On the Stable Perturbation and Nashed’s Condition for Generalized Inverses

© 2020 Taylor & Francis Group, LLC. Let T be a bounded linear operator from a Banach space to a Banach space with closed range and let ¯Τ = Τ + δΤ. Nashed’s condition is that (I + δΤΤ+)−1¯Τ maps the null space of T into the range of T. The stable perturbation means that the intersection of the range of ¯Τ and the null space of the generalized inverse of T is {0}. We show that the stable perturbation is the same as Nashed’s condition in the sense of duality

RELATIONSHIP BETWEEN THE HYERS–ULAM STABILITY AND THE MOORE–PENROSE INVERSE ∗

Abstract. In thispaper, a linkbetween the Hyers–Ulamstabilityand the Moore–Penrose inverse is established, that is, a closed operator has the Hyers–Ulam stability if and only if it has a bounded Moore–Penrose inverse. Meanwhile, the stability constant can be determined in terms of the Moore– Penrose inverse. Based on this result, some conditions for the perturbed operators having the Hyers– Ulam stability are obtained, and the Hyers–Ulam stability constant is expressed explicitly in the case of closed operators. In the case ofthe bounded linearoperators, some characterizations forthe Hyers– Ulam stability constants to be continuous are derived. As an application, a characterization for the Hyers–Ulam stability constants of the semi-Fredholm operators to be continuous is given

Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix

Convergence Analysis of Projection Methods for Frobenius-Perron Operators Based On Matrix Norm Techniques

Using matrix norm techniques, we give a unified convergence analysis of a general projection method for computing stationary density functions of Frobenius-Perron operators associated with piecewise C2 and stretching transformations of the unit interval

On stable perturbations of the generalized Drazin inverses of closed linear operators in Banach spaces

We investigate the stable perturbation of the generalized Drazin inverses of closed linear operators in Banach spaces and obtain some new characterizations for the generalized Drazin inverses to have prescribed range and null space. As special cases of our results, we recover the perturbation theorems of Wei and Wang, Castro and Koliha, Rakocevic and Wei, Castro and Koliha and Wei