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THESIS SECTION
SUMMARY OF
3niinitt ilatrites! anli Snbariant i$leansi
THESIS PRESENTED FOR THE DEGREE OF
Bottor of $t)tlosiopI)|) IN
MATHEMATICS OF THE
ALIGARH MUSLIM UNIVERSITY, ALIGARH
1^22-
Under the Supervision of
Dr. Z. U. Ahmad, M. SC , D. PWI., D. SC. READER
BY
SALEEN
DEPARTMENT OF MATHEMATICS ALIGARH MUSLIM UNIVERSITY
ALIGARH X 9 T e
l^fK AZAO Ui 2^^^-
L ^ r-v
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iit««ipr» itttn i«i3 H im^ mi^ cr Ci # • f i n a l f«iitiv»
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i t eiiOft 1^ iti9em Hist ^Im m%^ f - <i! ^ aiardOlvriit&a « th«
Cii) m(^\ •< m^ \ 11(^ UrtHlii ti J i l
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> ^ > ^ » ^ > ^
infinite iWatrites: mh 3Jnljariant Mtm$
THESIS SUBMITTED FOR THE DEGREE OF
Bottot of $f|ilQsiop{)p IN
MATHEMATICS
MURSALEEN
DEPARTMENT OF MATHEMATICS ALIGARH MUSLIM UNIVERSITY
ALIGARH i 8 7 e
2 2 JUN 1980
.00^
T1922 Gia^
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to the loving memory of my
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!ml:an J. pwc appl. Math., J0(4) ; 457-460, April 1979
ON INFINITE MATRICES AND INVARIANT MEANS
MURSALEEN*
Department of Mathematics, Aligarh Muslim University, Aligarh 202001
(Received 3 August 1978)
Schaefer (1972) has introduced the concept of cr-conservative, o-regular and <j-coercive matrices and obtained necessary and sufficient conditions to characterize these classes of matrices. The purpose of this paper is to characterize some more matrices in V^.
1. INTRODUCTION
Let loo and c respectively be the Banach spaces of bounded and convergent sequences x = {xk} with the usual norm || ;c 1| == sup | xt | .
k
Let (T be a mapping of the set of positive integers into itself. A continuous Hnear functional <j> on loo is said to be an invariant mean or a-mean if and only if (i) <^{x) > 0 when the sequence x = {x„} has x„ > 0 for all n; (ii) ^(e) = 1, where e = {1, 1, . . . } ; and (iii) <j>{{xa(n)}) = ^(x) for all x £ loo- Throughout this paper we deal only with mappings a are one-to-one and are such that a"'{n) 7^ n for all positive integers n and m, where (^"{n) denotes the mth iterate of the mapping at n. For such mappings, every cr-means extends the limit functional on c (Raimi 1963), in the sense that ^(x) = lim x for all x E c. Consequently, c C V,r, where V^ is the set of bounded sequences all of whose CT-means are equal. When a(n) = n + \, F<r is the set of almost convergent sequences (Lorentz 1948).
Schaefer (1972) has obtained necessary and sufficient conditions to characterize (^. y<T), (c, V^)rea, and (/<». Fff)-matnces. Quite recently Ahmad and Mursaleen (1979) have characterized the classes {c(p), V„), {c{p), V^)rea, a.nd(lj{,p), V,) which generalize the results of Schaefer (1972). In the present paper author obtains conditions which characterize (/(/>), V„), and {M^{p), V^) matrices.
2. PRELIMINARIES
If /Ji is real such that /)«;> 0 and sup/^t < 00, vve define (Simons 1965 and k
Maddox 1967, 1969)
looip) = {x : sup I Xt I "* < 00} k
*This work is supported by a Junior Research Fellowship of C.S.I,R., India, under Grant No. 7/112(451) 75-EMR.r.
458 MURSALEEN
c{p) = {A: : I x* - /1 ^* -» 0 for some /}
and
l{p) = {X : S I X;i I "ft < 00} k
^ o ( p ) = U {X : S I x t ( N'-'i^ic < 00} N>\ k
when pk = p V k, we write l„{p) — /«,, c{p) — c, l{p) = / , and ^/^(p) = / j , also
•''^o(/') = 1 for inf/)& > 0.
If X = {xn}, write 7x = {-JCffCn)}. The set K„ can be characterized as the set of
all bounded sequences x for which lim /m„fx) exists in I^ and has the form Le, where m
m L = tj — lim X and /m„(x) = S r^x/(m + 1).
y=o
Throughout this paper we shall use the notation a{n, k) to denote the element
a„ie of the matrix A. We write for all integers n, m ^ 1
Un{Ax) = (Ax + TAx + ... + T'^Ax)lim + 1)
= S /(«, A:, OT) Xt k
where
3. M A I N RESULTS
Theorem 3.1 — .4 £ {l{p), V<J) if and only if there exists B > 1 such that for
every n
f sup S I t{n, k, m) | "f^ B'^k < 00, (1 < p^ < 00) J "^ ^
^^' I sup I t{n,k, m) | "* < °o, (0 < /;» < 1) \^mk
CO
(ii) a(&) = {flnfc} £ Fa for each k n=\
i.e. lim t{n, k, m) — un uniformly in n. m
In this case, the o-limit of A is
(iii) lim S t{n, k, m) = S ii^Xk. m k k
ON INFINITE MATRICES AND INVARIANT MEANS 459
PROOF : Let 1 < / ^ < oo and ^ G (/(p), V„). Define e* = {0, 0, .... 0, l(/cth place), 0, . . . } . Since ek G /(/'), therefore (ii) holds. Put fmnix) = tmniAx), since tmn(Ax) exists for each m and x G /(/>)• Therefore {/mn(;c)}OT is a sequence of continuous real functional on /(/>) and further sup ] fmn{x) | < oo on l{p). Now condition
m
(i) follows by arguing with uniform boundedness principle. The case 0 < /?fc < 1 has similar proof.
Conversely, suppose that the conditions (i) and (ii) are true and x G i(p)- Now we have for every r "^ I
r 00
S I t{n, k, 777) I H B-n < sup S 1 t{n, k, m) | "k B-n . / t=] m k=l
Therefore
S I tik I "* B-n = lira lim S j tin, k, m) \ «* B~n k r m / c= l
< sup S I t{n, k, m) \ «* 5-«s < oo. m k—\
Thus the series S t{n, k, m) xt and E MjtXfc converge for each 7?7 and x G (z?). For a fc A;
given e > 0 and x G /(jP), choose k^ such that
( S i X;, I "* ) < e
where /f = sup pn. Since (ii) holds, therefore there exists m^ such that
(iv) S (r{77, /C, m) — Mfc) < € ( V m > Wo).
By condition (ii) it follows that
S (<(«, /c, m) — Mfc) /c=/f„+l
is arbitrary small, therefore condition (iii) follows i.e.,
lim S t(n. k, m) Xs — S u^xjc m k k
uniformly in n. This completes our proof.
Theorem 3.2 — /I G {M({p), V,) if and only if for every integer A' > 1
(i) MN = sup 1 f(n, k, m) | m'n < oo, ( v n) m,k
460 MURSALEEN
(ii) 0(4) G Va for each k.
In this case, the o-limit of Ax is same as in Theorem 3.1.
PROOF : Let ^ E (M^ip), K„). Since et G M^ip), (ii) must hold. Now on
contrary suppose that (i) is not true then there exists N > I such that M/f = oo.
Therefore by Theorem 3.1, matrix B = {bm) = (a„kN^l'>k) ^ (1^, Va), i.e. there exists
X G /i such that Bx ^ F„. Now y = ( js) = (m">jcXH) G M^ip). But /4:v = ^.v ^ F^,
which contradicts the fact that A G (•'^o(/'). ^a)- Hence (i) is true.
Conversely, suppose conditions (i) and (ii) are true and x G M„{p). Then
I S t{n, A;, ffj) Xi I < S 1 Xfc I A^-''"* | /(n. A;, m) | iVi/"* < oo. k k
Now arguing as in Theorem 3.1 we get
lim S t(n, k, m) Xh = !• utx^. m k k
Hence A G {.M^(p), V„).
ACKNOWLEDGEMENT
The author is grateful to Dr Z. U. Ahmad for his guidance and suggestions.
REFERENCES
Ahmad, Z. U., and Mursaleen (1979). On infinite matrices and invariant means. Read at the 66th Session of the Indian Science Congress Association at Hyderabad.
Lorentz, G. G. (1948). A contribution to the theory of divergent sequences. Acta Math., 80, 167-90. Maddox, I. J. (1967). Spaces of strongly summable sequences. Q. Jt. Math., Oxford (2), 18, 345-55.
(1969). Continuous and Kothe-Toeplitz duals of certain sequence spaces. Proc. Camb.
phil. Soc, 65, 431-35. Raimi, R. A. (1963). Invariant means and invariant matrix methods of summability. Duke Math.
/., 30, 81-94. Simons, S. (1965). The sequence spaces /(/>„) and m(p„). Proc. Land. math. Soc, 3(15), 422-36. Schaefer, P. (1972). Infinite matrices and invariant means. Proc. Am. math. Soc, 36 (1), 104-10.
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