The fundamental principle for invariant subspaces of analytic functions .1.

Let W be a differentiation-invariant subspace of the topological product H = H(G(1)) x...x H(G(q)) of the spaces of analytic functions in domains G(1),...,G(q) in C, respectively. Under certain assumptions there exists a sequence of complex numbers {lambda(i)}, i = 1, 2,..., and projection operators p(i): W --> W(lambda(i)) onto the root subspaces W(lambda(i)) subset of W corresponding to the eigenvalues lambda(i) of the differentiation operator. This enables one to associate with each element f is an element of W the formal series f similar to Sigma pi(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.