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ORTHOGONAL SPACES ASSOCIATED WITH EXPONENTIALS OF INDICATOR FUNCTIONS ON FOCK SPACE

Published online by Cambridge University Press:  24 March 2003

STÉPHANE ATTAL
Affiliation:
UFR de Mathématiques, UMR 5582 CNRS-UJF, Institut Fourier, Université de Grenoble I, BP74, 38402 St Martin d’Hères Cedex, France
ALAIN BERNARD
Affiliation:
UFR de Mathématiques, UMR 5582 CNRS-UJF, Institut Fourier, Université de Grenoble I, BP74, 38402 St Martin d’Hères Cedex, France
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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.

Type
Notes and Papers
Copyright
The London Mathematical Society, 2002

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