Aa a by pp a 2 A Ap xx A Ap x Ax0000 nco k Ap Ap 0 a a p A Let xy x complex numbers Then implies P and imply x and imply x ID: 849849
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fcth 0 k e [A^p]^ = ld S(d) U.&#x^000; m 1 2finite k b 2 \x U, 2 G 2 x e Let B=(b be any matrix of noughts and ones, p be any strictly positive sequence and If B is column finite and [B, [B there exists an integer such that of Theorem 1 holds, where s(ri)={k:b [B, [B^p]^ n, N=N(ri) 2 1�n(l)l 2 ^ 2 2" k e s(m). ir a ' R Z Q O R D G H G I U R P K W W S V Z Z Z F D P E U L G J H R U J F R U H - X Q D W V X E M H F W W R W K H & D P E U L G J H & R U H W H U P V R I X V H � N 1 () U) !**'* N (DIN** � NN Bb�
5 nn{2) \\) (n(2)) f Nl* � k(2)k
nn{2) \\) (n(2)) f Nl* � k(2)k(l) k(2) 2 Nl* (N (n(i)), (k(i)) U)2W*£1, l() N&#xni00; N b 0 1 k k(f), k(i+l) 2 x es(n(i+l)) i=0, ,� nl. n(i)() fc(fl)(࿀,;kk(i+l). x, 2 2 1 2 (n(i+i)) 2 2 2 x, 3 3 2kKkM xe[B,(r)]. x $ [B,p]^. w^ip). ' R Z Q O R D G H G I U R P K W W S V Z Z Z F D P E U L G J H R U J F R U H - X Q D W V X E M H F W W R W K H & D P E U L G J H & R U H W H U P V R I X V H Suppose that A is a column finite matrix which satisfies thecondition of Theorem 2. Let Then the following conditions a
6 re equivalent: is r-convex. [B, where
re equivalent: is r-convex. [B, where 0\ and a and b otherwise. There exists an integer� N 1 such that J N n s(n) where s(n) = {k:Q andp and 3 1 m(p)={x:sup \x is l-convex if and only if 0s\xpp 2 4 A w^(p): = = = ^ a k 2� n0. w^ip)c \ c D=(d d&#xn000; n\ rf C D w^ip)- w [D,p]^ n J The following statements are equivalent: w r-convex. ^ [B,/?]«,, ft 2 otherwise. and there is an integer� N1 such that 2 N* n s(n) where s(n)={k:2 ' R Z Q O R D G H G I U R P K W W S V Z Z Z F D P E U L G J H R U J F R U H - X Q D W V X E
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M H F W W R W K H & D P E U L G J H & R U H W H U P V R I X V H g aw^ip). 2 0 a X w^(p) 4 A = D. 5 1 [A^p]^ w^(p) C p k=2\ i=Q, . . p k^l*. s(n) C {k:l k=2 s(2 N 5 N=2, [B, [B,p]^ [A.p]^ A = C,[B, £ a1� (kl) a[A,p] [B,p] 2 p 7T k=2\ 3 w w^(p) The inclusion not necessary for the r-convexity p k=2* p w^ip) 2 w^ip)^ w^r) = k 2 [A^p]^ [A,p] [A, p]^ 4 [A,p] w w supp ' R Z Q O R D G H G I U R P K W W S V Z Z Z F D P E U L G J H R U J F R U H - X Q D W V X E M H F W W R W K H & D P E U L G J H & R U H W H U P V R I X
8 V H a Let and T={k\k e S and aLet [
V H a Let and T={k\k e S and aLet [A.p]^ {respectively [A,p] be paranormed. Then it is locally bounded if and only if inf (respectively [A,p]^. a=inf S(l) 6 1 a 6. N 0. a S(d)^N. X� \X\\ \À\~ sup x e S(\) \1/M 2a ! xjXe S(l)^XN, a [A^p]^ a B 0� d0 S(d) B a X XS(d) k e S x S(d) Id^suvaS^'e^. V a inf T S x e [A,p] n, 2 \x = 2 a T e [A,p] a� (n-co). c and m(p) are normable if and only if 0 (p) is normable if and only if p w and w^{p) are normable if and only if and where otherwise. 7 Linear functionals with Note on strong summability, ' R Z Q O R D G H G I U R P K W W S V
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Z Z Z F D P E U L G J H R U J F R U H - X Q D W V X E M H F W W R W K H & D P E U L G J H & R U H W H U P V R I X V H Matrix between some classes sequences, Spaces of summable On Kuttnefs Paranormed generated by infinite matrices, Some properties sequence 1 Maddox Absolute convexity in topological linear The £(p) and m(p), ' R Z Q O R D G H G I U R P K W W S V Z Z Z F D P E U L G J H R U J F R U H - X Q D W V X E M H F W W R W K H & D P E U L G J H & R U H W H U P V R I