ON THE GROWTH OF THE SPECTRAL MEASURE

We are concerned with the asymptotics of the spectral measure associated with a 
self-adjoint operator. By using comparison techniques we shall show that the eigenfunctionals of 
L2 are close to the eigenfunctionals L1 if and only if dΓ1≈dΓ2 as λ→∞.

1 INTRODUCTION   We would like to obtain a relation between the growth of the spectral measure of a self-adjoint operator and the behaviour of its eigenfunctionals.In this study we shall assume that we have two "close" self-adjoint operators acting in the same separable Hilbert space, H say. Without loss of generality we can assume that both operators have simple spectra.To this end, let us denote by (A) and y(A) the eigenfunctionals of L1 and L respectively.Recall that the spectrum of a self-adjoint operator is defined by VA 6 a, 3 ,,.6 D{L,} / I199,,,,II and IIL,9,,.-Aqo,,.l]0 where I 2. In case A is in the continuous spectrum the sequence is not compact in the Hilbert space H.For this we can assume the existence of a countab]y normed perfect space , such that where the embeddings are compact, for further details see [I] and [2].For the sake o[ simplicity we shall assume that the embeddings are given by the identities and so le #ell (l, ) =< l, # >(R)x, Since the sequence p. is bounded in H it is then compact in (1)', which implies and similarly for the operator L; Since both operators are acting in the same Hilbert space H, we shall assume that the space ' contains both systems of eigenfunctionals; i.e., {y(A)} C ' and Recall that the system {y(A)} helps define an isometry for L V f e f ]2()) < I where ](A)e L.r2() Similarly for f [ ]I(A)(A)dF(A) where L]rl() These transforms dee immetries, and Parsel equity elds where the nondecreing functions F() and F2() are 11 the spectral mees sociated with L and L, rpectively.It is these fctions that we wod like to estimate as In all that follows y() () means Vf and dr() dr2(A) mes that Vf e Lr() L() In this work, we shah try to answer the following problem: Statement of the Problem: der what contions y(A) () as dr() dr2() as In order to swer the above qution, we shl compare the selhadjoint operators L and L2, 'e [3].call that a sft operator or transmutation is defined by () yv() e a; Clearly the deition of V depen on a2 and al and we shall aee to set y(A)=0 if AChe, and (A)=0 if ACal Contion a C a inses that V0 0 and so deles an operator on the gebrc span of {(A)}.Thus it is clear that in order for V and Vto est nnear operator it is nessary that a C a and al C q2 a2 al.
It is really n that {(A)} form a mplete t in the refleve space (perfect) ', and so the spa generated by {(A)} is dense in '.Consequently V is densely deed.Ts in tns lows us to define the adjoint operator V .

MAIN RESULTS
r.l, such that We shall agree to say FI(A) is Abs-dF2 if there exists g(r) FI(A) g(n)dr(/) + FI(0) This fact shall be denoted by dF () =--7()e In this case the condition dFl() dF2(A) in the statement of the problem can be restated as g(A) x as A oo. Recall that due to reflexivity of the space (I), the operator V' is defined in (I) and since (I) -H, V is actually defined in H. Let us denote this extension to the space H by .S ince we are interested in the case where y(A) o() we can expect V to be bounded.In this regard we have the following result: Theorem 1: If the extension H H, is a bounded operator then Fl(A) is dF2-ABS continuous.
Proof: It is dear that for f .Dr, In other words < f,y(A) >oo' < f,Vm(A) >oo' < V'f,p(X) >oo' Equation 9..I obviously holds for f E H. Indeed let fn E Dr, C H such that fn _n f H.
Given that Q is a bounded operator in H, we obviously have Qfn Qf.Using the fact that Vn, ]nz(A) VA (A) d the imetries are bounded operators we have/ and QA V f Therefore /(A) @f (A) f H. (2.2) Thus each dF negligible set is a dF negnble t.Henceforth F (A) to be dF()-Abs continuo.
The above inequafity is exactly a scient contion for the don-Nikodym threm to hold, s [].
In all that follows we shall sume that dF() is dF2 Abs continuous w is denoted by dF g() (1. We now nd to dee a nction of an operator, namely g(L2) for the next rt: H f (L)f fO()p()()dr().
Theorem 2: Assume that V admits clure in O d F is Abs-dF2(A) then Priori From equation 2.1 and the fact that the embedngs e defined by identities, we dedu that Vf, Dr, C /()()drx() < I, Vv' However the left handside of equation 2.3 can rewritten as Observe that if we set f in equations 2.4 and 2.5 then we would obtain llx/g(L_fll llY'fll (2.6) from which we deduce that Dv, C D (v/ C , from we obtn V 6 Dv, g(L)'gCL) VV'. ( The next reset descfib the domn of '.Then use the fact that f ]2 is an isometry betwn H and Lr().
This work is bed on the following reset.Theorem 4: Aume that V admits closure in ' Proof: Notice that contions of Threm 2 hold and it follows that g(L)'g(L2) Y' in ' (2.9)By the above ntion we have that Y'(L2)'g(L)f e if f e Dw C .H owever since it is sumed thatests, then equation 2.8 yiel (g(L=))' g(L) V' in ' (2.10) -1 In order to proud further we nd to extend the operator V to .F or this observe that since is a bonded operator, V V is a bonded operator in .H ence V is defined for all elements in ', and in particar for y(A), thus V-1 v/g(2) v/g(L2)y(A) V'y(A).
Corollary 2 suggests to write V + K.In ts ce Threm 2 wod read a()V()-U()0 '()0 The question we wod like to answer now is der what contion wod K'y()0 as A.
First we ne to obrve that the above conrgence holds in .Indd by construction the fction y(A) is in d the operator K orinally w ded in must be extended to This is eily aceved if the operator K, i.e.V, is bonded in .
Theorem 5: Let Vbe a bounded operator.
-I, be such that H is densely defined in then K'()0 .
call that in order for the conclusion to hold we nd LK to be at least densely defined in Remark: The condition V bonded can replac by denly dned.This forces m to use Bre's Threm to obtn the density of Dv DLK in .Theorem 6: Let the conditions of Theorem 2 hold, and Vbe a bounded operator (g(L2) 1)-IK be a bounded operator in then Proof: (gCA) 1)yCA) 0 K'y(A) 0 as A co.
Since the [g(A)-1]y(A) 0 we obtain < f,K'y(A) >,--, 0 0 as Corollary 3: Assume that conditions of Theorem 4, hold and y(A) are bounded functionals for large A and so K'V(A) (g(L2) 1)-IK be a bounded operator in We shall see that the behaviour of w(x) 0 dictates the behaviour of the spectral function at infinity.Although this result is known, see [6], we shall provide a different treatment as it is stated in [7].For simplicity let the eigenfunctionals associated with Lland L2be defined by (0, ) , '(0, ) 0 (0, ) , '(0, 0. It is clear that qo(x, A) y(x, A) + R(x, t, )q(t)(t, )dt.
where R(x, t, A) is the Greens' function and it is shown, by the semi-classical approximation, see [8], that R(x, t, A) 0 as A oo. Therefore we have that o(x, A) y(x, A) The solution y(x, A) are known explicitly, y(x,A) vAJ_(( a + 2)x ). and A {2', where Therefore provided (Y'-1)y(x, A) --0,we shall have rl(,X) r(a) o.

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