Perturbation of least squares problem of dual linear operator in dual-Hilbert spaces

We introduce the dual-Hilbert space and study the basic properties of a dual operator and its generalized inverse on this space. We provide upper bounds on the perturbation of the dual Moore–Penrose inverse of the dual operator if the dual operator is injective or surjective. If the null space or range space of the perturbed dual operator is invariant, stable perturbations are used to give the perturbation bounds for the dual Moore–Penrose inverse. Additionally, given the aforementioned conditions, perturbation bounds for the least squares solution are provided. The upper bounds on the distance between the solution of a perturbed least squares problem and the set of all of its unperturbed solutions under the dual operator norm are also presented.