Orthogonal Spaces Associated with Exponentials of Indicator Functions on Fock Space

Parthasarathy and Sunder have proved that the set of coherentvectors associated with the indicator functions of Borel setsis total in the boson Fock space (L²(R⁺;C)). The paper studiesthe space generated by coherent vectors associated with theunion of n intervals. A complete characterization is given oftheir orthogonal space in terms of their chaos expansion. Parthasarathyand Sunder's result is recovered in a very simple way. In thecases of the Brownian motion or Poisson process interpretationof the Fock space, the result characterizes those random variablesthat are orthogonal to the exponential of any sum of n incrementsof the Brownian motion or Poisson process.