P-REPRESENTABLE OPERATORS IN BANACH SPACES

Let E and F be Banach spaces. An operator T∈L(E,F) is called p-representable if there exists a finite measure μ on the unit ball, B(E*), of E* and a function g∈Lq(μ,F), 1p

F such that Tx / <x,x*> dg-(x*), for all x C E* B(E*) where G is a vector measure on B(E*) with values in F and lifB(E,) q(x*)dG(x*)ll i(x*)IPd)l/P<-(JB(E*) for some finite measure on B(E*) and all continuous functions on B(E*) The representing vector measure for T need not be of bounded variation.Further, if G is of bounded variation and F doesn't have the RadonoNikodym property, then T need not be a kernel integral operator.
The object of this paper is to study operators which are in some sense kernel ingegral operators.Such operators is a sub-class of Pietsch p-integral operators.
Throughout this paper, if E is a Banach space, then E* is the dual of E and B(E) the closed unit ball of E If K is a set then 1K is the characteristic function of K If (,u) is a measure space, then LP(.c,,E) is the space of all p-Bochner integrable functions defr,e on with values in E, for p .. If p L'(',,E) is the space of es3ecia!]y bounded functions on L with values + Most of ir, E. The real q always denote the conjugate of p P q our terminology and notations are from Pletsch [6] and Diestel and Uhl [I].We refer to these texts for any notion cited but not defined in this paper.
2. Rp(E,F).It follows from the definition that every p-representable operator is Pietsch-p- integral operator, but not the converse.Let Rp(E,F) be the set of all p-repre- sentable operators from E into F.
THEOREM 2.5.Let H, E,: F and G be Banach spaces, and T Rp(E,F), A L(F,G) IIBIII:T I(p) and B L(H,E).Then ATB Rp(H,G) and PROOF.Let Tx fB(E*) <x'x*> g(x*)d(x*) for all x ( E and some finite measure u on B(E*) and some g ( Lq(B(E*),u,F).Then ATx =,

IIATII(p) <-llallllTlio(p)"
To show TB Rp(H,F), let gn be a sequence of simple functions converging to g be the associated operators in R (E F).So
Tf / f(t)g(t)d(t) if the function g is only Peztis q-integrable and the integral defining Tf is the Pettis integral, then T is known to be called veczor integral operator [I] Now using Theorem 2.5 we can prove- THEOREM 2.7.Let E,F be Banach spaces and T L(E,F).The following are equivalent" (i) T e Rp(E,F) (ii) There exists operators T e L(E,LP(2,)) and T 2 e L(LP(,),F) for some measure space (a,) such that T 2 is B-vector integral operator and T T2T1.(TlX)(X*) <x,x* >, Then T 2 is a B-vector integral operator and T T2TI.
If F has the Radon Nikodym property, then RI(E,F) II(E,F), and by using Corollary 5 in [i], we see that RI(C(c),F) II(E,F) NI(C(p,),F), where NI(E,F) is the class of nuclear operators from E into F.
Further if 5p(E,F) is the class of p-summing operators from E into F, then it follows from the Grothendieck-Pietsch represenation theorem [6], that Rp(E,F) IIp(E,F) We let R denote the operator ideal of a' p-representable operators.The fol- P lowing notions are taken from Pietsch [5] and Holu, 13].
(i) An operator ideal J is called regular if for all Banach spaces E and F, T J(E,F) if and only if KFT J(,F**), where K F is the natural embedding of F Into F**.
(ii) J is called closed if the closure of J(E,F) in L(E,F) is J(E,F) for all Banach spaces E and F.
(iii) J is called injective if whenever JF T ( J(E,=(B(F*))), then T (E,F) for all Banach spaces E and F.Here JF is the natural embedding of F into (B(F*)).
(iv) J is called stable with respect to the injective tensor product if T J(Ei,Fi), then T ( T 2 J(E E 2, F F2), for all Banach spaces EI,E2,FI,F 2. THEOREM 3.1.R is regular.P PROOF.Let E and F be any Banach spaces and let KFT Rp(E,F*), for T L(E,F).Then KFTX B(E*) <x,x*> g(x*)d(x*) for some and g as in Definition 2.1.Now g(x*) ( KF(F) for all x* B(E*).Since KF'F KF(F) is an isometric onto operator, the function g(x*) Kl(g(x*)) is well defined measurable and Hence T Rp(E,F).This ends the proof.
A negative result for R is the following- P THEOREM 3.4.R is not closed.P PROOF: Assume R is closed.Since the ideal of finite rank operator is con- P tained in Rp, one has the ideal of approximable operators is contained in Rp.By Lema 2.4, one gets R the ideal of approximable operators.Theorem 2.8, together

P
with the open mapping theorem we get that Ii ll(p) and I I are equivalent on Rp.This is a contradiction.Hence Rp is not closed.

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

DEFINITION 2 . 1 .
An operator T L(E,F) is called p-representable operator if there exists a finite measure defined on the Borel sets of B(E*) and a function ilg(x*)ll qd , and Txx,x* g(x*)d(x*) g'B(E*) --F such that iBfE,,, B(E*) for all x E.
It is not difficult to see that T [I converges in the operator norm to the operator Jy y,y*.S(y*)d (y*).n However T B TB In the operaLor norm.hence TBy =.f <y,y*.S(y*)d(y*), and n B(H*) PROOF.(i) (ii).Let T e Rp(E,F) andTxx,x* g(x*)d(x*)for some finite measure on B(E*) and g Lq(B(E*),,F).

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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