On the spectral properties of 2-isometric and related operators on a hilbert space

Abstract

We investigate the spectral properties of 2-isometric operators on a Hilbert Space.A bounded linear operator T is a 2-isometry if; T 2T2 - 2T T + I = 0 2-isometric operators arose from the study of bounded linear transformations T of a complex Hilbert space that satisfy an identity of the form, Pm k=0(ð€€€1)mð€€€k ð€€€m k

T kTk = 0 for a positive integer m,such operators are said to be m ð€€€ isometries. The case m = 1,gives rise to the class of isometries on a Hilbert space which has been widely studied due to its fundamental importance in the theory of stochastic processes,the intrinsic problem of modelling the general con- tractive operator via its isometric dilation and many other areas in applied mathematics. The case m = 2,is the class of 2-isometries on a Hilbert space,which contains the class of Brownian unitaries which play an essential role in the theory of non-stationary stochastic processes related to Brownian motion.Brownian motion or Pedesis(Greek for leaping) is the presumably random drifting of particles suspended in a uid(a liquid or a gas) or the mathematical model used to describe such random movements,which is often called particle the- ory. The mathematical model of brownian motion has several real world applica- tions .An often quoted example is the stock market uctuations. It has been shown in [1],that the general 2-isometry has the form,B = TjM,where B is the block form B =

V E 0 U ! where > 0 is constant,V is an isometry,U is unitary,E is a Hilbert space isomorphism onto kerV and M an invariant subspace for T.The operators B are refered to as Brownian unitaries of convariance