  We discuss various aspects of positive kernel method of quantization of the
one-parameter groups $\tau_t \in \mbox{Aut}(P,\vartheta)$ of automorphisms of a
$G$-principal bundle $P(G,\pi,M)$ with a fixed connection form $\vartheta$ on
its total space $P$. We show that the generator $\hat{F}$ of the unitary flow
$U_t = e^{it \hat{F}}$ being the quantization of $\tau_t $ is realized by a
generalized Kirillov-Kostant-Souriau operator whose domain consists of sections
of some vector bundle over $M$, which are defined by suitable positive kernel.
  This method of quantization applied to the case when $G=GL(N,\mathbb{C})$ and
$M$ is a non-compact Riemann surface leads to quantization of the arbitrary
holomorphic flow $\tau_t^{hol} \in \mbox{Aut}(P,\vartheta)$. For the above
case, we present the integral decompositions of the positive kernels on
$P\times P$ invariant with respect to the flows $\tau_t^{hol}$ in terms of
spectral measure of $\hat{F}$. These decompositions generalize the ones given
by Bochner theorem for a positive kernels on $\mathbb{C} \times \mathbb{C}$
invariant with respect to the one-parameter groups of translations of complex
plane.
