Stable Perturbations of Operators and Related Topics

The following sections are included:

• Preface

• Contents

In this chapter, we will overview some important results in linear functional analysis, Banach algebras as well as 𝐶^* ‒ algebras and will give a complete description of the reduced minimum modulus of a densely-defined operator and the gap function of linear subspaces. For convenience, we may assume that X, Y , Z are Banach spaces over complex filed ℂ and H, K, L are Hilbert spaces over complex field ℂ throughout the chapter.

The definitions and existences of various generalized inverses on infinite-dimensional spaces are fully discussed in [Nashed (1976b)]. In [Nashed (1976a)], Nashed proposed a condition that makes the perturbation of a bounded linear operator with a generalized inverse have a generalized inverse too. But it is not easy to verify this condition for a given operator and a perturbation. So, Chen and the author proposed a new notation so- called ‘stable perturbation’ of a bounded linear operator with a generalized inverse in [Chen and Xue (1997)], which is equivalent to Nashed's condition and is also easy to be checked. Recently, the stable perturbation of the closed operator with a bounded generalized inverse is established in [Wang and Zhang (2007)] and [Huang (2011)].

Tseng first introduced generalized inverses of unbounded operators on Hilbert spaces in his paper [Tseng (1949)]. Then a very useful theory of generalized inverses of operators on Hilbert spaces was established by means of Arghiriade's paper [Arghiriade (1968)], Hestenes' paper [Hestenes (1961)] and Petryshyn's papers [Petryshyn (1967)]. But there are two important problems remaining unsolved at that time. One is the continuity of the generalized inverse and the other is the error-estimations of the perturbation of the generalized inverses. The first problem is solved by Izumino in 1983 (cf. [Izumino (1983)]). The second problem is solved by Chen, Wei and the author in 1996 (cf. [Chen et al. (1996)]). The results in this paper generalize corresponding results in [Stewart (1977)].

The Drazin inverse of an element was originally defined by Drazin in [Drazin (1958)] in the context of semigroups and rings. It was later generalized to bounded linear operators whose resolvent had a pole, and more generally, any singularity at zero by [Koliha (1996)]. Applications to semigroups of operators require the Drazin inverse of a closed linear operator; a first account was give in 1992 by Nashed and Zhao [Nashed and Zhao (1992)] without proofs for the case of a resolvent pole. The fully general case is studied in [Koliha and Tran (2001)].

In this chapter, we present some applications of the stable perturbation of generalized inverses, Moore-Penrose inverses and Drazin inverses. These applications include …

(1) perturbation analysis of the generalized Bott-Duffin inverse.

(2) perturbation of the frame sequence in Hilbert space.

(3) the distribution of null spaces of a vector-valued holomorphic mapping.

(4) estimations of the condition numbers of a Moore-Penrose inverse and a Drazin inverse.

(5) the local linearization of a 𝐶¹-mapping.

This chapter consists of the perturbation of reduced minimum modulus, the density of Moore.Penrose invertible elements in some 𝐶^*-algebras, properties of reduced minimum modulus of an element in a 𝐶^*-algebra and the lifting of Moore.Penrose invertible elements from quotient 𝐶^*-algebras as well as Fredholm elements and their indices.

The following sections are included:

• Appendix A: Some Results From Topology And Analysis
  • Some topological results
  • Holomorphic vector—valued functions
  • Inverse Function Theorem

• Appendix B: More On C*—Algebras
  • Some basic results in C*—algebras
  • Lifting problems about projections
  • K—theory for C*—algebras

• Bibliography

• Index