TY - JOUR

T1 - Subspaces of C^infty invariant under the differentiation

AU - Aleman, Alexandru

AU - Baranov, Anton

AU - Belov, Yurii

PY - 2015

Y1 - 2015

N2 - Let L be a proper differentiation invariant subspace of C∞(a,b) such that the restriction operator has a discrete spectrum Λ (counting with multiplicities). We prove that L is spanned by functions vanishing outside some closed interval I⊂(a,b) and monomial exponentials xkeλx corresponding to Λ if its density is strictly less than the critical value , and moreover, we show that the result is not necessarily true when the density of Λ equals the critical value. This answers a question posed by the first author and B. Korenblum. Finally, if the residual part of L is trivial, then L is spanned by the monomial exponentials it contains.

AB - Let L be a proper differentiation invariant subspace of C∞(a,b) such that the restriction operator has a discrete spectrum Λ (counting with multiplicities). We prove that L is spanned by functions vanishing outside some closed interval I⊂(a,b) and monomial exponentials xkeλx corresponding to Λ if its density is strictly less than the critical value , and moreover, we show that the result is not necessarily true when the density of Λ equals the critical value. This answers a question posed by the first author and B. Korenblum. Finally, if the residual part of L is trivial, then L is spanned by the monomial exponentials it contains.

KW - Spectral synthesis

KW - Entire functions

KW - Paley–Wiener spaces

KW - Invariant subspaces

M3 - Article

VL - 268

SP - 2421

EP - 2439

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

ER -