Journal of Mathematics and Statistics 10 (2): 262-266, 2014 ISSN: 1549-3644 © 2014 Science Publications doi:10.3844/jmssp.2014.262.266 Published Online 10 (2) 2014 (http://www.thescipub.com/jmss.toc) Corresponding Author: Raed Hatamleh, Department of Mathematics, Jadara University, Irbid-Jordan 262 Science Publications JMSS STATIONARY CONNECTED CURVES IN HILBERT SPACES Raed Hatamleh, Ahmad Qazza and Hatim Migdadi Department of Mathematics, Jadara University, Irbid-Jordan Received 2014-03-06; Revised 2014-04-16; Accepted 2014-05-17 ABSTRACT In this article the structure of non-stationary curves which are stationary connected in Hilbert space is studied using triangular models of non-self-adjoint operator. The concept of evolutionary representability plays here an important role. It is proved that if one of two curves in Hilbert space is evolutionary representable and the curves are stationary connected, then another curve is evolutionary representable too. These curves are studied firstly. The structure of a cross-correlation function in the case when operator, defining the evolutionary representation, has one-dimensional non-Hermitian subspace (the spectrum is discreet and situated in the upper complex half-plane or has infinite multiplicity at zero (Volterra operator)) is studied. Keywords: Stationary Connectedness, Infinitesimal Correlation Matrix, Triangular Operator Model, Channel Operator Element 1. INTRODUCTION It is well known (Rozanov, 1967; Hannan, 2009; Pugachev and Sinitsyn, 2001) that if two stationary random processes of the second order 1 () t ξ and 2 () t ξ (in what follows we consider that () 0 M t α ξ = ) are stationary connected, then in the corresponding space ( 29 { } [ 29 , , ( , 1, 2; 1, ; 0, =∨ = = ∞ = ∞ k k k ka H H C t k t α ξ ξ α ξ α is the value space of random processes) they correspond to the stationary curves of the form 0 () t t U α α ξ ξ = , where t U is a one-parameter group of unitary operators which always can be represented as itA t U e = , where A is a self-adjoint and, in general, unbounded operator in H ξ . If ( 29 ( 1, 2 t α ξ α = are nonstationary random 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), i.e., may be expressed as (29 ( 29 0 1, 2 itA t e α α α ξ ξ α = = , where A α are linear bounded operators in H ξ . For simplicity, we assume that 1 2 , H H H H ξ α = = are Hilbert spaces ( 29 , k k H C t α α α α ξ =∨ . 2. EVOLUTIONARY REPRESENTABLE STATIONARY CONNECTED CURVES IN H ξ ξ ξ Let us introduce an infinitesimal correlation matrix (Pugachev and Sinitsyn, 2001) with components: (,) ( ) (,) W ts t sK ts αβ αβ =- ∂ +∂ For evolutionary representable curves one can easily derive the following expression: * (,) ( ), () H A A W ts t s i α β αβ α β ξ ξ ξ - = (1)