  The paper is devoted to the study of Gamma white noise analysis. We define an
extended Fock space $\Gama(\Ha)$ over $\Ha=L^2(\R^d, d\sigma)$, and show how to
include the usual Fock space ${\cal F} (\Ha)$ in it as a subspace. We introduce
in $\Gama(\Ha)$ operators $a(\xi)=\int_{\R^d} dx \xi(x)a(x)$, $\xi\in S$, with
$a(x)=\dig_x+2\dig_x\di_x+1+\di_x +\dig_x\di_x\di_x$, where $\dig_x$ and
$\di_x$ are the creation and annihilation operators at $x$. We show that
$(a(\xi))_{\xi\in S}$ is a family of commuting selfadjoint operators in
$\Gama(\Ha)$ and construct the Fourier transform in generalized joint
eigenvectors of this family. This transform is a unitary $I$ between
$\Gama(\Ha)$ and the $L^2$-space $L^2(S',d\mu_{\mathrm G})$, where
$\mu_{\mathrm G}$ is the measure of Gamma white noise with intensity $\sigma$.
The image of $a(\xi)$ under $I$ is the operator of multiplication by
$\la\cdot,\xi\ra$, so that $a(\xi)$'s are Gamma field operators. The Fock
structure of the Gamma space determined by $I$ coincides with that discovered
in {\bf [}{\it Infinite Dimensional Analysis,
  Quantum Probability and Related Topics} {\bf 1} (1998), 91--117{\bf ]}. We
note that $I$ extends in a natural way the multiple stochastic integral (chaos)
decomposition of the ``chaotic'' subspace of the Gamma space. Next, we
introduce and study spaces of test and generalized functions of Gamma white
noise and derive explicit formulas for the action of the creation, neutral, and
Gamma annihilation operators on these spaces.
