ON THE SOLUTION OF NEVANLINNA PICK PROBLEM WITH SELFADJOINT EXTENSIONS OF SYMMETRIC LINEAR RELATIONS IN HILBERT SPACE

The representation of Nevanlinna Pick Problem is well known, see (71, (8) and (111. The aim of this paper is to find the necessary and sufficient condition for the solution of Nevanlinna lick Problem and to show that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of symmetric linear relation in Hilbert space. Finally, we define the resolvent matrix which gives the solutions of the Nevanlinna


INTRODUCTION
The Nevanlinna Pick Problem consists of finding a function f(), E , analytic in Im > 0, Im (t) having a fixed sign there, to take assigned values at an infinite sequence of points in Im(t) > 0, see [13].
We define the Nevanlinna class IN" as the class of all n x n matrix functions N(t), which are holomorphic in \ IR, satisfying N(t)* N(t), IR, and for which the kernel: N(t)-N()* , A iR # X, is nonnegative. K(e,) t-X It is well known that for each N(e) CIN" there exist n x n matrices A and B with A A" and B B" 0, d a nondecreasing n x n matrix function E onlR with [ (t + 1)-dE(t) < such that: N(e) A + Be + t-t + With this so-cled Riesz-Herglotz reprentation the kernel K(e, ) tak the form K(e, a) R dC (t)   + ( e)(t )' e,A e .This is the general ce, but now our reprentation for Nevanlinna function takes such different cases.
To construct the solution function of the Nevanlinna Pick Problem in this paper we define the suitable Hilbert space which we need and define on it the linear relation which is symmetric and find the minimal selfadjoint extension of that relation in order to show that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of the symmetric linear relation we have.This will form the contents of the first two sections.In the last section, we give a description of the resolvent matrix which gives the solutions of the Nevanhnna Pick Problem and shows that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and all Nevanlinna pairs (PL) defined in [8], [9] and [10].

PRELIMINARIES
In this section, we collect several basic observations concerning our subject.If 7-//is a Hilbert space over the complex numbers , we let 7-/-7"/q) 7-/, considered as the Hilbert space of all pairs {f, g with f, g e 7t.A linear relation in H is a linear manifold T in 7/,.The domain of T, D(T), is defined by D(T) {f 7"ll{f, g} T for some g E 7-/}, and the range of T, R(T), is given by R(T) {g E 7-/[{f,g} E T for some f E }.For f E R(T), we let T(f) {g]{f,g} E T}, and thus R(T) is the union of all T(f), for f e R(T).The inverse of T, T-1, is the linear relation T -1 {{g,f}]{f,g} G T}.The null space of T, v(T), is defined by v(T) {f e 7/]{f, 0} e T}.
A hnear relation T in 7-/is a (linear) operatorin 7"l if T(0) {0}.If T is an operator in 7-/we shall abbreviate T(f) by the usual Tf.
If S and T are linear relations in 7/, we define their product ST by ST {{f,k}l{Y,g} T, {g, k} S, for some g 7}.For each c E we can associate an operator in 7-/(which can be thought of as a times the identity operator I) given by {{f, af}[f 7}.If we identify this operator with a, then we have aT {{f, ag}[{f,g} T}.
The linear relations which are of most interest to us are those which are closed linear manifolds in 7"/2, and we shall call these just subspaces in 7/2.Unless otherwise stated, all of the linear relations we consider in the remainder of this work will be subspaces.
The adjoint T" of a subspace T in 7"/2 is defined as the linear relation T" {{h,k} 72119, hi [f, k], for all {f,#} T}.
Here [-,-] denotes the inner product defined in 7. A convenient way to analyze the properties of the adjoint is to introduce the two operators J, U defined on all of 7"/2 as follows: J{f,g} {g,-f}, U{f,g}-{g,f}.
The subspace T C H is called symmetric if T C T" and selfadjoint if T T'.Recall that if T C 7"/ is a subspace, the resolvent set p(T) of T is defined by p(T) {e I(T-e)-' and the resolvent operator RT(g)'p(T) [7"/] of A is defined by RT(g.) (T-g)-, p(T).
A symmetric subspace S in f2 always has selfadjoint extensions in a suitably larger Hilbert space, but there exist selfadjoint extensions of S in 2 if and only if for some g + (and hence for all t +) the defect numbers of S are equal.Let '4 be selfadjoint extension of S in a larger Hilbert space K:, K: D /'/with nonempty resolvent set p(A).Then on p('4) we study the locally holomorphic [H]-valued function R defined by R,(e) P(A-e)-'l {{g gf, Pf}l{f,g} '4,g gf TI}, g p('4), where P denotes the orthogonal projection from K: onto (K: D 7"/).This function R1 is called the compressed resolvent of ,4 in 7"/.If S is symmetric, one can easily verify that R(e) is a holomorphic mapping with values in [7-/], with domain of holomorphy Dnl which is symmetric with respect to the real axis: Dal D.1, also RI(e)* R().Finally (R(e)f, eR(e)f + f} s" for see [6].In this case the compressed resolvent R (g) of .4 in is called a generalized resolvent of S, or the generalized resolvent of S associated with .4,see [6] and [7].
Finally, if T D 7"/2 is a closed linear relation, v (go.We define the Cayley transform C(T) of T and the inverse Cayley transform F(T) of T with respect to u by: C,(T) {{g-vf, g--fff}l{f,g} T) F,(T) {{g-f, vg--gf}[{f,g} T}.
We refer to [1], [4] and [14].We may see that the description of Coddington for all selfadjoint extensions in possibly larger spaces is based on the corresponding results for unitary extensions of isometric operators, see [6].
3. SOLUTION OF THE NEVANLINNA PICK PROBLEM Our interest will be in selfadjoint extensions of a given symmetric subspace (closed linear rela- tion) and this has been discussed, for instance see [3], [5], and [6], when one of that extensions is minimal and its connection with Resolvent has been discussed in [9], [10, [12] and [13].Then taking into consideration [5], [61 and [7l, we come to the following.THEOREM 3.1.If we have two sets of points Zo and Wo defined as: Zo {zi]zi E 17, +, 1, 2,..., n}, Wo {wi]w, E (13+, 1, 2,..., n} and the matrix G, defined as G0 w -_% if 2; Z the matrix G 0 >_ 0, we can find an f defined from + up + such that f(zi) o, that will solve the Nevanlinna pick problem. PROOF.
2. Sufficient.We build the Hilbert space 1"/as and two inner products on 7 defined by i=l,...,n} Straightforward calculations will show that S is symmetric.
This implies that for x and y e D(S) that is (z, e) (y, e) O, that [z, ]-[, z] 0.

f(g) is analytic for
This follows immediately from the fact that A is selfadjoint.This completes the proof.THEOREM .2.There exists a ontne correspondence betwn all solutions of the Nevan- linna Pick Problem constructed in THEOREM 3.1 and all minimal selfdjoint extensions of S. PROOF.Assume f(t) is a solution of the Nevanlinna Pick Problem There exists a Hilbert spce (, (, }), an element u and an unitary operator U from to such that f(g) s + (z ,)((I + p(g)U)(I p(g)U)-'u, u) with 61R and p(g) ( z,)(g ,)-', s [S].
We can take K: minimal in which case U is unique up to isomorphism f(z) =wl = s Re@) and (u,u)= (wl 1)(zl t) -1.
Take A Fz1(U), the inverse Cayley transform of U.
=>. f(g) wl + (e-zl)(u,u) + (-zl)(g-I)((A- We wish to show that 3 F isomorphism 7"/ K: such that FS C_.A F. A(e el) 5ej ,el F(3e, q) FS(e -e) for j 2,...n FS C AF [ei, %] (e,, %), for these cases we have" (b) (el,e3) can be constructed as follows: f(%) % = (c) (e,,%) can be constructed as follows: From the above, we get From (a), (b) and (c) and the above, every solution can be written in the form f() wl + (g- zl)Gn + (l z)(g-:)[(A $)-1u, u] with A a selfadjoint extension of S. As we have seen before every selfadjoint extension gives a solution of the Nevanlinna Pick Problem This completes the proof of the one-to-one correspondence of f(g) and A.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

DEFITION 3 . 1 .
If the function f(z)is defined as f(g) w+(g-z)G +(-z)(g-,)W() where W(g) is defined in mma 3.2, then we define the solution matrix of the Nevanlinna Pi Problem :