On matrix transformations concerning the Nakano vector valued sequence space

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ON MATRIX TRANSFORMATIONS CONCERNING

THE NAKANO VECTOR-VALUED

SEQUENCE SPACE

SUTHEP SUANTAI

(Received 15 August 2000 and in revised form 20 October 2000)

Abstract.We give the matrix characterizations from Nakano vector-valued sequence space(X, p)andFr(X, p)into the sequence spacesEr,∞,_∞(q),bs, andcs, where

p=(pk)andq=(qk)are bounded sequences of positive real numbers such thatpk>1 for allk∈Nandr≥0.

2000 Mathematics Subject Classiﬁcation. 46A45.

1. Introduction. Let(X, · )be a Banach space,r ≥0 andp=(pk)a bounded sequence of positive real numbers. We writex=(xk)withxkinXfor allk∈N. The X-valued sequence spacesc0(X, p),c(X, p),∞(X, p),(X, p),Er(X, p),Fr(X, p), and _∞(X, p)are deﬁned as

c0(X, p)=

x=xk: lim k→∞xk

p_k =0,

c(X, p)=x=xk: lim

k→∞xk−a

pk₌₀_,_{for some}_a_∈_X_,

∞(X, p)=

x=xk: sup k

xkpk_<_∞_,

(X, p)=

x=xk: ∞

k=1

xkpk_<_∞

,

Er(X, p)=

x=xk: sup k

xkpk kr <∞

,

Fr(X, p)=

x=xk: ∞

k=1

kr_xkpk_<_∞

,

_∞(X, p)=

∞

n=1

x=xk: sup k

_xk_n1/p_k_.

(1.1)

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sequence spaces were first introduced by Cooke [1]. The space_∞(p)was first defined by Grosse-Erdmann [2] and he has given in [3] characterizations of infinite matrices mapping between scalar-valued sequence spaces of Maddox. Wu and Liu [10] gave necessary and sufficient conditions for infinite matrices mapping fromc0(X, p)and ∞(X, p)intoc0(q)and∞(q). Suantai [8] has given characterizations of infinite ma-trices mapping(X, p)into∞and∞(q)whenpk≤1 for allk∈Nand he has also given in [9] characterizations of those infinite matrices mapping from(X, p)into the sequence spaceEr whenpk≤1 for allk∈N.

In this paper, we extend the results of [8,9] in casepk>1 for allk∈N. Moreover, we also give the matrix characterizations from(X, p)andFr(X, p)into the sequence spacesbsandcs.

2. Notations and definitions. Let(X, · )be a Banach space, the space of all se-quences inXis denoted byW (X), andΦ(X)denotes the space of all finite sequences inX. WhenX=K, the scalar field ofX, the corresponding spaces are written asw andΦ.

A sequence space inXis a linear subspace ofW (X). LetEbe anX-valued sequence space. Forx∈Eandk∈N,xkstands for thekth term ofx. Fork∈N, we denote by ekthe sequence(0,0, . . . ,0,1,0, . . .)with 1 in thekth position and byethe sequence (1,1,1, . . .). Forx∈X andk∈N, letek_(x)_{be the sequence}₍₀_,₀_{, . . . ,}₀_{, x,}₀_{, . . .)}_with x in thekth position and let e(x)be the sequence(x, x, x, . . .). We call a sequence space E normal if(tkxk)∈E for all x =(xk)∈E and tk∈K with |tk| =1 for all tk∈N. A normed sequence space(E, · )is said to benorm monotoneifx=(xk), y=(yk)∈Ewith xk ≤ ykfor allk∈Nwe havex ≤ y. For a ﬁxed scalar sequenceµ=(µk), the sequence spaceEµis deﬁned as

Eµ=x∈W (X):µkxk∈E. (2.1)

LetA=(f_kn)withf_kn inX, the topological dual ofX. Suppose thatE is a space ofX-valued sequences andF a space of scalar-valued sequences. ThenA is said to map E intoF, written byA:E→F, if for eachx=(xk)∈E, An(x)=∞k=1fkn(xk) converges for eachn∈N, and the sequenceAx=(An(x))∈F. Let(E, F )denote the set of all inﬁnite matrices mapping fromEintoF.

Suppose that theX-valued sequence spaceEis endowed with some linear topology τ. ThenEis called aK-space if for eachk∈N, thekth coordinate mappingpk:E→X, deﬁned bypk(x)=xk, is continuous onE. If, in addition,(E, τ)is a Fréchet (Banach) space, thenEis called an FK- (BK-) space. Now, suppose thatEcontainsΦ(X). ThenE is said to havepropertyAB if the set{n

k=1ek(xk):n∈N}is bounded inEfor every x=(xk)∈E. It is said to havepropertyAK ifnk=1ek(xk)→xinEasn→ ∞for every x=(xk)∈E. It haspropertyAD ifΦ(X)is dense inE.

It is known that the Nakano sequence space(X, p)is an FK-space with property AK under the paranormg(x)=(∞k=1xkpk)1/M, whereM=max{1,supkpk}. Ifpk>1 for allk∈N, then(X, p)is a BK-space with the Luxemburg norm deﬁned by

_xk₌_inf ε >0 :

∞

k=1

xk

ε

pk

≤1

 

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3. Main results. We ﬁrst give a characterization of an inﬁnite matrix mapping from (X, p)intoEr whenpk>1 for allk∈N. To do this, we need the following lemma.

Lemma_3.1. _Let_E_{be an}_{X-valued BK-space which is normal and norm monotone and}

letA=(fn

k)be an inﬁnite matrix. ThenA:E→Erif and only ifsupn

∞

k=1|fkn(xk)|/nr <∞for everyx=(xk)∈E.

Proof. _{If the condition holds true, it follows that}

sup n

∞ k=1fkn

xk nr ≤sup_n

∞

k=1

_fn k

xk

nr <∞ (3.1)

for everyx=(xk)∈E, henceA:E→Er.

Conversely, assume that A : E → Er. Since E and Er are BK-spaces, by Zeller’s theorem,A:E→Er is bounded, so there existsM >0 such that

sup n∈N (x_k)≤1

∞ k=1fkn

xk

nr ≤M. (3.2)

Letx=(xk)∈Ebe such thatx =1. For eachn∈N, we can choose a scalar sequence (tk)with|tk| =1 andfn

k(tkxk)= |fkn(xk)|for allk∈N. SinceEis normal and norm monotone, we have(tkxk)∈Eand(tkxk) ≤1. It follows from (3.2) that

∞

k=1

fn k

xk

nr =

∞ k=1fkn

tkxk

nr ≤M, (3.3)

which implies

sup n∈N

∞

k=1

f_knxk

nr ≤M. (3.4)

It follows from (3.4) that for everyx=(xk)∈E,

sup n∈N

∞

k=1

fn k

xk

nr ≤Mx. (3.5)

This completes the proof.

Theorem_3.2. _Let_p=_(pk)_{be a bounded sequence of positive real numbers with}

pk>1for allk∈Nand1/pk+1/qk=1for allk∈N, and letr≥0. For an inﬁnite matrixA=(fn

k),A∈((X, p), Er)if and only if there ism0∈Nsuch that

sup n

∞

k=1

fn k

qk_n−r qk_m−qk

0 <∞. (3.6)

Proof. _Let_x=_(xk)∈_{(X, p)}_{. By (}_3.6_{), there are}_m₀∈N_and_{K >}_{1 such that}

∞

k=1

fn k

qk_n−r qk_m−qk

0 < K, ∀n∈N. (3.7)

Note that fora, b≥0, we have

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It follows by (3.7) and (3.8) that forn∈N,

n−r

∞

k=1 fkn

xk

=n−r

∞

k=1 fkn

m−01·m0xk

≤

∞

k=1

n−r_m−1

0 fknm0xk

≤

∞

k=1

n−r qk_m−qk 0 fkn

qk₊_mα 0

∞

k=1

_xkpk

≤K+mα 0

∞

k=1

xkpk_, _where_α₌_sup k

pk.

(3.9)

Hence supn−r_|∞

k=1fkn(xk)|<∞, so thatAx∈Er.

For necessity, assume thatA∈((X, p), Er). For eachk∈N, we have supnn−r|fkn(x)| <∞for allx∈Xsincee(k)_(x)_∈_{(X, p)}_{. It follows by the uniform bounded principle} that for eachk∈Nthere isCk>1 such that

sup n n

−r_fn

k≤Ck. (3.10)

Suppose that (3.6) is not true. Then

sup n

∞

k=1

f_knqk_n−r qk_m−qk= ∞_, ∀_m∈N_. _(3.11)

Forn∈N, we have by (3.10) that fork, m∈N,

∞

j=1

f_jnqjn−r qj_m−qj= k

j=1

f_jnqjn−r qj_m−qj+ j>k

f_jnqjn−r qj_m−qj

≤ k

j=1

C_jqjm−qj+ j>k

f_jnqjn−r qj_m−qj_.

(3.12)

This together with (3.11) give

sup n

j>k

fjn

q_j

n−r qj_m−qj= ∞_, ∀_{k, m}∈N_. _(3.13)

By (3.13) we can choose 0=k0< k1< k2<···,m1< m2<···, mi>4i and a

subsequence(ni)of positive integers such that for alli≥1,

k_i−1<j≤ki

fni

j qj

n−_ir qjm_i−qj>2i. (3.14)

For eachi∈N, we can choosexj∈Xwithxj =1, forki−1< j≤kisuch that

k_i−1<j≤ki

fni

j

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For eachi∈N, letFi:(0,∞)→(0,∞)be deﬁned by

Fi(M)=

k_i−₁<j≤k_i

fni

j

xjqjn−_ir qjM−qj_. _(3.16)

ThenFiis continuous and non-increasing such thatF (M)→0 asM→ ∞. Thus there existsMi>0 such thatMi> miand

FMi= ki−1<j≤ki

fni

j

xjqjn−_ir qjM_i−qj=2i_. _(3.17)

Put

y=yj, yj=4−iMi−(qj−1)n −r q_j/p_j

i f

ni j

xjqj−1xjforki−1< j≤ki. (3.18)

Thus

∞

j=1

_yjpj₌

∞

i=1

k_i−1<j≤ki

4−ipj_M−pj(qj−1)

i n

−r qj i f

ni j

xjpj(qj−1)

≤

∞

i=1

4−i k_i−₁<j≤k_i

M_i−qjn−_ir qjfni j

xjqj

=

∞

i=1

4−i_·₂i

=

∞

i=1

1 2i =1.

(3.19)

Thusy=(yj)∈(X, p). Since(X, p)is a BK-space which is normal and norm mono-tone under the Luxemburg norm, byLemma 3.1, we obtain that

sup n

∞

k=1

_fn k

yk

nr <∞. (3.20)

But we have

sup n

∞

j=1

f_jnyj nr ≥sup

i ∞

j=1

fni

j yj nr i ≥sup i

k_i−1<j≤ki

fni

j yj nr i =sup i

k_i−1<j≤ki

4−iMi−(qj−1)n

−r (qj/pj+1)

i f

ni j

xjqj

=sup i

k_i−₁<j≤k_i

4−i_M−(qj−1)

i n

−r qj i f

n_i j

xjqj

=sup i

k_i−1<j≤ki

_fni j

xjqjn−ir qjM −q_j i

4−iMi

≥sup i

2i= ∞, becauseMi>4i.

(3.21)

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Theorem_3.3. _Let_p=_(pk)_{be a bounded sequence of positive real numbers such}

thatpk>1for allk∈N,1/pk+1/qk=1for allk∈N,r≥0ands≥0. Then for an inﬁnite matrixA=(fn

k),A∈(Fr(X, p), Es)if and only if there ism0∈Nsuch that

sup n

∞

k=1

k−r qk/pk_fn k

qk_n−sqk_m−qk 0

<∞. (3.22)

Proof. SinceFr(X, p)=(X, p)_(k_{r /pk}₎, it is easy to see that

A∈Fr(X, p), Es⇐⇒k−r /pk_fn k

n,k∈

(X, p)Es. (3.23)

ByTheorem 3.2, we have(k−r /pk_fn

k)n,k∈((X, p)Es)if and only if there is m0∈N

such that

sup n

∞

k=1

k−r qk/pk_fn k

qk_n−sq_k_m−qk 0

<∞. (3.24)

Thus the theorem is proved.

SinceE0=∞, the following two results are obtained directly from Theorems3.2 and3.3, respectively.

Corollary_3.4. _Let_p=_(pk)_{be a bounded sequence of positive real numbers with}

pk>1for allk∈Nand let1/pk+1/qk=1for allk∈N. Then for an inﬁnite matrix A=(fn

k),A∈((X, p), ∞)if and only if there ism0∈Nsuch that

sup n

∞

k=1

_fn k

qk_m−qk

0 <∞. (3.25)

Corollary3.5. Letp=(pk)be a bounded sequence of positive real numbers with

pk>1for allk∈Nand let1/pk+1/qk=1for allk∈N. Then for an inﬁnite matrix A=(fkn),A∈(Fr(X, p), ∞)if and only if there ism0∈Nsuch that

sup n

∞

k=1

k−r qk/pk_fn k

q_k m−qk

0

<∞. (3.26)

Theorem _3.6. _Let_p=_(pk)_and _q=_(qk) _{be bounded sequences of positive real}

numbers withpk>1for allk∈Nand let1/pk+1/tk=1for allk∈N. Then for an inﬁnite matrixA=(fn

k), A∈((X, p), ∞(q))if and only if for eachr∈N, there is mr∈Nsuch that

sup n,k

∞

k=1

rtk/qn_fn k

tk_m−tk

r <∞. (3.27)

Proof. Since_∞(q)= ∩∞_r₌₁_∞

(r1/qk), it follows that

A∈(X, p), _∞(q)⇐⇒A∈(X, p), ∞(r1/qk)

, ∀r∈N. (3.28)

It is easy to show that forr∈N,

A∈(X, p), ∞(r1/qk)

⇐⇒r1/qn_fn k

n,k∈

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We obtain byCorollary 3.4that forr∈N,(r1/qn_fn

k)n,k∈((X, p), ∞)if and only if there ismr∈Nsuch that

sup n

∞

k=1

rtk/qn_fn k

t_k m−tk

r <∞. (3.30)

Thus the theorem is proved.

Theorem _3.7. _Let_p=_(pk)_and _q=_(qk) _{be bounded sequences of positive real}

numbers withpk>1for allk∈Nand let1/pk+1/tk=1for allk∈N. For an inﬁnite matrixA=(f_kn),A∈(Fr(X, p), _∞(q))if and only if for eachi∈N, there ismi∈N such that

sup n

∞

k=1

itk/qn_k−r tk/pk_fn k

tk_m−tk

i <∞. (3.31)

Proof. SinceFr(X, p)=(X, p)_(k_{r /pk}₎, it implies that

A∈Fr(X, p), _∞(q)⇐⇒k−r /pk_fn k

n,k∈

(X, p), _∞(q). (3.32)

It follows fromTheorem 3.6thatA∈(Fr(X, p), _∞(q))if and only if for eachi∈N, there ismi∈Nsuch that

sup n

∞

k=1

itk/qn_k−r tk/pk_fn k

tk_m−tk

i <∞. (3.33)

Theorem 3.8. Letp =(pk)be bounded sequence of positive real numbers with

pk>1for alln∈Nand let1/pk+1/qk=1for allk∈N. Then for an inﬁnite matrix A=(f_kn),A∈((X, p), bs)if and only if there ism0∈Nsuch that

sup n

∞

k=1

n

i=1 f_ki

qk

m−qk

0 <∞. (3.34)

Proof. For an inﬁnite matrixA=(f_kn), we can easily show that

A∈(X, p), bs⇐⇒

 n

i=1 f_ki

 

n,k

∈(X, p), ∞. (3.35)

This implies by Corollary 3.4 thatA∈((X, p), bs) if and only if there ism0∈N

such that

sup n

∞

k=1

n

i=1 f_ki

qk

m−qk

0 <∞. (3.36)

Theorem_3.9. _Let_p=_(pk)_{be a bounded sequence of positive real numbers with}

pk>1for allk∈Nand let1/pk+1/qk=1for allk∈N. Then for an inﬁnite matrix A=(fn

k),A∈((X, p), cs)if and only if (1) there ism0∈Nsuch thatsupn

∞ k=1

n

i=1fkiqkm −qk

0 <∞and

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Proof. _{The necessity is obtained by}_{Theorem 3.8}_{and by the fact that}_e(k)_(x)∈

(X, p)for everyk∈Nandx∈X.

Now, suppose that (1) and (2) hold. ByTheorem 3.8, we haveA:(X, p)→bs. Letx= (xk)∈(X, p). Since(X, p)has the AK property, we havex=limn→∞nk=1e(k)(xk).

By Zeller’s theorem,A:(X, p)→bsis continuous. It implies that

Ax=lim n→∞

n

k=1

Ae(k)xk. (3.37)

By (2),Ae(k)_(xk)_∈_cs_{for all}_k_∈_N_{. Since}_cs_{is a closed subspace of}_bs_{, it implies that} Ax∈cs, that is,A:(X, p)→cs.

Acknowledgement. The author would like to thank the Thailand Research Fund

for the ﬁnancial support during the preparation of this paper.

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Suthep Suantai: Department of Mathematics, Faculty of Science, Chaing Mai Uni-versity, Chiang Mai,50200, Thailand