Changing the spectrum of an operator by perturbation (Q1187512)

Let \(A\) be a linear operator in complex n-dimensional space having arbitrary spectrum and \(M\) be a finite set of complex numbers. The author proves a criterion on existence of an one-dimensional linear operator \(K\) acting in the same space such that the spectrum of the operator \(A + K\) is \(M\). He investigates two cases of different and multiple eigenvalues of \(A\). He studies this question for normal, self-adjoint, and unitary operators and gives the explicit form for the operators \(K\) which transform the spectrum of a given operator \(A\) into the given spectrum \(M\), if such operators exist. He notes that analogical problems arise in system theory: given a matrix \(A\) and a column vector \(b\), one can find (under some weak assumptions) a row vector \(f\) such that \(A + bf\) has arbitrary spectrum.