Description of invariant subspaces of multiple differentiation operators
\(H\) denotes the space of entire functions of one complex variable endowed with the topology of uniform convergence on compact subsets of the complex plane. The author considers \(W\) a closed \(D^n\)-invariant subspace, where \(D^n\) denotes a multiple differentiation operator. He uses a result of \textit{S. G. Merzlyakow} [Mat. Zametki 33, No. 5, 701-713 (1983; Zbl 0541.47006)], to expand \(W\) in the topological direct sum \(W= W_1+ W_2\), where \(W_1\) is the closed span of a certain set of functions \(A_1\). The main result is as follows. There exists a sequence of functions \((f_s)\), such that each function in this sequence is a linear combination of functions in \(A_1\), and such that any function \(f\) in \(W\) has a unique representation of the form \[ f(z)= \sum_{s=1}^\infty c_s f_s (z)+ (Qf) (z), \qquad z\in \mathbb{C}. \] \(Q\) is here the projection of \(W\) on \(W_2\) and the coefficients \(c_s\) are in a space topologically isomorphic to \(W_1\).
space of entire functions of one complex variable
topology of uniform convergence on compact subsets
scientific article; zbMATH DE number 687486