  In twistor theory, the canonical quantization procedure, called twistor
quantization, is performed with the twistor operators represented as
\hat{Z}^{A}=Z^{A}(\in C) and \hat{\bar{Z}}_{A}=-\frac{\partial}{\partial
Z^{A}}. However, it has not been clarified what kind of function spaces this
representation is valid in. In the present paper, we try to find appropriate
(pre-)Hilbert spaces in which the above representation is realized as an
adjoint pair of operators. To this end, we define an inner product for the
helicity eigenfunctions by an integral over the product space of the circular
space S^{1} and the upper half of projective twistor space. Using this inner
product, we define a Hilbert space in some particular case and
indefinite-metric pre-Hilbert spaces in other particular cases, showing that
the above- mentioned representation is valid in these spaces. It is also shown
that only the Penrose transform in the first particular case yields
positive-frequency massless fields without singularities, while the Penrose
transforms in the other particular cases yield positive-frequency massless
fields with singularities.
