SOME SPECTRAL INCLUSIONS ON D-COMMUTING SYSTEMS

Some spectral inclusions for the Taylor joint Browder spectra of D-commuting systems are estabished. The obtained results are applied in generalizing the spectral mapping theorems.

For two complex IIilbert spaces X and Y, let L(X,Y) denote the set of all closed linear operators, defined on linear subspaces of X, with values in Y. Let C(X,Y) denote the subset of those operators from L(X,Y) which are defined everywhere, and hence continuous.
For L(X,X) and C(X,X), we shall simply write L(X) and C(X), respectively.For any operator A e L(X,Y), D(A), N(A), and R(A) denote, respectively, the domain, null space, and range of A. Let [XP}pe Z (where Z is the ring of integers) be a family of Hi[bert spaces, and (p) (p+) let a (p) e L(xP,Yp) be a family of operators such that R(a cN(a ), for all p.Let the following sequence represent the complex of Hilbert spaces: Let {HP(X'a)}peZ denote the cohomology oF the complex (X,a)=(XP,a (p) p, that s, HP(x,a)=N(a (p) )/R(a(P-1)).
), for all (p) A complex (X,a) (XP,a is sald to be Fredllolm If inf{r(a (p))} > 0 (where r(a (p)) is the reduced minimum modulus of a(P)), dim(HP(X,a)) < for each pZ, and HP(X,a) # 0 only for a finite number of indices.
Note that for a Fredholm complex (X,a), the quotient space IIP(X,a) is isomorphic to the subspace N(a (p)) 0 R(a(P-I)), for all pZ, and hence HP(x,a) will carry this meaaing i.n the sequel.le also assume without loss of generality that D(a (p)) is dense in X p for each pEZ so that this emphasis is attached to the complex (X,a) (XP,a(P)).
We recall some definitions [I].
Let s--(Sl,...,s be a system of n n indeterminates, and let E(s) be the exterior algebra over the complex field C, generated by the indeterminates Sl,...,s For two Htlbert spaces X and Y, there is a natural identification of the space E(X,s) E(y,t) with space E(X @ Y,(s,t)), where (tl,...,t) is another system of indeterminates.
It is obvious that HjHk+HkHj=O, for j,k 1,.Then a=(al, ..,a is said to be a complex of H[bert spaces (EP(:K,s), a p=O" n Fredholm if the corresponding complex is Fredholm.
DEFINITION 1.5.The joint spectrum of a D-commuting system L n C n a (a ...,a n (X) is the set of those points h such that a-hi is slngular, denoted T(a, X).
DEFINITION 1.6.The Fredholm spectrum of a D-commuting system a=(al,. .,an Ln(X) t the set of those points such that a -hi is not Fredholm, denoted DEFINITION 1.7.The joint Browder spectrum of a D-commuting system L n a=(a an) U{accumulation points of i(a) }, for i=I,D,P,T, where I,D,P and T, respectively, denote the commutant spectrum, Dash spectrum, polynomial spectrum, and Taylor spectrum.For detail, see [3], [I] and [4].
The prime aim of this paper Is to establish some inclusions for the Taylor joint Browder spectra of D-commuting systems.
The obtained results are also applied In generalizing the spectral mapping theorems on tensor products.Unlike the case of the bounded linear operator systems, the situation is quite different in the case of the closed densely-defined operator systems [3].
We state and prove (in certain case) some results for our main theorems.We are about to establish tle ,nain results.And, if (%,)c oT(ab,XY)\ oT(ab,X Y), then (%,I)is not an e isolated point of oT(aOb,XY).Thus, in turn, it implies, by Lemma 2.1, that either k or is not an isolated point.This further implies that T T (,,U) c [oB(a,X) oT(b,Y)!O[or(a,X) OB(b,Y)l, and this completes the proof.
REMARK 3.3.The question whether the inclusion (3.1) is true for the cases of the Commutant, Dash, and Polynomial spectra is still open.Let XI'''''Xn be lli|bert spaces and let Sj E L(X]), for j--l,...,n, be densely-defined operators.
Let X=X O ..,O X be tle completion of the algebraic n tensor product XIO...O X with respect to the canonical ttilbert norm, and let n S.
For n=2, it reduces to a special case Let us assume that the theorem holds for any n-I operators, for n>3.Thls completes the proof.
THEOREM 3.5.Let ff(fl'''" f be a family of functions which are holomorphtc in a netghbourhood of the Taylor spectrum of the operator family a @ b (al@ I,...,an [, I @ bl,..., PROOF.Apply Lemma 2.4 to Lemma 2.3.
PROOF.The proof follows from an application of the Lemma 2.5 to the Theorem 3.1.

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

3 . 4 .
Note that to each D-commuting system we can associate a complex of Hilbert spaces (EP(x s) d(p)Let a=(a ...,a (X) be a D-commuting system associated wth (p))n 1.(Grosu and Vasilescu [211.Let a=(a ,a (X) be D -commuting n a system, and b=(b l,...,bm Lm(y) be a Db-COmmuting system.Then T o (a @ b,X I} Y)=o'(a,X) o-(b,Y).LEMMA 2.2.(Grosu and Vasilescu [211.Let (X,a) and (Y,b) be two complexes of Hilbert spaces.Then their tensor product (X Y,a b) is Fredholm and exact either (X,a) or (Y,b) is Fredholm and exact.order to prove the [nclusion, let us take a point (h, IJ) ' l : [Tea X) JOb Y)] u ((a x) ](b,Y)].e eWe may assume without loss of generality that h=O, and =0.If 0 o:(a,X) and 0 (a,X), thon from Lemma 2.2, t follows that a b ts Fredholm.The same thing happens when 0 :(b,Y).Therefore, LEMIA 2.4.(Fa[qshtein[5]).Let a=(a ..,a c Lq(X) be a I) -commuting system.Let be a family of functions, which are ho!,),nocphie in a neighbolrhood of o(a l) x...x o(a ).
shall donote the tensor product XE(s) (resp.xEP(s)).An element EP(X,s) wll] be n is the eector subspace of E(s) coutatning all homogeneous oxter[or forms of degree p (p=O,l,.,n) in s] s For any Hilbert space X, E(X,s) (resp.EP(X,s))

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