Number Operators On The Generalized Functionals Of Discrete-Time Normal Martingales And Their Applications

Quantum stochastic analysis is a new research branch in the field of mathe-matical physics.It is a new theory of classical stochastic analysis and quantum mechanics interpenetrating.The main research objects include various quantum s-tochastic processes.In this theory,the annihilation operator,creation operator and number operator in Fock space play an important role in quantum noise.In recen-t years,the functional of discrete-time normal martingale has received more and more attention.However,a series of research work by scholars belongs to the cate-gory of "commutative stochastic analysis",so the development of "non-commutative stochastic analysis" based on discrete-time normal martingale functional is undoubt-edly very natural and necessary.Let M be a discrete-time normal martingale that has the chaotic representation property.In this article,we show that the number operator on the space of square integrable functionals of M can be extended to a continuous operator on the generalized functionals space of M.First the number op-erator on generalized functionals of discrete-time normal martingales of M is defined via Fock transform,then we discuss the basic properties of the number operator,finally the operator is expanded further.The paper is organized as follows:In chapter 1,we briefly describe the background and the state-of-art concerning our research topics.Some general concepts and fundamental knowledge are also presented.In chapter 2,the construction of the generalized functionals of discrete-time normal martingales of M and related conclusions are introduced.In chapter 3,we use the Fock transform to define number operator and weighted number operator on generalized functionals of discrete-time normal martingales of M and discuss its basic properties,then the related operator semigroup is established.