Criterion of analytic continuability of functions in principal invariant subspaces on convex domains in $\mathbb{C}^{n}$

Abstract

Abstract. Subspaces invariant under differentiation are studied for spaces of functions analytic on domains of a many-dimensional complex space. For a wide class of domains (in particular, for arbitrary bounded convex domains), a criterion of analytic continuability is obtained for functions in arbitrary nontrivial closed principal invariant subspaces admitting spectral synthesis. §1. Notions and notationFor the reader's convenience, here we collect the main notions used in the paper and fix the notation for them.C n is the n-dimensional complex space. S is the sphere of radius 1 and centered at the origin. B(z, δ) is the ball centered at z = (z 1 , . . . , z n ) and of radius δ. {z k } is a sequence of points in C n . The coordinates z = (z 1 , . . . , z n ) of a vector z are denoted similarly, but the specific meaning of the symbol {z k } will be clear from the context each time.For ς ∈ S and a ≥ 0, we denote by Φ a (ς) the set of sequences {z k } ⊂ C n with the following properties:Passing to a subsequence, on each sequence {z k } ∈ Φ a (ς) we can also impose the additional condition saying that {|z k |} is monotone decreasing. Indeed, put k 1 = 1, and for j ≥ 2 choose k j to be the minimal k with |z k

Basis in invariant subspace of analytical functions

“…The problem on such continuation of the functions in the invariant spaces has a rich story. The most general results on this problem both for the case of one and several variables were obtained in works [23]- [25]. These works provide also a historical survey on the continuation problem.…”