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We discuss various aspects of positive kernel method of quantization of theone-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 $\vartheta$ onits 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 ageneralized Kirillov-Kostant-Souriau operator whose domain consists of sectionsof 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 arbitraryholomorphic flow $\tau_t^{hol} \in \mbox{Aut}(P,\vartheta)$. For the abovecase, we present the integral decompositions of the positive kernels on$P\times P$ invariant with respect to the flows $\tau_t^{hol}$ in terms ofspectral measure of $\hat{F}$. These decompositions generalize the ones givenby Bochner theorem for a positive kernels on $\mathbb{C} \times \mathbb{C}$invariant with respect to the one-parameter groups of translations of complexplane.

Some aspects of positive kernel method of quantization