Incompleteness and closure of the multiplicity system {t keλj t } in the weighted Banach space

We investigate the completeness, closure and minimalty of the multiplicity system M(Λ) = {tkeλnt : k = 0, 1, 2, . . . , μn − 1; n = 1, 2, . . .} in the weighted Banach space Cα, where the sequence Λ = {λn,μn}∞ n=1 and {λn} (n = 1, 2, . . .) are distinct complex numbers in the right half-plane satisfying |λn| < |λn+1| (n = 1, 2, . . .) and increasing to infinity, each λn appears μn times with μn not necessarily bounded. © 2007 Elsevier Inc. All rights reserved.