Invariant subspaces in half-plane

Abstract

We study subspaces of functions analytic in a half-plane and invariant with respect to the differentiation operator. A particular case of an invariant subspace is a space of solutions a linear homogeneous differential equation with constant coefficients. It is known that each solution of such equations is a linear combination of primitive solutions, which are exponential monomials with exponents being possibly multiple zeroes of a characteristic polynomial. The existence of such representation is called Euler fundamental principle. Other particular cases of invariant subspaces are spaces of solutions of linear homogeneous differential, difference and differential-difference equations with constant coefficients of both finite and infinite orders as well as of more general convolution equations and the systems of them. In the work we study the issue on fundamental principle for arbitrary invariant subspaces for arbitrary invariant subspaces of analytic functions in a half-plane. In other words, we study representation of all functions in an invariant subspace by the series of exponential monomials. These exponential monomials are eigenfunctions and adjoint functions for the differentiation operator in an invariant subspace. In the work we obtain a decomposition of an arbitrary invariant subspace of analytic functions into a sum of two invariant subspaces. We prove that the invariant subspace in an unbounded domain can be represented as a sum of two invariant subspaces. Their spectra correspond to a bounded and unbounded parts of a convex domain. On the base of this result we obtain a simple geometric criterion of the fundamental principle for an invariant subspace of analytic functions in a half-plane. It is formulated just in terms of the Krisvosheev condensation index for the sequence of exponents of the mentioned exponential monomials.

Necessary condition of fundamental principle for invariant subspaces on unbounded convex domain

“…In work [6] there was obtained a criterion of representation (1.1) in the case 𝐷 = C. Such representation holds if and only if the inequality 𝑆 Λ < ∞ is satisfied. The case, when the domain 𝐷 is a half-plane was studied in works [7] and [8]. The criterion of the representation was formulated only in terms of the index 𝑆 Λ .…”