of Corollary 2.2 as follows. Recall that ∆v tˆnv = tˆnv − tˆn,v+1 .
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Thus tˆnv = ∆v tˆnv + tˆn,v+1 . Using (i), (ii) and (iv) of Corollary 2.2, n−1 X
X n−1 |tvv | tˆnv = |tvv | ∆v tˆnv + tˆn,v+1
v=0
v=0
≤
n−1 X
X n−1 |tvv | tˆn,v+1 + |tvv | ∆v tˆnv
v=0
=
n−1 X
v=0 n−1 X ∆v tˆnv |tvv | tˆn,v+1 + O (1)
v=0
v=0
= O (|tnn |) + O (|tnn |) = O (|tnn |) , and condition (ii) of Theorem 2.1 is satisfied. Using (i), (iii) and (v) of Corollary 2.2, ∞ X
k−1
(n |tnn |)
∞ X k−1 tˆnv = (n |tnn |) ∆v tˆnv + tˆn,v+1
n=v+1
≤
n=v+1 ∞ X
k−1
(n |tnn |)
n=v+1 ∞ X
+
∆v tˆnv
k−1
(n |tnn |)
tˆn,v+1
n=v+1
k−1 k−1 = O (v |tvv |) |tvv | + O (v |tvv |) k−1 = O (v |tvv |) , and condition (iii) of Theorem 2.1 is satisfied. We remark that conditions (i)–(iv) of Theorem 2.2 imply conditions (i)–(v) of Corollary 2.2. Now we will show the remarks. Condition (i) of Theorem 2.2 with tnv nonnegative implies condition (i) of Corollary 2.2. Since ∆v tˆnv = −∆n (tn−1,v ) , and using conditions (ii) and (iii) of Theorem 2.2, we have n−1 X
X n−1 ∆v tˆnv = |tnv − tn−1,v |
v=0
v=0
=
n−1 X v=0
tn−1,v −
n−1 X
tnv
v=0
= t¯n−1,0 − t¯n0 + tnn = tnn ,
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230
and condition (ii) of Corollary 2.2 is satisfied. By (iii) of Theorem 2.2 m+1 X
m+1 X ∆v tˆnv = |tnv − tn−1,v |
n=v+1
n=v+1
=
m+1 X
(tn−1,v − tnv )
n=v+1
= tvv − tm+1,v ≤ tvv . Thus
∞ X ∆v tˆnv = O (|tvv |). n=v+1
Using condition (i) of Theorem 2.2 ∞ ∞ X X k−1 ∆v tˆnv = O v k−1 |tvv |k , (n |tnn |) ∆v tˆnv =O (1) n=v+1
n=v+1
and condition (iii) of Corollary 2.2 is satisfied. Obviously, conditions (i)–(iv) of Theorem 2.2 imply conditions (iv) and (v) of Corollary 2.2. Corollary 2.3. If {pn } is a positive sequence satisfying npn Pn ,
(2.1) ¯ , pn ∈ B (Ak ) . then N
¯ , pn . Condition (i) of Theorem 2.1 is clearly Proof. In Theorem 2.1 set T = N satisfied. n−1 X pv pn Pv−1 Pn n−1 1 X |tvv | tˆnv = |tnn | v=0 pn v=0 Pv Pn Pn−1 = O (1)
1
n−1 X
Pn−1
v=0
pv = O (1)
and condition (ii) of Theorem 2.1 is satisfied. k−1 X k−1 ∞ ∞ X 1 npn pn Pv−1 Pv k−1 ˆ (n |t |) t = nn nv k−1 vpv Pn Pn Pn−1 v |tvv | n=v+1 n=v+1 = O (1) Pv−1
∞ X
O (1)
n=v+1
= O (1)
pn Pn Pn−1
Pv−1 = O (1) Pv−1
and condition (iii) of Theorem 2.1 is satisfied. From Das [1], (C, 1) ∈ B (Ak ). For completeness, we show that Corollary 2.4 follows from Corollary 2.3.
Triangles Which are Bounded Operators on Ak
231
Corollary 2.4. (C, 1) ∈ B (Ak ) . Proof. Set each pn = 1 in Corollary 2.3. Then the condition (2.1) is satisfied. ¯ , pn and (C, 1) are nonnegative and have each row sum equal to one, Since N we also obtain Corollary 2.3 and Corollary 2.4 from Theorem 2.2. Remark 2.1. Condition (i) of Theorem 2.1 is necessary. To see this, let e(j) denote the jth coordinatePsequence; i.e., the sequence with 1 in position j and zeros n elsewhere. Then Yn = v=0 tˆnv av satisfies 0 , n j . Since T ∈ B (Ak ) there exists a positive number m such that ∞ ∞ X X k k m≥ nk−1 |Yn | = nk−1 |Yn | n=1
≥j
k−1
n=j k
|Yj | = j
k−1
k
|tjj | ,
which implies that |tjj | = O(1), and condition (i) is necessary. References [1] G. Das, A Tauberian theorem for absolute summability, Proc. Cambridge Philos. Soc. 67 (1970), 321–326. ` [2] M. Fekete, Zur theorie der divergenten reihen, Math. ` es Termezs Ertesit¨ o (Budapest) 29 (1911), 719–726. [3] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. (3) 7 (1957), 113–141. [4] B. E. Rhoades, Inclusion theorems for absolute matrix summability methods, J. Math. Anal. Appl. 238 (1999), no. 1, 82–90. [5] B. E. Rhoades and E. Sava¸s, General inclusion theorems for absolute summability of order k ≥ 1, Math. Inequal. Appl. 8 (2005), no. 3, 505–520. [6] E. Sava¸s and H. S ¸ evli, On extension of a result of Flett for Ces` aro matrices, Appl. Math. Lett. 20 (2007), no. 4, 476–478. [7] M. Stieglitz and H. Tietz, Matrixtransformationen von Folgenr¨ aumen. Eine Ergebnis¨ ubersicht, Math. Z. 154 (1977), no. 1, 1–16.
