ORTHOGONAL SPACES ASSOCIATED WITH EXPONENTIALS OF INDICATOR FUNCTIONS ON FOCK SPACE

Abstract

Parthasarathy and Sunder have proved that the set of coherent vectors associated with the indicator functions of Borel sets is total in the boson Fock space $\Gamma(L^2({\bb R}^+;{\bb C}))$ . The paper studies the space generated by coherent vectors associated with the union of $n$ intervals. A complete characterization is given of their orthogonal space in terms of their chaos expansion. Parthasarathy and Sunder's result is recovered in a very simple way. In the cases of the Brownian motion or Poisson process interpretation of the Fock space, the result characterizes those random variables that are orthogonal to the exponential of any sum of $n$ increments of the Brownian motion or Poisson process.