MEASURE OF NONCOMPACTNESS OF OPERATORS AND MATRICES ON THE SPACES c AND c 0

whenever the series are convergent for all n≥ 1. For any given subsets X , Y of s, we will say that the operator represented by the infinite matrix A = (anm)n,m≥1 maps X into Y that is A∈ (X ,Y), if (i) the series defined by An(x)= ∑∞ m=1 anmxm are convergent for all n≥ 1 and for all x ∈ X ; (ii) Ax ∈ Y for all x ∈ X . If c ⊂ cA = {x : Ax ∈ c}, A is conservative. Well-known necessary and sufficient conditions for A to be conservative are


m=1
a nm x m , n = 1,2,..., ( 1 ) whenever the series are convergent for all n ≥ 1.For any given subsets X, Y of s, we will say that the operator represented by the infinite matrix A = (a nm ) n,m≥1 maps X into Y that is A ∈ (X,Y ), if (i) the series defined by A n (x) = ∞ m=1 a nm x m are convergent for all n ≥ 1 and for all x ∈ X; (ii) Ax ∈ Y for all x ∈ X.If c ⊂ c A = {x : Ax ∈ c}, A is conservative.Well-known necessary and sufficient conditions for A to be conservative are for all x ∈ c, then A is called regular.A conservative matrix is regular if and only if a 00 = 1 and a 0m = 0 for all m [5,6].
Let B(c) be the set of all bounded linear operators on c.It is well known (see [6, Theorem 4.51-D]) that each bounded linear operator A on c into c determines and is determined by a matrix of scalars a nm , n = 1,2,...,m = 0,1,2,..., y = Ax, is defined by where x = (x n ) in c, and lim n x n = x 0 .In this case, the norm of A is defined by and for A ∈ B(c), the additional conditions are (4) and Let X, Y be Banach spaces, and let B(X,Y ) be the set of all linear bounded operators from X to Y .If Q is a bounded subset of X, then the Hausdorff measure of noncompactness of Q is denoted by q(Q), and The function q is called the Hausdorff measure of noncompactness (ball measure of noncompactness); it was introduced by Gohberg et al. [4], later studied by Goldenstein and Markus in 1968, Istrǎt ¸esku in 1972, and others.Let us point out that the notation of the measure of noncompactness has proved useful results in several areas of functional analysis, operator theory, fixed point theory, differential equations, and so forth (see, e.g., [1,2,4]).Let us recall that q(Q) = 0 if and only if Q is a totally bounded set.For A ∈ B(X,Y ), the Hausdorff measure of noncompactness of A, denoted by A q , is defined by A q = q(AB 1 ), where Let us recall that if X is a Banach space with a Schauder basis {v 1 ,v 2 ,...}, Q a bounded subset of X, P n : X → X the projector onto the linear span of {v 1 ,v 2 ,...,v n }, and μ(Q) = limsup n→∞ (sup x∈Q (I − P n )x ), then the following inequality holds: where a = limsup n→∞ I − P n [1,2,4].Now, we can state the following main result.

Corollary 2. Let A ∈ B(c). Then A is compact if and only if
Let us recall that if A ∈ B(c), y = Ax, then y 0 = x 0 for every choice of x if and only if α 00 = 1 and a 01 = a 02 = ••• = 0 (see, e.g., [6]); in this case A is called regular.Now, by Corollary 2, we have the next well-known result of Cohen and Dunford [3, Corollary 3].

Corollary 3. Let A ∈ B(c) be regular transformation. Then A is compact if and only if
Corollary 4. Let A ∈ (c,c), let a 00 be as in (3), and let a 0m , n = 1,2,..., be as in (4).Then and A is compact if and only if Let us remark that Corollary 4 implies that compact conservative matrix is conull.
If we recall the characterizations of the sets (c,c 0 ) and (c 0 ,c 0 ) [5,6], and remark that in this case the projector P n (x) = (x 1 ,x 2 ,...,x n ,0,...) maps c 0 into c 0 , and I − P n = 1, then by the proof of Theorem 1, we have the next result.
and A is compact if and only if and A is compact if and only if

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

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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