GLIDING HUMP PROPERTIES AND SOME APPLICATIONS

AIISrl’llAC’I ’. in this not(, xe consider several types of gliding h,.p properties for a sequence space E and we consider he various implications between these properties. By means of exa.=ples we show that mos! of the implit-ations are strict and they afford a sort of structure between solid sequence spaces and those wit h weakly sequentially complete /-(tuals. Our/nain result is used to extend a result of Bennett and Kalton which characterizes the class of sequence spaces E with the properly that E C S, whenever F is

Over the past eighty years the "gliding hump" technique has been a frequently used tool to esta- blish results in summability and sequence space theory.Among the more familar examples would be the Silverman-Toeplitz theorem which gives necessary and sufficient conditions for the regularity of a sum- mability method [22], the Mazur-Orlicz bounded consistency theorem ([6], [12] and [13]), the theorem of KSthe and Toeplitz on the weak sequential completeness of the KSthe dual of a solid sequence space Ill] and the theorems of Schur on the characterization of coercive matrices and the equivalence of weak and strong convergence in [19].Whereas the first three of these have subsequently been argued using functional analytic techniques (see e.g.[24] and [10]) no such "soft" proofs of Schur's theorems are known.
In section 3 of this note we introduce various types of gliding hump properties and discuss the im- plications between them.We give examples in section 5 to show that most of these implications are strict and they are, in some sense, affording a structure to the set of sequence spaces between the solid spaces and those with weakly sequentially complete -duals.
2. NOTATION AND Ptll';I,IMINAlllES.I,et c denote the linear space of all scalar (real or complex) sequences.By a sequence space /:' we shall mean any linear subspace of .A sequence space E endowed with a locally conw,x topology is called a h'-space if the inclusion map E cv is continuous where has the topology of coordinatewise conwrgence.A h'-space E with a Fr6chet topology is called an FK-space.If, in addition, the topology is normable then E is called a BK-space.We assune throughout this note familarity with the standard sequence spaces and their natural topologies (see e. g. [24], [9]).
For a sequence space E the multiplier space of E and the fl--dal of E are given by .M(E) {xw xy E for each y E} and where xy denotes the coordinatewise product.For x w, u Ibl the n h section ot .r is x [n]   x e k-1 where e ($,) is the k t coordinate vector.For any positive term sequence # (/) let If (E, F) is a dual pair then a(E, f), r(E, F) denotes the weak topology and the Mackey topology respectively.For a sequence space E and a linear subspace F of E (E,F) is a dual pair under the natural bilinear form {x, y) , If E is a K-space containing , the space of finitely non-zero sequences, we let where E' denotes the topologicM dual of E. A K-space E containing with E SE is ced an AK-space.
If A (a,) is an infinite matrix with scMar entries the convergence domMn adnits a tal,ral I"h lOl>oiogy [2 I]. i"or .rE ca we wrile li .rli ,I.r.
We begin by introducing several types of gliding hump properties.
DEFINITION 3.1.,A sequence (y(n)) in w \ {0} is called a block sequence if there exists an index sequence (kj) such that y") 0 for any n,k IN with k ]k,,,_,k,,] where k0 := 0, and it is called a 1-block sequence if furthermore y(k ' for each k ]k,,_,k,] and n Let E be a sequence space containing . E has the gliding hump property (ghp) if for each block sequence and any monotonicly increasing sequence (n) of integers there exists a subsequence (m) of (n) with y(,) _ E (pointwise sum).E has the pointwise gliding hump property (p_ghp) if for each z E, any block sequence (y(")) satisfying sup I1-11 < o0 nd any monotonicly increasing sequence (n) of integers there exists nell subsequence (mk) of (n) with E zy(",) E (pointwise sum).
E has the uniform gliding hump property (u_ghp) if the sequence (ro,) in the definition of the p_ghp may be chosen independently of z E. E has the pointwise weak gliding hump property (p_wghp) if the definition of the p_ghp is fulfilled for each 1-block sequence.
E has the uniform weak gliding hump property (u_wghp) if the definition of the u_ghp is fulfilled for each 1-block sequence.
We say that E has the strong p.ghp (u_ghp, p_wghp or u_wghp) if xy (',) .E (pointwise sum) holds =1 for any subsequence of (rn) in the above definitions; in this case, we use the notation sp_ghp, su_ghp, sp_wghp and su_wghp, respectively.REMARKS 3.2.Let E be a sequence space containing (a) Obviously, the definition of the ghp corresponds with the definition given in [20], [4] and the definition of the p_wghp corresponds to the weak gliding hump property considered by D. Noll [14].
(f) I. [.1] T. l,eiKer an! tl' first, author proved the validity of theore, of Nlazur Orlicz tyi' u.der the asnunptio thal ,I is a seqence space such that .(M)has the glp, that is, I has the _ghp.Actally, each instance only the fact lhat ;(,1) has the p_ghp was used in theargl.ents.
"I'IIEOI{EM 3.3.l,et E be an Fh" space containing @.Then S.: has the stro,g php; in particular, E is an l"h'-A h space then E has the 8[rong p_ghp.
i>ROO!:.The Fh" topology of E may be generated by seminorms p,.(rE IN) such that p,(x)_<p,.+l(x)(rEIN and x E E). (o) Si,ce S: is an l"h'--Ah" space we may assume that E is an FK-AK-space.Now, let x E E be givcn.Then supp,.( xke 0 (n oc and 'rEIN).
On account of (o) it is sucient to prove zy O) 0 in E. For that end let r IN be given.Then we P"(xYO)) P " ( -] x ' Y ( ' i ) e ' ) \ k = , , have by (,) which proves xyO) In general, WE fails the p_wghp. [Example: Let E be the summation matrix and E := c-,.Then WE fails the p_wghp since x := e Z es E WE (pointwise sum) and (ns) (2/) does not have any subsequence (m) such that , := e (pointwise sum) E since E-fi m0 c.] THEOREM 3.5.Let E be a sequence space contning , and let B be a matrix such that E C cn.
Then E C Sn if E h the p_wghp.PROOF.Suppose E has the p_wghp.We know from Threm 6 of D. Nell [14] and Remark 3.2(a) that (E , a(Ea, E)) is weakly sequentiy complete.Therefore, by an inclusion threm of G. Bennett Since E has the p_wghp we may assume that yx E (otherwise we switch over to a subsequence (y('")) and adapt the chosen index sequences).For a proof of Theorem a.5 it is sufficient to prove x .
(ii) If b' is any separable FK-space with E C F then E C (iii) If A isany matrix with ECca then EC I'ROOF.The equivalence (i)o(ii) is Theorem 6, (i)o(ii) of G. Bennett and N. J. Kalton [2].The implication (ii) (iii) is obviously vafid since domains ca are separable FK-spaces.
Assume, (E,r(E, Ea))is not AK.Thus, we may choose an z E and an absolutely convex a(E,E) compact subset K of E a such that pK(x [hI -z)O(n) where p(z) := sup Iaz*[ (z ).
ah" Therefore we may choose an index sequence (n,) and a sequence (aO)) in h" such that Since K is (a, -compct, (,) nd (a,) coincide on K nd (Ea,)is metrizable we may nssume that (a(')) is a(Ea, E)-convergent to an a K. (Otherwise we switch over to subsequence of (a0)) .)If A denotes the mtrix given by a, a ') (i, IS) then -in summability language-the last assumption tells us ECCA (even EcA).
From (,) we get z Sa which contradicts the assumption that (iii) is true.
[] COROLLARY 3.7.Let E be a sequence space containing and F be a separable FK-space with E C F. If E has the p_wghp then E C PROOF.Theorem 3.6 and 3.5.[] COROLLARY 3.8.Let Y be a sequence space and E be an FK-space with C Y 91 E and B be a matrix with Y 91SE C co. Then Y 71SE C So if Y has the p_wghp.
The statement remains true if we replace co by any separable FK-space F. IRO()I:.Corollary 3.7 anl the fact that Y ,%: has lhe p_wghp.COI{()I,I,ARY 3.9 l,et E I)e a separable l"E space ontaining such that S: WE Then fails the p_wghp (whereas SE has the slrong p-ghl).I)IO()1'.'l'hooren 3.3 and Corollary 3.    n{l k=l PROOF.The implication (a)(b) comes from the AK-property of Co and the monotonicity of FK-topologies.(This statement follows also by Theorem 3.5 since Co obviously has the p_wghp.)Using standard estimations we may prove (c):: (a).We are going to prove the essentiaJ part (b) (c).Let Co C Sa. Therefore, we can apply the above remark to any z Co.If ]IA]] oo wemay choose asequence (Ha)in IN and index sequences (%) and (/3j) with (j I) such that Defining y Co by we get la.,,I> j (j e IN).sgna., if% _<k_< :--0 otherwise E a,,,.y[ -E la",']>-j (J ,N)" k=o k=o 0 Thus a,y does not converge uniformly in n I which contradicts c0 C Sa.
k=l Using the same method we get Mso a proof of a threm contorting a theorem of Hahn (equivMence of (a) and (c)).However, we should mention that the proof of '(a) (c)' presented in [18, Threm 4.1,  p. 110] is more elegant.n,k PROOF.(a) = (b) follows fron lhe coniinity of the in('lusion map and lhe facl lhat is an FK .IK-- space and the monotoniciy of l"h" lop<flogies whereas (c) (a) may be prove<l with classical estination.
'(b)(c)': l:et t C S. Ths.obviosly, C ca is lrue.We assume up ]a,.] . In the next step we use this method to reprove both the well-known Schur theorem and the Hahn theorem.(The Schur theorem characterizes the matrices summing all bounded sequences, the Hahn theorem tells us that a conservative matrix which sums all x X sums also all bounded sequences where X denotes the set of all sequences with 0 and .)Moreover, we take an extended version of Schur's theorem (see [3]) into consideration.In ce of conservative matrices the equivalence (a)(c) is Hahn's theorem.
The implications (1) and ( 5) are immediate corollaries of Theorem 3.5 since m, and m0 have the p_wghp.
l,'or a proof of () al,l (1(I) we refer to [3].Now, we give a proof of (6).l"or thal we ass.me thai ,1 is a ,alrix with real entries.[in the general case of complex entries we have to .ore that [a,.] converges .niformly in .E I if and o,,ly if this is true for the real part of a,., and the imaginary part of a. .] Let (c*) be true.Then C c,t.
We define y E m0 by SiIice sgna.,.ifo <k :--0 otherwise the series a,,y does not converge uniformly in n I. Therefore y S which contradicts X C S. The aim of this section is the presentation of some examples distinguishing almost all of the gliding hump properties.For that purpose we collect known connections between gliding hump and related properties of sequence spaces in the following graphic.
Figure 1: Each arrow stands for 'implies' and the corre- sponding number in the circle gives the number of the example in 5.1 pro- ving the strictness of the implication.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning (e) su_ghp == su_wghp u_wghp p_wghp; su_ghp sp_ghp : sp_wghp =: p_wghp; su_ghp u_ghp p_ghp ==:, p_wghp; and N. J. Kton[2,  Theorem 5]  we get E C W, in particar E C Ln and E C A. Now, sume E C Wu and E Sn, that is, there ests x fi E C Wn L A with x Sn, thus ma loroforo wo .,avI,oo.,,o . . .! > 0 a.! i,lex seq.e.es ().(,) a.l (.) wil I, b n.cl lhal Now we onil)lov a gli(lin hulli I) arRlimoni, l,(,l k()"-and ('li()os(, n; sllcll lhal Thon tliere exisi a Yl IN with n, > n; and a kl >/J.such tllltt (nolo m /inductively, we get index seque.ces(k.).(3).(') w it h define a subsequence (y()) of a 1-block sequence by if %
.1.l,et A-(an) be amatrix with C ea and let x ( ca.Then z Sa e== a.x converges uniformly in n This observation gives us a short proof of the following theorem containing a, Toeplitz-Silvernan theorem.

THEOREM 4 . 2 (
matrices being conservative for Co ).For matrices A (a,,) the following state- ments are equivalent: (a) Co C ca.(b) Co C (c) p C CA and Ilall := sup y] la.l < .

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation