PERTURBATIONS OF ONE RANK AND THE POLE ASSIGNMENT
Liu J · 1982
Let (?) be a Banach space.A linear discrete operator A on (?) into (?) is said to be of D_(?)-class if there are a nonconditional basis of (?),{(?)}_1~∞,a sequence of complex numbers{λ_(?)}_1~∞and an integer N such thatλ_n≠λ_m,(?)n,mN,A(?)=λ_n(?)_n,|λ_n-λ_m|≥δ|λ_(?)|~(?),n,mN,for someδ0,(?)and the spectrum of the restriction of A on (?) is {λ_1,…,λ_N}.Let f be a linear functional on (?) with f A~(?) being bounded.Then,for any b∈(?),theone-rank perturbation operator T=A+〈f,·〉b is again of D(?)-class.This result can be applied to pole assignment problems.Given a sequence of complexnumbers{μ_n},we can find an element b in (?) such that the operator T=A+〈f,·〉bwill have a sequence{μ_n}as its spectrum,if and only if the series∑(μ_b-λ_n)/(f((?)_n))(?)is convergent.An explicit expression for b is also given.