STATIONARY CONNECTED CURVES IN HILBERT SPACES

In this article the structure of non-stationary cur ves which are stationary connected in Hilbert space is studied using triangular models of non-self-adjoint operato r. The concept of evolutionary representability pla ys here an important role. It is proved that if one of two curves in Hilbert space is evolutionary representab le nd the curves are stationary connected, then another curve is evolutionary representable too. These curves ar tudied firstly. The structure of a cross-correlation funct ion in the case when operator, defining the evoluti nary representation, has one-dimensional non-Hermitian s ubspace (the spectrum is discreet and situated in t he upper complex half-plane or has infinite multiplicity at zero (Volterra operator)) is studied.


INTRODUCTION
It is well known (Rozanov, 1967;Hannan, 2009;Pugachev and Sinitsyn, 2001) that if two stationary random processes of the second order 1 ( ) processes, then the question concerning stationary connectedness, i.e., the cross-correlation function dependence upon difference t s − , is still opened. The solution of problem may be found in the framework of the Hilbert approach to the construction of the correlation theory of random processes. We will restrict our consideration to the case when corresponding curves in H ξ are evolutionary (linearly) representable (Pugachev and Sinitsyn, 2001;Livshits and Yantsevich, 1979) Let us introduce an infinitesimal correlation matrix (Pugachev and Sinitsyn, 2001) with components: For evolutionary representable curves one can easily derive the following expression:

JMSS Lemma 1.
In order for two random processes to be stationary connected, it is necessary and sufficient that The necessity is evident, since, in the case of stationary connected processes,

Theorem 1.
If the random processes are stationary connected and are evolutionary representable, then Proof.
From the other hand: it follows from the stationary connectedness that: Proceeding to a closure of L , we derive that Hence For the stationary and stationary connected curves  (Rozanov, 1967).
In the subsequent discussion, we restrict our consideration to the most interesting case for application, when the subspace Im AH ξ is finite-dimensional and let r be the subspace dimension.
If the nonstationarity rank of the vector curve in H ξ is defined as maximal rank of quadratic forms: are complex numbers, then, in the case of dim Im AH r = , one may show that the nonstationarity rank does not exceed 2r .
Really, if the operator A is included in the operator complex (Livshits and Yantsevich, 1979;Zolotarev, 2003), where:

Theorem 2.
If two curves in H ξ are stationary connected and one of them is evolutionary representable, then another curve is also evolutionary representable.  .

( )
If the dissipative operator A has discreet spectrum:

∑ ∑
Then there exists the evolutionary representable Gauss vector process with matrix Equation 6 and 7: with:

Proof.
Let us build a triangular model of operator Â acting in the space 2 ℓ with the help of the set of 1 2 , , λ λ ⋯ (Livshits and Yantsevich, 1979;Zolotarev, 2003). Then consider in 2 ℓ the model curves: Taking into account the expressions (5) for ( ) ∫ the contour γ encloses the spectrum of operator B (Pugachev and Sinitsyn, 2001), we obtain the expressions (6) and (7). The existence of Gauss vector processes results from the possibility of the correlation matrix , t t ξ ξ may be reconstructed, but not uniquely defined, by the correlation matrix (Hannan, 2009). In our case, the most interesting is the structure: where, ĝ is a channel element of dissipative operator A with discrete spectrum. Then using results from chapter V (Livshits and Yantsevich, 1979) we derive the expression:  and (10) changes to the expression obtained in the reference (Pugachev and Sinitsyn, 2001). Taking into consideration (10), for