ON CERTAIN SEQUENCE SPACES
PIYAPONG NIAMSUP AND YONGWIMON LENBURY Received 15 July 2004 and in revised form 20 June 2005
We study the multiplicativity factor and quadraticity factor for near quasinorm on cer- tain sequence spaces of Maddox, namely,l(p) andl∞(p), where p=(pk) is a bounded sequence of positive real numbers.
1. Introduction
LetXbe an algebra over a fieldF (RorC). AquasinormonXis a function| · |:X→R such that
(i)|0| =0,
(ii)|x| ≥0, for all x∈X, (iii)| −x| = |x|, for all x∈X,
(iv)|x+y| ≤ |x|+|y|, for all x,y∈X,
(v) iftk∈F,|tk−t| →0, andxk,x∈X,|xk−x| →0, then|tkxk−tx| →0.
If| · |satisfies only properties (i) to (iv), then we call| · |anear quasinorm. If the quasinorm satisfies|x| =0 if and only ifx=0, then it is said to betotal.
Aquasinormed linear space(QNLS) is a pair (X,| · |) where| · |is a quasinorm onX.
If (X,| · |) is a quasinorm space, then the map| · |:X→Ris continuous. For p >0, a p-seminorm onXis a function · :X→Rsatisfying
(i)x ≥0, for all x∈X,
(ii)tx = |t|px, for all t∈F, for all x∈X, (iii)x+y ≤ x+y, for all x,y∈X.
A seminorm is called a norm if it satisfies the following condition:
(iv)x =0 if and only ifx=0.
Ap-seminormed linear space (p-semi-NLS) is a pair (X, · ) where · is a semi- norm onX.p-normed linear spaces (p-normed-LS) are defined similarly.
In [1,2],multiplicativity factors(orM-factors) andquadrativity factors(orQ-factors) for seminorms on an algebraXhave been introduced and studied in detail. A numberµ >
0 is said to be a multiplicativity factor for a seminormSif and only ifS(xy)≤µS(x)S(y), for all x,y∈X. Similarly, a numberλ >0 is said to be a quadrativity factor forSif and
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:15 (2005) 2441–2446 DOI:10.1155/IJMMS.2005.2441
only ifS(x2)≤λS(x)2, for all x∈X. The necessary and sufficient conditions for existence ofM-factor andQ-factor forSare answered in the following results.
Theorem1.1. LetXbe an algebra and letS=0be a seminorm onX. Then (a)ShasM-factors onXif and only ifKerSis an ideal inXand
µinf≡supS(xy) :x,y∈X,S(x)=S(y)=1<+∞, (1.1) (b)ifShasM-factors onXandµinf>0, thenµinfis the best (least)M-factor forS,
(c)ifShasM-factors onXandµinf=0, thenµis anM-factor forSif and only ifµ >0.
Theorem1.2. LetXbe an algebra and letS=0be a seminorm onX. Then
(a)ShasQ-factors on X if and only ifKerSis closed under squaring (i.e.,(KerS)2⊂ KerS) and
λinf≡supSx2:x∈X,S(x)=1<+∞, (1.2) (b)ifShasQ-factors onXandλinf>0, thenλinfis the best (least)Q-factor forS,
(c)ifShasQ-factors onXandλinf=0, thenλis aQ-factor forSif and only ifλ >0.
IfSis a norm, then KerS= {0}. If in additionX is finite-dimensional, then a sim- ple compactness argument shows thatµinf is finite. Therefore, byTheorem 1.1, norms on finite-dimensional algebras always haveM-factors. If Sis a seminorm on a finite- dimensional algebraX, thenShasM-factors onXif and only if KerSis a (two-sided) ideal inX. In [1,2] several examples of seminorms havingM-factors andQ-factors are given.
In [3], scalar multiplicativity factors for near quasinorms on certain sequence spaces of Maddox are studied. Motivated by these results we defineMr-factorsandQr-factorsfor a near quasinormqon an algebraXas follows.
A numberµ >0 is anMr-factorforqif and only ifq(txy)≤µ|t|rq(x)q(y) , there exists r >0, for allt∈F, for all x,y∈X.
A numberλ >0 is aQr-factorforqif and only ifq(tx2)≤λ|t|rq(x)2, there existsr >0, for all t∈F, for all x∈X.
Let
µinf≡sup
q(txy)
|t|rq(x)q(y):t∈F− {0},x,y∈X−Kerq
, λinf≡sup
qtx2
|t|rq(x)2 :t∈F− {0},x∈X−Kerq
.
(1.3)
2.Mr-factors andQr-factors for near quasinorms In this section, we will prove the following theorems.
Theorem2.1. LetXbe an algebra over a fieldF (F=CorR). Letqbe a near quasinorm onX. Then
(a)qhasMr-factors onXif and only ifKerqis a (two-sided) ideal inXandµinf<+∞, (b)ifqhasMr-factors onXandµinf>0, thenµinfis the best (least)Mr-factor forq,
(c)ifqhasMr-factors onXandµinf=0, thenµis anMr-factor forqif and only ifµ >0.
Theorem2.2. LetXbe an algebra over a fieldF (F=CorR). Letqbe a near quasinorm onX. Then
(a)qhasQr-factors onXif and only ifKerqis closed under squaring (i.e.,x2∈Kerq, for allx∈Kerq) andλinf<+∞,
(b)ifqhasQr-factors onXandλinf>0, thenλinfis the best (least)Qr-factors forq, (c)ifqhasQr-factors onXandλinf=0, thenλis aQr-factors forqif and only ifλ >0.
Proof ofTheorem 2.1. (a) Suppose thatqhas anMr-factorµonX. Clearly, Kerqis a sub- space ofX. Now take anyx∈Kerqandy∈X. Thenq(xy)≤µq(x)q(y)=0 which implies thatxy∈Kerq. Similarly, yx∈Kerq, so Kerqis a (two-sided) ideal inX. Now fort∈ F− {0}andx,y∈X−Kerq, we haveq(txy)≤µ|t|rq(x)q(y) orq(txy)/|t|rq(x)q(y)≤µ which implies thatµinf≤µ <+∞. Conversely, suppose that Kerqis a (two-sided) ideal inX andµinf<+∞. Ift=0,x∈Kerq, or y∈Kerq, thentxy∈Kerq, so 0=q(txy)= µinf|t|rq(x)q(y). If t=0 andx,y /∈Kerq, thenq(txy)/|t|rq(x)q(y)≤µinf orq(txy)≤ µinf|t|rq(x)q(y). Therefore,q(txy)≤µinf|t|rq(x)q(y), for all t∈F and for all x,y∈X which implies thatqhasMr-factors onX.
(b) Letµbe an Mr-factor forqonXandµinf>0. Thenq(txy)≤µ|t|rq(x)q(y) for all t∈Fand for all x,y∈X. Therefore,q(txy)/|t|rq(x)q(y)≤µ, for allt∈F− {0}and for all x,y∈Kerq, soµinf≤µ.
(c) This part follows directly from definition ofµinfandMr-factors forqonX. Proof ofTheorem 2.2. The proof of this theorem is a simple modification of the proof of
Theorem 2.1and will be omitted.
3.Mr-factors andQr-factors for near quasinorm on certain sequence spaces of Maddox
Let p=(pk) be a bounded sequence of positive real numbers. The sequence spaces of Maddoxl∞(p) andl(p) are defined as follows:
l∞(p)= xk
:xk∈C, sup
k
xkpk<∞ ,
l(p)=
xk
:xk∈C,
k
xkpk<∞
.
(3.1)
With the usual multiplication (i.e., (xk)(yk)=(xkyk)), bothl∞(p) andl(p) are algebras overC. We define near quasinormsq1onl∞(p) andq2onl(p) as follows:
q1
xk
=sup
k
xkpk/M, xk
∈l∞(p), q2
xk
=
k
xkpk1/M
, xk
∈l(p), (3.2)
whereM=max{1, supkpk}. We observe thatq1andq2may or may not be quasinorms.
For example, when (pk)=(1/k), then q1 is a near quasinorm but not a quasinorm; if (pk)=(1−1/(k+ 1)), thenq1is a quasinorm.
In this section, we give necessary and sufficient conditions for sequence spacesl∞(p) andl(p) to haveMr-factors andQr-factors.
Theorem3.1. Let p=(pk)and letMbe defined as above. Then the following are equiva- lent.
(a) p0=pk=pk+1for allk≥0wherep0is a positive real number.
(b)q1hasMr-factors onl∞(p).
(c)q1hasQr-factors onl∞(p).
(d)q1is ap0/M-seminorm onl∞(p).
Theorem3.2. Let p=(pk)and letMbe defined as above. Then the following are equiva- lent.
(a) p0=pk=pk+1for allk≥0wherep0is a positive real number.
(b)q2hasMr-factors onl(p).
(c)q2hasQr-factors onl(p).
(d)q2is ap0/M-seminorm onl(p).
Proof ofTheorem 3.1. (a)⇒(b) Ifp0=pk=pk+1for allk≥1, then q1(txy)=sup
k |txy|pk/M=sup
k |txy|p0/M≤ |t|p0/Mq1(x)q1(y) (3.3) for allx,y∈l∞(p), soq1has anMr-factor onl∞(p).
(b)⇒(a) Assume thatq1hasMr-factors onl∞(p). This implies that µinf=sup
q1(txy)
|t|rq1(x)q1(y):t∈F− {0},x,y∈X−Kerq1
<+∞. (3.4) We shall show thatr=supkpk/M=infkpk/M which implies that pk=pk+1 for all k≥1. To this end we observe that
µinf=sup
q1(txy)
|t|rq1(x)q1(y):t∈F− {0},x,y∈X−Kerq1
≥sup
q1(txy)
|t|rq1(x)q1(y):t∈F− {0},x,y=(1, 1, 1,...)
≥sup
supk|t|pk/M
|t|r :t∈F,|t| ≥1
=sup|t|supkpk/M:t∈F,|t| ≥1
(3.5)
so that
µinf≥sup
|t|supkpk/M
|t|r :t∈F,|t| ≥1 . (3.6) If r <supkpk/M, then µinf =+∞ which is a contradiction. Therefore, r≥supkpk/M.
Similarly, we can show that r≤infkpk/M from which it follows that r=supkpk/M= infkpk/Mand the proof is complete.
(a)⇒(c) The same proof as (a)⇒(b).
(c)⇒(a) The same proof as (b)⇒(a).
(d)⇒(b) This is obvious.
(b)⇒(d) Assume that q1 hasMr-factors. Then, by (a), p0=pk=pk+1for all k≥0 wherep0is a positive real number. Moreover, we have
q1(txy)=sup
k
t· xk
ykp0/M= |t|p0/Msup
k
xkykp0/M= |t|p0/Mq1(xy) (3.7)
for allx=(xk),y=(yk)∈l∞(p) and allt∈F. Puttingy=(1, 1, 1...) we see that q1(tx)= |t|p0/Mq1(x) (3.8)
and the proof is complete.
Proof ofTheorem 3.2. The proof is almost the same as inTheorem 3.1and will be omit-
ted.
Remark 3.3. If the algebraXhas an identity elementx0 for multiplication andq=0 is a near-quasinorm onXwhich has anMr-factor onX, then we obtainq(x0)>0,µinf≥ 1/q(x0) and
1
q(x0)µinf|t|rq(xy)≤q(txy)≤µinf|t|rq(x)q(y) (3.9) for allx,y∈Xand allt∈F.
References
[1] R. Arens and M. Goldberg,Multiplicativity factors for seminorms, J. Math. Anal. Appl.146 (1990), no. 2, 469–481.
[2] ,A class of seminorms on function algebras, J. Math. Anal. Appl.162(1991), no. 2, 592–
609.
[3] S. Suantai, Scalar multiplicative factors for near quasi-norms, Bull. Calcutta Math. Soc. 90 (1998), no. 3, 183–190.
Piyapong Niamsup: Department of Mathematics, Faculty of Science, Chiangmai University, Chi- angmai 50200, Thailand
E-mail address:[email protected]
Yongwimon Lenbury: Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
E-mail address:[email protected]
