Full invariants on certain class of abstract dual systems

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R E S E A R C H

Open Access

Full invariants on certain class of abstract dual

systems

Fubin Wang

1*

and Cuixia Hao

2

* Correspondence:

[email protected]

1_{Department of Mathematics,}

Heilongjiang College of

Construction, Harbin 150025, China Full list of author information is available at the end of the article

Abstract

In this article, we study the convergence of sequences of operators on some classical vector-valued sequence spaces in the frame of abstract dual systems. Several

consequences of full invariants are obtained.

Keywords:sequential-evaluation convergence, abstract dual system, full invariant, uniformly exhaustive, essentially bounded

1 Introduction

A dual system in linear analysis consists of a linear space and a collection of linear functionals defined on the linear space. Theory of locally convex spaces is exactly about theory of dual systems which plays a crucial role in many fields of mathematical analysis. In fact, many people working in different fields of mathematics have been devoting themselves on the research of some special dual systems such as measure sys-tem (Σ,Ca(Σ,X)), abstract function syssys-tem (Ω,C(Ω,X)) and operator system (X,L(X,Y)) as well as fuzzy system (U,F(U)) etc.

In recent years, a few consequences have enlarged the known invariant ranges of boundedness, subseries convergence, and sequential-evaluation convergence etc., and theory of spaces has achieved substantial development. It is no doubtful that invariant principles make great influence on the trend of different fields of mathematical analysis.

This article is motivated by the problems considered by the authors in [1-7]. And the purpose of this article is to study full invariant of the convergence in

(Ylp(X), lp(X))-topology on the dual system(Ylp(X), lp(X)), (0< p ≤∞). We obtain that the above mentioned convergence is full invariant. Throughout the article, we adapt the definitions and notations from [8].

We denote (X, Y) is a dual system as usual.

Definition 1.1 [8]]. Letτbe a locally convex topology onX. If (X,τ)* =Y, then the topologyτis called a compatible topology onXwith respect to the dual system (X, Y).

It is easy to know that the weakest compatible topology onX is the weak topologys (X, Y) onXand the strongest compatible topology is Mackey topology [1].

Definition 1.2[8]. Letℬ be a family ofs(Y, X)-bounded sets ofX, and letb(X, Y) be aℬ convergent topology onX,s(X, Y) be the weak topology onX. Ifb(X, Y)⊇s (X, Y), thenb(X, Y) is called an admissible polar topology onX with respect to the dual system (X,Y).

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It is not difficult to know that not every admissible polar topologyb(X,Y) is a compa-tible topology onXwrt the dual system (X, Y).

Definition 1.3 [9]. If a property P ofX is shared by all admissible polar topologies lying between s(X,Y) andb(X,Y), then P is called a full invariant.

We denote (E, F) an abstract dual system, whereE=0, F ⊂XE and X is a locally convex space.

A topology τ onE is called a (E, F)-topology if there existsℱ⊂ 2Fsuch that ∪_M _Î

ℱM = Fandτis a topology of uniform convergence on each set MÎ ℱ, which means

that a sequence {u_n}⊂(E,τ) is said to converge to a pointuÎ (E,τ) if and only if for everyM Î ℱ, {f(u_n)} converges uniformly tof(u) for allfÎ M [10]. We uses(E,F) to represent the weakest (E,F)-topology, i.e., in the topological space (E,s(E,F)), a sequence {un}⊂(E,s(E,F)) converges to a pointuÎ(E,s(E, F)) if and only if for every

fÎF, {f(un)} converges tof(u).

Similarly, for an abstract dual system (F,E), we can also give the concept of (F, E)-topology, denoted by τ*(F, E). The weakest (F, E)-topology is denoted by s*(F, E), called weak star topology. For the convenience, the sequence {fn} converges tofin (F,

s*(F, E)) and in (F,τ*(F, E)) is denoted by _f_n_{−−−−→}σ∗(F,E) _f and _f_n_{−−−−→}τ∗(F,E) _f, respectively. In addition, (F, E)-topologyb*(F,E) is called strong star topology if fn β

∗₍_F_,_E₎

−−−−→f implies

thatfn® ffor all (F, E)-topology, sayτ*(F, E) onF. Obviously,b*(F, E) is the strongest

(F, E)-topology on F. Full invariant of (F, E) are analog to those of (E,F).

2 Full invariant on the dual system(Ylp(X), lp(X))(0< p <∞)

In the sequel, XandYwill always denote Banach spaces unless specified otherwise. Let XNbe the set of allX-valued sequences,

lp(X)= ⎧ ⎨ ⎩

xj∈XN:

j

xjp<+∞ ⎫ ⎬ ⎭,

YX _and

Ylp(X)represent the sets of Y- valued operators with domainsX and lp(X), respectively. Let

lp(X)βY = ⎧ ⎨

⎩(Aj)⊂YX :

j

Aj(xj) is convergent for all (xj)∈lp(X)

⎫ ⎬ ⎭,

lp_(X)bY_{is called the}_b_{-dual space of}_lp_{(X) [4].}

Suppose that (Aj)Î lp(X)bY, for each (xj)Îlp(X), we define f(Aj),_n:l

p₍_X₎_→_Y_,_n_∈_N

and f(Aj):l

p₍_X₎_→_Y by

f(Aj),n

xj= 1≤j≤n

Ajxj,

and

f(Aj)

xj=

j

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Obviously,{f(Aj),n} ⊂Y lp₍_X₎

, f(Aj)∈Y lp₍_X₎

and

f(Aj),n

σ∗_Ylp(X)_,_lp₍_X₎

−−−−−−−−→f(Aj)(n→ ∞)

in the dual system(Ylp(X), lp(X)).

In the following, we shall first characterize a collection of sets in lp(X) generating strong star topology_β∗(Ylp(X), lp(X)).

Definition 2.1 [2]. IfM⊂lp(X) and lim

n→∞

j≥nxjp= 0uniformly for (xj)Î M, that

is, for allε>0, there existsjε ÎNsuch that

sup (xj)∈M

j≥jε

xjp≤ε,

then Mis called uniformly exhaustive set.

Denote M[lp(X)]={M ⊂lp(X) :Mis uniformly exhaustive}.

Lemma 2.1[2]. Let M ⊂ lp (X), 0 <p< ∞, (Aj)Î lp(X)bY. If ΣjAj(xj) is uniformly

convergent for all (xj)Î M, thenM is uniformly exhaustive.

Theorem 2.1. Let(Ylp(X), lp(X))be a dual system,_τ∗(Ylp(X), lp(X))be a topology on Ylp(X) generated by ℳ[lp(X)]. Then for every (Aj) Î lp(X)bY,

f(Aj),n

σ∗₍_Ylp(X)_,_lp₍_X₎₎

−−−−−−−−→f(Aj)(n→ ∞)if and only if

f(Aj),n

τ∗_Ylp(X)_,_lp₍_X₎

−−−−−−−−→f(Aj)(n→ ∞).

Proof Necessity. Otherwise, we suppose that

f(Aj),n

σ∗₍_Ylp(X)_,_lp₍_X₎₎

−−−−−−−−→f(Aj)(n→ ∞)

but

f₍_A_j₎_,_n τ

∗_Ylp(X)_,_lp₍_X₎

−−−−−−−−→f(Aj)

doesn’t hold. By the definition of(Ylp(X), lp(X))-topology, there must existMÎ ℳ[lp (X)] such thatlim

n f(Aj),n

xj=f₍Aj)

xj_{not uniformly for all (x}_j)Î M, i.e., the series

∑jAj(xj) is not uniformly convergent with respect to all (xj) ÎM. Thus, there isε>0

such that for any m0 ÎN, there existm > m0and (x_j)ÎM satisfying

j≥m

Aj(xj) ≥2ε.

Since (A_j)Îlp(X)bY,ΣjA_j(x_j) is convergent and there existsn > msuch that

j≥n+1

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Hence

m≤j≤n

Aj(xj) =

j≥m

Aj(xj)−

j≥n+1

Aj(xj) ≥ j≥m

Aj(xj) −

j≥n+1

Aj(xj)

>2ε−ε=ε.

Since Mis uniformly exhaustive, there existsj1 ÎNsuch that for every (x_j)Î M,

j≥j1

xjp< 1

2.

Thus, there exist integers n1 > m1 > j1and (x1j)Î Msuch that

_m1≤j≤n1

Aj(x1j)

> ε.

Again using the uniform exhaustion of M, we can getj2 > n1 such that for every (xj)

Î M,

j≥j2

xjp< 1

22.

So there existn2 > m2> j2and (x_2j)Î Msuch that

m2≤j≤n2

Aj(x2j)

> ε.

In the same procedure, we can get a sequence of integers m1< n1< m2< n2 <... and

{(xkj)∞_j₌₁:k∈N} ⊂M: satisfying

mk≤j≤nk

xkj p

≤

j≥jk

xkj p < 1 2k and mk≤j≤nk

Aj(xkj)

> ε, k= 1, 2,. . .

Let

yj=

xkj,mk≤j≤nk,k= 1, 2,. . .,

0, otherwise.

Then (yj)Îlp(X). In fact,

j

yjp=

k

mk≤j≤nk

xkjp< k

1

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However

mk≤j≤nk

Aj(yj) =

mk≤j≤nk

Aj(xkj)

> ε, k= 1, 2,. . ..

This is a contradiction with (Aj)Îlp(X)bY.

Sufficiency. It is trivial, we omit it.

Remark. For any(Ylp(X), lp(X))-topology, let N[lp(X)]be a collection of sets generat-ing the topology. Then by Lemma 2.1 we can show that for any N∈N[lp(X)], N is uniformly exhaustive, that is, N[lp(X)]⊂M[lp(X)]. So _τ∗(Ylp(X), lp(X))-topology is exactly the strongest_β∗(Ylp(X), lp(X))-topology. Therefore, the following consequence of Theorem 2.1 is immediate.

Corollary 2.1. Let (Ylp(X), lp(X)) be a dual system, _τ∗(Ylp(X), lp(X)) a

(Ylp(X), lp(X))-topology on _Ylp₍_X₎

generated by ℳ[lp(X)],(Aj) Î lp(X)bY, then

f(Aj),n

τ∗₍_Ylp(X)_,_lp₍_X₎₎

f(Aj),n

β∗₍_Ylp(X)_,_lp₍_X₎₎

−−−−−−−−→f(Aj)(n→ ∞).

The Corollary 2.1 shows that the convergence of_τ∗(Ylp(X), lp(X))is equivalent to that of _β∗(Ylp(X), lp(X)). Combining this with Theorem 2.1, we obtain the next conse-quence which means that the convergence of{f(Aj),n}is full invariant.

Theorem 2.2. Let (Ylp(X), lp(X)) be a dual system, _τ∗(Ylp(X), lp(X)) be a

(Ylp(X), lp(X))-topology on _Ylp₍_X₎

generated by ℳ[lp(X)], (A_j) Î lp(X)bY, then f(Aj),n

σ∗₍_Ylp(X)_,_lp₍_X₎₎

f(Aj),n

β∗₍_Ylp(X)_,_lp₍_X₎₎

−−−−−−−−→f(Aj)(n→ ∞).

3 Full invariant on the dual system (Yl∞(X),l∞(X)) Let XNbe the set of allX-valued sequences,

l∞(X) =

(xj)∈XN: sup

j∈N

xj<+∞

and YX,_Yl∞(X)_{represent the sets of} Y-valued operators with domainsX _and l∞(X),

respectively. Let

l∞(X)βY = ⎧ ⎨

⎩(Aj)⊂YX :

j

Aj(xj) is convergent for all (xj)∈l∞(X)

⎫ ⎬ ⎭,

then l∞(X)bYis called theb-dual space of l∞(X) [4].

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f(Aj),n

xj= 1≤j≤n

Aj(xj)

and

f(Aj)

xj=

j Aj(xj).

Obviously,f(Aj),n

⊂Yl∞(X)_, _f (Aj)∈Y

l∞(X)_{, and}

f(Aj),n

σ∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞)

in the dual system(Yl∞(X)_,_l∞₍_X₎₎_.

Next we shall first find a collection of sets in l∞(X) generating strong star topology

β∗₍_Yl∞(X)_,_l∞₍_X₎₎_{. And then we prove that} _f

(Aj),n

β∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞). Further-more, we specify that f(Aj),n→f(Aj)(n→ ∞) (Yl

∞₍_X₎_, _l∞(X)

)-topology admit full invariant.

Definition 3.1 [2].M ⊂l∞(X) is called essential bounded set if there isj0 ÎNsuch that

sup (xj)∈M,j≥jo

xj<∞

Denote ℳ[l∞(X)]={M⊂l∞(X) :Mis essential bounded}.

Lemma3.1 [2]. LetXand Ybe Banach spaces,M⊂l∞(X), (A_j)Îl∞(X)bY. If∑jA_j(x_j) is uniformly convergent for all (x_j)Î M, thenM is essential bounded.

Theorem 3.1. Let (_Yl∞(X)_, l∞₍X_{)) be a dual system,} _τ∗₍_Yl∞(X)_,_l∞₍_X₎₎ _{be a} (Yl∞(X),l∞(X))-topology on _Yl∞(X) _{generated by} ℳ[l∞(X)], (A_j₎ Î l∞(X)bY_{, then}

f(Aj),n

σ∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞)if and only if

f(Aj),n

τ∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞).

Proof. Necessity. Otherwise, we suppose that

f(Aj),n

σ∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞)

but

f(Aj),n

τ∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞)

doesn’t hold. Then there must exist M Î ℳ[l∞(X)] such that

lim

n f(Aj),n[(xj)] =f(Aj)[(xj)]not uniformly for all (xj)ÎM, i.e., the series∑jAj(xj) is not uniformly convergent with respect to (x_j)Î M. Thus, there exist ε>0, a sequence of integersm1 < n1< m2< n2<... and {(x_kj) :kÎ N}⊂Msatisfying

mk≤j≤nk

Aj(xkj)

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Let

yj=

xkj,mk≤j≤nk,k= 1, 2,. . .,

0, otherwise.

Since Mis essential bounded, there isj0 ÎNsuch that

sup

j≥j0

yj≤ sup (zj)∈M,j≥j0

zj<+∞.

Then (y_j)Îl∞(X). But we have

mk≤j≤nk

Aj(yi) =

mk≤j≤nk

Aj(xkj)

≥ε,k= 1, 2,. . ..

This is a contradiction with (A_j)Îl∞(X)bY. Necessary. Trivial.

The Lemma 3.1 leads us to the following consequences.

Corollary 3.1. Let (_Yl∞(X)_, l∞₍X_{)) be a dual system,} _τ∗₍_Yl∞(X)_,_l∞₍_X₎₎_{be a (}

Yl∞(X), l∞₍_X_{))-topology on}

Yl∞(X) generated by ℳ[l∞(X)], (Aj) Î l∞(X)bY, then

f(Aj),n

τ∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞).if and only if

f(Aj),n

β∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞)

Theorem 3.2. Let (_Yl∞(X)_, l∞₍X_{)) be a dual system,}_τ∗₍_Yl∞(X)_,_l∞₍_X₎₎_{be a (}

Yl∞(X), l∞_{(X))-topology on}

Yl∞(X) generated by ℳ[l∞(X)], (Aj) Î l∞(X)bY, then

f(Aj),n

σ∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞)if and only if

f(Aj),n

β∗₍_Yl∞(X)_,_l∞₍_X₎₎

−−−−−−−−−→f(Aj)(n→ ∞).

4 Conclusion

In a frame of some certain class of dual system, we discuss a few problems on conver-gence of the sequences of operators in classical vector-valued sequence spaces and we obtain some consequences on full invariants. Precisely, sequences of operators in the spaces _Ylp₍_X₎

and_Yl∞(X)_{are studied, respectively. We not only find the strongest}

sequential-evaluation convergence but also obtain a series of theorems about full invar-iants on the sequential evaluation convergence.

Acknowledgements

We would like to express our deep gratitude to the referee for meaningful advice which makes a great improvement of the original manuscript, and gave many thanks to professor Shusen Ding for his careful suggestions. FW was greatly supported by Heilongjiang College of Construction. CH was supported by a grant from Ministry of Education of Heilongjiang Province supporting overseas returned scholars (1055HZ003), and by a grant from Ministry of Education ofHeilongjiang Province (11551366).

Author details

1

Department of Mathematics, Heilongjiang College of Construction, Harbin 150025, China2Department of Mathematics, Heilongjiang University, Harbin 150080, China

Authors’contributions

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Competing interests

The authors declare that they have no competing interests.

Received: 26 October 2011 Accepted: 26 March 2012 Published: 26 March 2012

References

1. Junde, W, Ronglu, L: An Orlicz-Pettis theorem with applications to A-spaces. Studia Sci Math.35, 353–358 (1999) 2. Ronglu, L, Fubin, W, Shuhui, Z: The strongest intrinsic meaning of sequential-evaluation convergence. Topol Appl.154,

1195–1205 (2007). doi:10.1016/j.topol.2006.12.002

3. Ronglu, L, Junming, W: Invariants in abstract mapping pairs. Austral Math Soc.76, 369–381 (2004). doi:10.1017/ S1446788700009927

4. Ronglu, L, Yunyan, Y, Swartz, C: A general Orlicz-Pettis theorem. Studia Sci Math Hungar.42(4):63–76 (2005)

5. Fubin, W, Hongzhang, J, Ronglu, L, Baoling, W: The Matrix transformation theorem on a type of vector-valued sequence space. Adv Math.40(2):161–167 (2011)

6. Agarwal, P, Ding, S, Nolder, CA: Inequalities for Differential Forms. Springer, New York (2009)

7. Ding, S:Lφ(μ)-averaging domains and the quasi-hyperbolic metric. Comput Math Appl.47(10-11):1611–1618 (2004).

doi:10.1016/j.camwa.2004.06.016

8. Wilansky, A: Modern methods in topologiacal vertor spaces. McGraw-Hill, New York (1978)

9. Ronglu, L, Longsuo, L, Min Kang, Shin: Invariants in duality. Indian J Pure Appl Math.33, 171–183 (2002) 10. Rolewicz, S: On unconditional convergence of linear operators. Demostratio Math.21, 835–842 (1988)

doi:10.1186/1029-242X-2012-71

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