On spaces o! strongly summable sequences with an Orlicz metric

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1968)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1968)

A.

(Poznań)

On spaces o! strongly summable sequences with an Orlicz metric

1. In order to built up a general theory of modular spaces it is ad­

visable to investigate concrete examples of modular spaces which may be applied in various problems of mathematical analysis. Modular spaces of integrable functions (i. e. Orlicz spaces) have been investigated fairly well. This cannot be said about spaces whose elements are sequences.

Therefore in this paper I develop the theory of spaces of strongly summable sequences, thus continuing investigations started by J. Musielak and W. Orlicz [ 8 ], by J. Musielak [ 6 ] and by W. Orlicz [10]. The above quoted authors limited themselves to investigation of the case of strong sum- mability to zero by means of the first arithmetic means. In this paper we shall make more general assumptions on methods of the strong summabil- ity. I shall deal in an other paper with similar problems concerning methods of strong summability of double sequences, integral methods, etc. (*).

I am very indebted to Professor W. Orlicz for his kind remarks and suggestions and to Dozent J. Musielak for his remarks in course of pre­

paration of this paper.

1.1. Notation. Let T , Tb, TQ, Tf denote spaces of all real sequences, bounded real sequences, real sequences convergent to zero, “finite”

sequences (i.e. sequences with a finite number of elements different from zero), respectively. Sequences belonging to T will be denoted by x — {/„}, у = {sv} } etc. We shall write <9 for the zero sequence 0, 0, . . . ;| a?|

= {|<J}; xn, en, en will mean the sequences tx, t2, ..., tn, 0 , 0 , ...; 1 , 1 , ...

. . . , 1 , 0 , 0 , . . . (with 1 at the first n places); 0 , 0 , . . . , 0 , 1 , 0 , 0 , . . . (with 1 at the nth. place), respectively. Finally, e% will denote the sequence 0 , 0 , . . . , 1 , 1 , . . . , 1 , 0 , 0 , . . . with 1 at the ptii, (p-\-l )st, ..., (p-\-q— l)st place. If x x, x 2e T , x x = {tl}, x 2 = {tl}, the inequality x x > x 2 will mean tl > tl for v = 1 , 2 , ... The relation > is a semiorder relation in T and

(*) The results were presented on a Seminar of Professor W. Orlicz on modular spaces, October 10th, 17 th, 31th, 1966.

230 A. W a s z a k '

T is a linear lattice complete with, respect to this relation, i.e. an arbitrary non-empty set of elements from T bounded from above has a least upper bound belonging to T. The supremum (infimum) of two elements x x, x 2eT will be denoted by x x v x 2 (xx л x 2) . As regards larger sets of elements, the supremum will be denoted by x x v x 2 v ... v xn, У xn, etc.

1.2. By a ^-function we understand a continuous non-decreasing function <p(u) defined for u ^ O and such that 99(0) = 0 , <p(u) > 0 for и > 0 and y(u)

as и oo. 99 -functions will be denoted by Greek letters 99, гр, ..., and the inverse functions by q>_x, ip_ x, ..., respectively.

Moreover, in this paper A = (anv) denotes a non-negative matrix (anv > 0 for n , v = 1 , 2 , ...) which contains no column consisting of zeros only.

Let the matrix A and a 99 -function 99 be given. We adopt the following notation:

for n = 1 , 2 , . . . ,

v = 1

T v0 = {xeT: Gn(x) <

for n — 1 , 2 , . . . and o%,(x) -> 0 for n ->

T ę = {xeT: XxeTv0 for an arbitrary A > 0 },

T* — {xeT: Xx€Tę,0 for a certain Я > 0}.

In these notations we did not mention the dependence on the matrix A, since in our consideration we shall deal with a fixed matrix A only.

Sequences x belonging to T* are called strongly (A , <p)-summable to zero. The notion of strong ( A , 99 )-summability is a generalization of strong summability with an exponent a if we take <p(u) = ua, a > 0 . In the definition of the space T* there are no restrictions on the matrix A with the only exception of the previously made. In order to develop a theory of these spaces, we shall need in particular theorems some addition­

al assumptions on A . Bor example, it is easily shown that Tf a T* if and only if anv 0 as n ->

for v — 1, 2, ... If T% contains at least one sequence whose ftth term is different from 0 , then ank -> 0 as n ->

1.3.1. By the usual definition of addition of sequences and multipli­

cation of a sequence by a number, T * is a linear space, and Tę ,0 is a convex set in Tę. In particular, if A 0 a?eTę ,0 for some Я 0 > 0 then XxeTę,0 for all 0 < Я < Я0.

1.3.2. Т% is a linear lattice and a sublattice of the lattice T with the same order relation as in T. The space T*ę is an order cr-complete lattice (conditionally complete in the terminology of G. Birkhoff), in fact, even an order-complete lattice.

We may restrict ourselves to the case X i ^ i 9, i — 1 , 2 , . . . , у > 0.

Let Xi < у for i = 1 , 2 , ..., Xi, у eTę, where = {tv}, у = {«„}.

Spaces of strongly summable sequences 231

We have to show the existence of x 0 = \Jх^Т%. We have tv^ s r for every v.

Let snp^ = t°v, then x 0 — {t°v} < y. Since ?/eT*, there exists a number

Я > 0 such that XyeTy0. Hence

CO 0 0

ol{Xy) = ^ a nv(p{X\sv\) > anv(p(X\fv\) = о%{Ы0).

" V =1 V =1

Thus we obtain о^(Яж0) 0 as n -> oo, because On(Xy) -> 0 as n -> oo.

Consequently, ж 0 еТ* and a ?0 is the least majorant of щ in T%.

1.3.3. We define the following functional in T*:

I

On (x) for ХеТуц, n

OG for a?eT*\Tę,0.

The functional Qę{x) is a modular in T* in the sense of the definition adopted by Orlicz [9], [11], i.e. satisfies the following conditions:

A. Qę(x) = 0 if and only if x = 0 , B. ev(a?i) <

{

2) if \хх\ < \x2\,

C. ev(a?i v a?2) < Q vM +

{

2) if aq > 0 , a ?2 > 0 , D. ^(Яа?) 0 as Я 0.

We limit ourselves to the proof of the property D only. Let xeT*, i.e. let Х0хеТу0 for some Я 0 > 0. Given e > 0, there exists an index m such that Gn{X0x) < e for n ^ m. Hence аЦХх) < e for 0 < Я < Я0. Since an{Xx) < e + sup an(Xx)

n < m

and the sums al(X0x) are finite for n = 1 , 2 , . . . , (m — 1), we have sup ог£(Яж) < 2e for sufficiently small Я.

П

1.4.1. Let us suppose that the matrix A = (anv) satisfies the inequali­

ties апг + ап2+ ... < К for n = 1 , 2 , ... Then T* rs Tb = T* ^ Tb for arbitrary two ер-functions cp and ip.

We need only to prove T* Tb a T* Tb. If х е Т ^ г л Т ь, then Xbx e T ę(i for some Я0. Moreover, given rj > 0, we have Я0|£„| < mr\ for v = 1 , 2 , ... and for sufficiently large m > 0. But (see [10]) there holds the inequality

r(AICI) < w(v) + ip(mr])

<p(y) <p(h\tv\)-

Multiplying both sides of this inequality by anv we obtain after summation o%(XQx) < ip (rj)

2

aUv + ip(mrj)

<

Z{X0

).

232 A. W a s z a k

But Оп{Хйх) -> 0 as п — > оо. Hence limsup(r£(/l 0 a;) < Ky(r]). Thus, п

о%(Хох ) 0 as п -»

1.4.2. I f the matrix A possesses the property + a n2 + • • • < К for n = 1 , 2 , . . . , then T* Ть — T v Tb for an arbitrary y-function y.

Evidently, we have T v T b cz T* о Tb. In order to show that Ту T b cz T ę rs T b, we choose xeT* r-, Tb. Let X0x e T ę0 for some X0 > 0, and let x e T b. Denoting y{u) = у{Х^1и), 1.4.1 yields x e T v, because 0 ДО 0 #) = el{x).

1.5.1. It follows from 1.3.3 that T * is a semiordered modular space in the sense of the definition of [ 1 1 ] and one can define a norm ||*||p in Tę as follows:

(*) INI* = inf{e > 0: < e}.

Then И*Up is an E-norm. If у is an s-convex ^-function (0 < s < 1), i.e.

y(au-\-(3v) < ay{u)-\- ffy{v) for u, v > 0 , a + = 1 , then a norm can be defined in the space T* by means of the following formula:

(**) INIs,, = inf{e > 0 : Q^x/e118) < 1 ).

This norm is s-homogeneous and equivalent to the norm (*). If s = 1, i.e. if у is a convex ^-function, the norm defined in (**) is denoted by II* ||p; it is a homogeneous norm. 1.3.3.В implies monotony of ||*||p and of ||*l|Sp, immediately, that is if H x-± x% then ||«ri|jęj INI* and Hajillsp

< ||a? 2 ||Sp, respectively. Coordinates tn of the sequence x = {tv} are linear functionals over T* with respect to the norm ||*||p or ||*||Sp, respectively.

Moreover, the following relations are equivalent:

(a) M * - * 0 as n - ^ o o ,

(b) Qę{^Xn) -> 0 as n oo . for every X > 0 .

Still we show that T v is a linear subspace of the space T*, closed with respect to the norm (*).

Let xn€ T ę ,x 0€T*ę,\\xn — x 0 ||p-*0 as n -* oo. By 1.5.1. (a), (b), Qv {H®n — x 0)) -> 0 for every X. But the property 1.3.3.B.C implies

6<р(Лхo) < Qę(2X(yn—x Q)) + Qę{2Xxn),

Hence we have 2X(xn—x0)j <

for every X for sufficiently large n.

But Qv(2Xxn) <

Consequently,

(X

0) <

i.e. x 0eTv .

In the sequel, writing T*, T v, ... we shall understand the respective

spaces of sequences as normed spaces with the norm (*) (generated by y)

or some other equivalent norm.

Spaces of strongly summable sequences 233

1.5.2. Let anv ->

as n -> oo for v =

,

, ... Then ew, belong to T*. Let us denote supunv = A v and &Щ){апр-\г аПгР+х-\~...-\-апр+а_1)

П П

— A p. If cp is an «-convex 99-function, then cp is strongly increasing and calculation gives the following values of the «-homogeneous norms of the above sequences: .

1.5.3. Let anv-> 0 as n->oo for all v. I f x e T ^ then ||ж —a?*|| ->0 as Tc-^oo.

Let xeTy', given an e >

, there exists n

such that о%,{х/е) < e

for every n > n 0. Moreover, one can choose and index v0 (depending on e) such that

anv

<p(\tv

\It)+ ^ + 1 1 ( 1 ^ + ill£) + •■■ < «/2 for n ^ n 0.

Hence we obtain

(compare [8j, [6J). However, one should take into account the fact that spaces Xm in [6], [

] are identical with spaces T v, and spaces X*, X * defined in [6] are not studied in this paper. The space X* contains T*, for it consists of all xeT such that supo^(a?) < °°-

П

1.5.4. Let 99 denote a convex 99-function satisfying the conditions

( O x )

4>(u ) 0

и as ₀ , (°°i)

(p{u)

--- >00 as

и U -> oo.

The complementary function 9?* is defined by the formula

<p*(v) = sup(w — <p{u)).

M >0

It is also a 99-function satisfying the conditions (Ox), (ocq). The following Young inequahty is satisfied

uv < 99 (u) +q>*(v) (compare e.g. [2]).

Let us suppose in

.

and

.

that the matrix A — {anv) satisfies

the condition anv -> 0 as n -> 00 , v = 1 , 2 , ...

2 3 4 A . W a s z a k

Let x €Tę, then

= sup sup У anvtvs„

where the supremum is taken over all y e T ^ satisfying the inequality Q<r*{y) ^ I» is a monotone Б -norm in T*.

Let X0x e T 9Q. Applying the Young inequality to ^-functions <p and Ф* we obtain

A0tvsv < (p(A0tv)+(p*(sv) and

1«- = X

®nv^P (^0 ^v) I anv<P*(sv

Hence ||- ||J is finite. The homogeneity and the subadditivity of ||-||* in T * are easily shown. Evidently, x = 0 implies ||a?||* = 0. In order to prove that ||ж||* = 0 implies x = 0 let us take у = ev. This sequence belongs to T,,* and moreover, ^(Яе,,) < 1 f°r Я = 99 * i^A,,)-1). Since

^ = INI? ^ K|A„, we have tv = 0 .

We define in T* another norm by means of the modular

q

I(

) = sup al(x) for xeT*.

П

It is easily verified that

I(

) satisfies the axiom 1.3.3.A-D. Hence, one may define a norm

INl£ = inf

0:

< 1}.

Evidently, we have ||®||J < INC, but the sign of equality does not need to hold. However, it is clear that |[a?||J = IN|J f°r because Qę(x)

= qI (x)

for

Let us remark that the following situation may occur.

It may hold <г£(Яа?) -> 0 as n -> oo for 0 < Я < 1, but supo^O») < 1 , and П

the sequence cr%,(x) is not convergent to 0. Then

< 1 but Q<p(x)

еДЛж) <

for 0 < Я < 1 .

1.5.5.

The following conditions hold:

(a) lim lla ^ H j = IN C ,

£—>•00

(b) Hm ||/|j*

*

Жii

k— *x>

Spaces of strongly summdble sequences 2 3 5

Proving this lemma we may suppose x > 0. Since 0 < x1 < x 2

< . . . < x, \\x% is a non-decreasing sequence bounded from above by

||a?||J. Let lim ||a?*||J. = y. By the definition of the norm |[*||J,

k—> oo

super*

П II®'

< 1 , i.e.

< 1

for к —

and n = 1 , 2 , ... Hence it follows with к

and k0 fixed that

Consequently, o ^ x ly ) < 1 for all n. Thus ||ж||® < rj. Hence

Irt-HNJ as

In order to prove (b) let us remark that the sequence ||&fc||* is non­

decreasing and bounded from above by ||a?||*. Let lim||a?fe||* = rj. Let us

choose a sequence y e T ^ so that gv*(y) < 1. Then the inequality O O

ll®*llj > anv%sv

v = l

is satisfied for every к and n. Taking к ->

we obtain

OO

f] > ^ anvtvsv for n = 1 , 2 , . . . ,

Г —

1 i.e.

OO

Г] > sup Y a n ,tv8p.

n 7=1

Since the last inequality is satisfied for every y e T ę* for which gv*{y) < 1, we get rj > ||a?||*. Hence ||®*||J -> ||a?||J as к oo.

1.5.6.

The norm

satisfies the following inequalities:

236 A. W a s z a k

It follows from the definition of the norm [|-||* that

o©

£ anytvsv < ||a?||J if Qę*(y) < 1 ,

v = l

oo

^

< IN*

(

if

> 1 , v=X

where n = 1 , 2 , ... First, let us suppose x eT f , x Ф 0. Let # > 1. We choose (Ty ^ 0 in such a manner that

Then

\ty\ К

+<p*{sy) , * = 1 , 2 , , . .

2

i \tv\s, __

anv m \ i ~

oo oo

n — 1 ,2 t • • r

But we have ty = 0 for sufficiently large v. Hence also 1ГУ = 0 for sufficiently large v. If we denote у

{^}, then y e T ^ , Qę*{y) <

Now, we consider two cases, Q^{y) > 1 and Qę*{y) < 1. In the first case we have

2 -

\ty\ Sv 1

Поо\\1 ^ n

-y

for n — 1 , 2 , . . . Since # > 1, we may choose n 0 in such a manner that

00 _ 1

JT anQV<p*(sy) > — Qę*(y)-

V =1 ”

Hence if n — n 0

OO

V=1

\ty\

m \ ;

i _ i + ^Q<P*iy)

and this gives finally

Spaces of strongly summable sequences 237

This inequality implies tw = 0 for v — 1 , 2 , ..., v0. Hence we have also

= 0 for v = 1, 2 , r0. Consequently, Q&iy) = 0 . We obtained a contradiction.

Now, let Qę*{y) < 1 . Then

Hence

V7 !<>!*> _ 1 _ Z V = 1 щх\\; ^ * '

n 1 , 2 , . . .

« = 1 , 2 , . . . and finally

Moreover, we have

Я | Н С Г ° - as

because of x e T f . Hence

^

/||

||*) < 1. By the definition (**) with s = 1 we obtain ||aj||* < ||я?Ц* for x e T f . Thus we proved the inequality \\x\\cv < к for the sequence xk. Applying Lemma 1.5.5. (a), (b) and the inequality

INIJ < ||a?||J and taking Tc -> oo we observe that the inequality ||a?||J < ||a?|jj is satisfied for every xeT*.

In order to prove the inequality ||ж||* < 2 ||ж||® it is sufficient to remark that for an arbitrary в > 0

у* %\ Kl

Z ^ ( \\x\\l+e)^

«пу^*(!«у|)

„0

Qq>

X

||ж||®+ e

+

^ 2

if

1. Hence we obtain ||ж||* ^2||a?||®.

2.1. The space T * is complete with respect to the norm И*^.

We apply the following theorem (see [11]): If the axiom C.II is satisfied in a modular space X (

), then this space is complete with respect to every norm generated by

. (

Hence it is sufficient to verify that the axiom C.II is satisfied in the

space T*, i.e. if xn e T * ,x n ^ 0 for n — 1 , 2 , . . . and

238 A. W a s z a k

+ . . . <

then there exists x 0 — \/ xneT*. We consider the sequence у к — v x 2 v ... v xk, where xn > (9, n — 1 , 2 , . . . We have

oo

о1{Ук) = £ anv(p (sup (tl,t2 v, tk v)) V— 1

oo

^ У? anyjyjtl) +<p(t2 v) + . . . +<p(ty))-

V = 1

O O O O 00

= ^ an»<p(tl) + ^nv<p(€) +• • • + UnvVitv)

v=l v—1 V=X

— + o'w(^2)+• • • + ° rw(#fc);

hence

fc O O 00

(+ ) вп(Ук) ^ ^ ^ У1 Qq>{^l) <-^ 00

г=х г=х г=х

for any fixed w. But yk form an increasing sequence. Hence lim tk — t°v.

fc->oo

Let {/”} = x 0, where x Q = V We show x QeT*. It is easily seen that liniowy*) = (£(я0) <

By ( + ),

&—>oo

oo

ffn(®o) < °n {xi) •

Thus there holds

0%,{x o) ^ (a?i) ~b Ora (^ 2 ) "t • • • “f" OVi i^k— 1 ) ~b Qq> {^k) “b Qtp +• • •

for an arbitrary n. By the assumption, we have Qę{xk) + ev(a?*+1) +. . . < e for sufficiently large k. Hence

<Jn{X0) < Gn(®l) + ffn(^2) +• • • + °«(%-x) + e

for arbitrary w and sufficiently large A;. Since Оп{хг) -> 0 as n -> 00 , we may choose n 0 so large that

+ + + < e for n > n 0.

Consequently, cr„(«0) < 2e for n > n 0, i.e.

O^(a?0) “> 0 as П OO.

E em ark 1. The above argument shows that Q<p{®o) ^ Qq>{x i) + Q<p{x v) + •• •

E em ark 2. The space T v is also complete with respect to the norm

II* H«p, because it is closed with respect to this norm (see 1.5.1).

Spaces of strongly summable sequences 2 3 9

2.2. A function <p is called поп-weaker than a function ip for large u, and we write ip

<p, if there are constants c, b, l, к, u0 > 0 such that

cip(lu) < b(p(ku) for и ^ u0.

Functions ip and cp are called equivalent for large u, ip ~ <pf if there

are constants a, b, с, кг, k2, l, u 0 > 0 such that

acpik-^u) < cip(lu) < bcp{k2u) for и > u 0.

i i i

It is clear that cp ~ ip if and only if ip -3 cp and cp ip, simultaneously (see [3], [4]).

2.3. Let us suppose the matrix A satisfies the condition

anl + а

+ • • • < A for n = 1 , 2 , ... I f ip 99 then

c

and

c

. The proof is analogous to the argumentation in [ 8 ], [10]. By the assumption, there exist constants k , b , u 0 > 0 such that ip(u) ^ b y ( k u ) for и > u 0. Let a?eT*, i.e. X^xeT^ for some A 0 > 0. Hence there holds ip{X0k - l tv) ^ bcpi^X^tf} for v satisfying the inequality X

к tv ^ ^ 0 * "W"e define x — {t'v} as follows:

% for XQk~l tv < u 0,

0 for remaining values of v.

Evidently, x'

By 1.4.1, x ' eT* r\

=

i.e. x'

However, the inequality

00 00

^ a nvip(XQk - l \tv\) < Ъ ^ а тср{ХМ) for n = 1 , 2 , . . .

V = 1 V = 1

implies x" = x —x ' еТ\. Finally, x = x'-\-x"eT*w.

The inclusion T v c: Tv is proved, analogously.

2.4. (a). I f I j c l j , then ||a><||,, -> 0 implies for щ е Т 1 ^.

(b). I f T ę cz T v, then ||a?ill<p 0 implies ||%||v 0 for XieTę.

The proof follows from the closed graph theorem. The operator A x — x from T* into T* is additive and homogeneous and its graph is closed. Indeed, x{ -> x 0 in T* means that \\Xi—x 0\\v -> 0, Xi~+y0 in T*

means that \\Xi — y 0\\v -> 0. Hence x 0 = y 0, since the terms of the sequences are linear functionals with respect to the norms ||*||v, ||*||v. The proof of (b) runs the same lines taking into account the fact that T V, T V are complete with respect to the norms ||«||ff, ||*||v, respectively.

2.5.

Let f{y) be positive-valued and defined for positive in­

tegers y. Moreover, let f(y) —>

as у -> oo and

(r ) limsup

У—>00

f ( y + 1 )

f(y)

240 A. W a s z a k

There exist constants c > 0 and A 0 such that for every A > A 0 the inequality

cf{y) < Я <f(y)

is satisfied for some у = у (Я).

The condition (r) implies а Ф 0 and the inequality

for у > y0. Hence we obtain

(+) —7~.— :/(У + 1 ) < / ( У) for a( l + r?)

Let us take A 0 == /( y 0) * and let Я ^ Я 0 = /( y 0) • There exists a non-negative integer Tc such that f ( y 0-\-7c) < A < / ( y 0+fc + l), because f ( y 0 + h) -> oo as 7c oo. Taking у = Уо+&? the previous inequality can be written in the form f(y ) < Я < / ( y + l).

Applying ( + ) we get

- T i V -r/Cy + l) < / ( ? / ) < Я < / ( y + l).

Denoting y-f-1 = u, (a(l + ?y ))_1 = c, we obtain cf(u) < Я

2.6. Leź ws suppose that a non-negative matrix A has the following properties:

(a) anv

0

n

for

1

2

(b) Av = supanv -> 0

->

П

(c) (A „)-1 is a function satisfying the condition 2.5.(r).

I f \\Xi\\v -> 0 implies \\Xi\\4, -> 0 /or an arbitrary XieTf then y> i q>.

We prove this theorem indirectly. Let us suppose y> i <p does not hold, i.e. for every system of constants 7c, b, u0 there exists и > u 0 such that гр(7си) > b<p(u). We take into account the assumptions on the matrix A and we apply Lemma 2.5 taking/(v) = (A ,)-1 and = Я, where и is chosen so large that e~lp(u) = Я > A0; here, s > 0 is a prescribed number and A 0 is the constant in Lemma 2.5. By 2.5, the inequality

£

( + ) A,

Spaces of strongly summable sequences 241

holds for some v — v0, where

is a constant defined in Lemma 2.5. We may choose the number и in such a manner that <p(u) > s)l 0 and y>(su)

> q p ( u ) j c e .

Taking into account ( + ) we obtain

By the definition of the norm (*), we get ||ewe ||v > 1. Now, we calculate that

and the definition of the norm (*) gives Weue^ < s. Since s is arbitrary, one can choose a sequence XieTf such that Ца^Н,, -> 0 but \\Xi\\y, > 1 , a con­

tradiction.

2.6.1. The assumptions of Theorem 2.6 may be slightly generalized as follows. Let e denote the set of natural numbers, %e = {^„}, where

?„ = 1 if vee and tv = 0 if vie, and let

in case of % eeT*. Let us remark that if %ezT* then 1.4.2 implies Xe€T,p.

We shall say that T* possesses the property (S>f) if there exist numbers p 0 > 0 and 0 < с < 1 such that for every 0 < p < p0 there exists a set e(p) for which £e(jU)eT* and cp < m[e(u)) < p.

Let us remark that there exists some connection between the con­

dition ( S x) after reformulation in terms of [ 12 ], and the condition (St) in [12], p. 169. One may consider the condition ( S x) in case of spaces of sequences strongly (A , ę>)-summable to zero as in some sense more general than the condition (S).

If a matrix A satisfies the assumptions of Theorem 2.6.(a)-(c), then T * possesses the property (Sf).

Indeed, let f(v) = (-А *)-1 = [m(e(r))]_1, where e(v) = {r}. Denoting p 0 = 1[X0, p = 1/Я, we see by Lemma 2.5, that the condition (Sf) is satisfied.

2.6.2. I f T * satisfies the condition (Sf) than the statement of Theorem 2.6 remains valid.

We choose £0 > 0 in such a manner that ЦжЦ,, < e 0 implies ||a?||v < 1 if xeTf. Let р й be the number defined in the property (Stf). Choosing w 0 so large that £ 0 /ę?(w/£0) < p 0 for и > u 0, the property ( S x) implies existence of a set e(p) such that

£

<p(l) m(e) = supOn(Xe) = Qg>(Xe) n

(+) C

<p(u l £o) <T m (e(p)j <:

<p(ufe0)

Roczniki PTM — Prace M atem atyczne XI.2 16

242 A . W a s z a k

for и > u 0, where 0 < c < 1 is a constant independent of the set e(p).

How, we calculate that

By ( + ),

e

o*

Hence the definition of the norm (*) yields the inequality \\xe^)u \\q> ^ £o?

which implies Wxe^Wy, < 1* Thus,

Qv(Xe(n)U) = гр.(и) m(e{[i)) < 1 .

By (4-)j we obtain

1 ^ ipi^) m[e{p)^ ^ w(u) —:—

—

< p { u h o)

and finally,

ip(u) --- (p

1

C£0

for

for и > u0,

и ^ u0.

2.7. The following theorem follows from 2.3, 2.4, 2.6:

I f a matrix A has the properties (a)-(c) of 2.6 and the property (d) anx + an2 + ... < К for n-= 1 , 2 , . . . ,

then the following conditions are mutually equivalent:

(a) ip 4 (p, (P) К c Tj, (y) T v c T v, (8) -> 0 implies ||я^||у 0 for arbitrary Xi<zTf .

Co r o l l a r y.

By the assumptions of

on the matrix A —

the following conditions are mutually equivalent:

(«) «Р ~ ( P ) 3 ^ = TJ,

{ y ) T r = T „ (8) the norms Ц»!^ and ||»||v are equivalent in Tf .

2.8. Let anv^ 0 as n -^ o o for all v. The space T ę is separable with respect to the norm ||*||v.

This follows from 1.5.3, immediately (compare also [ 8 ], [ 6 ]).

2.9. I f a matrix A satisfies the assumptions of 2.7, then the following conditions are mutually equivalent:

(a) (p satisfies the condition ( d2) for large u, (P) T l = T 9,

(у) the space T* with the norm ||*||v is separable.

(a) => (P). If (A2) holds, given any Я > 1 there exists cx > 1 such

Spaces of strongly summable sequences 2 4 3

that

< ся

for

>

(A). It is easily observed that arguments analogous to those of the proof of the Theorem 2.3 yield T* c T9.~ Since the inclusion T ę с T* is obvious, we get T * = T v .

((3) => (y)- The proof follows from 2.8 (compare [ 8 ], [ 6 ]).

(y) => (a). The proof runs the same lines as in [10] with the only exception that we apply the possibility of a construction of sequences of the form Xi = щвщ, where щ Ф щ for i Ф j, \\Xi\\9 < 2~г, \\х^\9 ^ 1.

Existence of such a sequence {xf} follows from the arguments of the proof of Theorem 2.6.

3. Examples of methods of a strong (A , <p)-summability of sequences, where the matrix A = (anv) possesses the following properties:

(a) anv -» 0 as n oo, v — 1 , 2 , ..., (b) A v = supaM V — > 0 as v ->

П

(c) limsup

a

V—> 0 0

(d) апг + аш + ... < К for n = 1 , 2 , . . .

3.1. Cesaro method of order k, where k > — 1,

\n+Tc— v Tc-

П

(“Л o,

n > V,

n < V,

where

w+fc\ _ Ц п + k + l ) (fc+l)(fc+2)- . . . • (k+n )

к / “ Г (к + 1 )Г (п + 1 ) ~ nl

In order to find A v we calculate which n satisfy the inequality anv ^ an+\,vi v being fixed. This inequality is equivalent to the following one:

j n+lc — v —

') ^ГП > {n+Z ⁷ )[

i.e. to the inequality

n + k + 1 n + k —v

--- > --- , n ф kv— 1.

n + 1 n —v+1

244 A. W a s z a k

Hence anv decreases for n > 7cv — 1 , where v is fixed. Similarly, we observe that anv increases for v < n < kv— 1. Thus anv attains its maximum for n = kv— 1. Hence -

lv (k -i)+ k —2\

\ fc-l /

^kv+Jc— 1 |

Now, the property (b) follows from

(7c

1 )

1 ) •

•

2 )

A v = --- --- ;---> 0 as v -> oo.

(lev + 1 ) (lev + 2 ) •... • (lev + k — 1 ) and the property (c) from

A v (le — 1) (vie—r+1) •... - (vie—v+le— 2 ) (lev+le) (kv+le-h 1 ) •... • (kv+2k— 1 ) A v+1 le(lev-\-1) •... • (kv+k— 1) (vk—v+k — l)(v k —v+k) •... '(vk—v+2k+3)

( k - l f - ' J c * kk( k - l ) k~l 3.2. Euler method S r (r — 1 , 2 , . . . )

= 1 as v -> oo.

dnv

Г\П J

n < v.

In order to calculate A v we solve the inequality anv > an+1>v with fixed v.

We obtain an equivalent inequality

( 2 T +1 (™)( 2 r- l f ~ ’' > ( 2 T ( ^ 1 )( 2 r- l ) n_v+1, i.e.

2r( n - v + l) ^ ( 2 r- l ) ( n + l ) , n > 2 rv - l .

Hence anv decreases for n > 2rv — l . Similarly, we prove that anv increases for v < n < 2rv— 1 , i.e. anv attains its maximum for n = 2rv — l.

Thus

A v = (2rv~ i y 2 ri- l f v- ’'- 1( 2 y - 2r\

Spaces of strongly summable sequences 245

Now, the property (b) is obtained applying Stirling formula A < ( 2 ^ - ^ ^ rjj ( 2 r~ l f v- v- 1

__ ( 2 r— 1 f *-’-■} (2 rv f v V 2n2rv е1еге2Гр- р (2r)2 *’_1 e2 V / 2 uv £ 2 ( 2 *V— v)2 v~v V2n(2rv— v)es

(2 r)3/4

= --- -= z = --- > 0

as

( 2 r— 1)3/2 V2

V£2£3

because ex -» 1 , £2 -> 1 , s 3 -» 1 ; and the property (c) follows from A v __ ( 2 r)2r(v + l ) ( 2 r—1)(2*V—v + l)*...*(2rv—r + 2r—2) A+ i ~ ( 2 r— l ) 2 r~ 12 r( 2 rv + l) •... • ( 2 *V + 2 r— 2 ) ( 2 rr + 2 r— 1 )

( 2

) 2 Г( 2

- 1)2Г- 1 ( 2 r- l ) 2 " 1 ( 2r)2

1 for

3.3. Nórlund method, ( N ,p n) . Let us suppose that p n > 0 for n = 0 , 1 , 2 , . . . and p о +Pi +.. • +pn

as №

We take

t t f l v

P n —V

Po + 1L + • • • +Pn 1

0 ,

for n > V, for n < v.

An argumentation as in 3.1 and 3.2 depends on the sequence {pn} and in general the formulae for A v are rather complicated. In particular, if Pn~v ^-'pn+i-v for every n ^ v , then anv form a non-increasing sequence and

A v

Po+Pl + ‘--+Pv

Then the condition (b) holds, i.e. A v -> 0 as v ->

If, moreover, we suppose that limp n+ilpn = where g > 0 , then the condition (c) is satisfied also.

n—>oo

However, in case of the Cesaro means this condition holds for 1c = 1 only.

3.4. Eiesz method ( B , p , l ) . Let us suppose that p n > 0 for n — 1 , 2 , . . . and Po+PiĄ- • • • A p n ->

as n ->

We define

Gtn

_____ Pv______

Po+Pl + -"+Pn ’ 0 ,

for n ^ v,

for n < v.

246 A. W a s z а к

It is easily verified that

A v Pv

Po+Pi + ---+Py '

If, moreover, we suppose that limpn_1lpn = 1, then the properties (b)

n—>oo

and (c) are satisfied.

An example of means satisfying the above conditions are the C'x-means;

also logarithmic means

v l n ( l

+n) ’ 0 ,

for

for

above form with p v — l[v.

References

[1] Gr. H. H a rd y , Divergent series, Oxford 1949.

[2] M. A. K r a s n o s e l’sk ii, Ya. B. R u t ic k ii, Convex functions and Orliez spaces. Groningen 1961. '

[3] W. M a tu sz e w sk a , On generalized Orliez spaces, Bull. Acad. Polon. Sci., Ser. sci. math., astr. et phys., 8 (1960), pp. 349-353.

[4] — Przestrzenie funkcji p-eałkowalnych. Część I. Własności ogólne q>-funkcji i klas funkcji p-całkowalnych, Prace. Mat. 6 (1961), pp. 121-139.

[5] — and W. O rliez, A note on modular spaces, V, Bull. Acad. Polon.

Sci., Ser. sci. math., astr. et phys., 11 (1963), pp. 51-54.

[6] J. M u sie la k , On some modular spaces connected with strong summability, Math. Student 27 (1959), pp. 129-136.

[7] — and W. O rliez, On modular spaces, Studia Math. 18 (1959), p. 49-65.

[8] — On modular spaces of strongly summable sequences, ibidem 22 (1962), pp. 127-146.

[9] W. O rliez, A note on modular spaces, IV , Bull. Acad. Polon. Sci., Ser.

sci. math., astr. et phys., 10 (1962), pp. 479-484.

[10] — On some spaces of strongly summable sequences, Studia Math. 22 (1963), pp. 331-336.

[11] — A note on modular spaces, VII, Bull. Acad. Polon. Sci., Ser. sci.

math., astr. et phys., 12 (1964), pp. 305-309.

[12] — On some classes of modular spaces, Studia Math. 26 (1966), pp. 165-192.

[13] A. W a s z a k , Some remarks on Orliez spaces of strongly (A , p) - summable sequences, Bull. Acad. Polon. Sci., Ser. sci. math., astr. et phys., 15 (1967), pp. 265-269.

DEPARTMENT OF MATHEMATICS I, A. MICKIEWICZ UNIVERSITY

POZNAN (KATEDRA MATEMATYKI I, UNIWERSYTET IM. A. MICKIEWICZA, POZNAN)