The fundamental principle for invariant subspaces of analytic functions. I

Let $W$ be a differentiation-invariant subspace of the topological product $H=H(G_1)\times \dots \times H(G_q)$ of the spaces of analytic functions in domains $G_1,\dots ,G_q$ in $\mathbb C$, respectively. Under certain assumptions there exists a sequence of complex numbers $\{\lambda _i\}$, $i=1,2,\dots$, and projection operators $p_i\colon W \to W(\lambda _i)$ onto the root subspaces $W(\lambda _i)\subset W$ corresponding to the eigenvalues $\lambda _i$ of the differentiation operator. This enables one to associate with each element $f\in W$ the formal series $f\backsim \sum p_i(f)$. The fundamental principle is the phenomenon of the convergence of this series to the corresponding element $f$ for each $f$ in $W$. The existence of the projections $p_i$ depends on a particular property of the annihilator submodule of $W$: its stability with respect to division by binomials $z-\lambda$. Stability questions arising in establishing the fundamental principle are considered.