## 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(1)mk
m
k
T kTk = 0
for a positive integer m,such operators are said to be m 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

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