Abstract:
Let W be a differentiation-invariant subspace of the topological product H=H(G1)×⋯×H(Gq) of the spaces of analytic functions in domains G1,…,Gq in C, respectively. Under certain assumptions there exists a sequence of complex numbers {λi}, i=1,2,…, and projection operators pi:W→W(λi) onto the root subspaces W(λi)⊂W corresponding to the eigenvalues λi of the differentiation operator. This enables one to associate with each element f∈W the formal series f∽∑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 pi depends on a particular property of the annihilator submodule of W: its stability with respect to division by binomials z−λ. Stability questions arising in establishing the fundamental principle are considered.