The representation and approximation of the Drazin inverse of a linear operator in Hilbert space

Abstract

We present a unified representation theorem for the Drazin inverse of linear operators in Hilbert space and a general error bound. Five specific expressions, computational procedures, and their error bounds for the Drazin inverse are uniformly derived from the unified representation theorem.

Introduction

In 1958, Drazin [7] introduced a generalized inverse of an element in an associative ring or a semigroups. It is defined as follows.

Let S be an algebraic semigroup (or associative ring). Then an element a∈S is said to have a Drazin inverse, or to be Drazin invertible [7] if there exists x∈S such that

$$\text{a}{}^{\mathrm{k+1}}\text{x=a}{}^{\mathrm{k}}\text{,}\text{xax=x,}\text{ax=xa.}$$

If a has a Drazin inverse, then the smallest non-negative integer k in (1.1) is called the index, denoted by Ind(a), of a. For any a, there is at most one x such that (1.1) holds. The unique x is denoted by a^D and called the Drazin inverse of a. The Drazin inverse does not have the reflexivity property, but it commutes with the element as shown in (1.1).

In particular, when Ind(a)=1, the x satisfying (1.1) is called the group inverse of a and denoted by x=a^#.

The Drazin inverse has various applications in the areas such as singular differential and difference equations and Markov chain [2], [3], [4], [6], [10], [12], [15], [16], [18], [19], [20], [21].

In this paper, we present a unified representation of the Drazin inverse in Hilbert space and a general error bound. For a sequence of continuous real valued functions converging to 1/x, we can represent the Drazin inverse T^D as the limit of a sequence of corresponding functions of the operator T. Then, we apply the unified representation and the error bound to five special cases and derive five computational methods for the Drazin inverse and their error bounds. These results are analogous to those of the Moore–Penrose inverse [8].

We conclude this section by introducing some notations and basic properties of the Drazin inverse.

Let X and Y be infinite-dimensional complex Hilbert spaces. We denote the set of bounded linear operators from X into Y by B(X,Y). In this paper, we consider linear operators in B(X)=B(X,X). For T in B(X), N(T) and R(T) denote the null space and the range space of T, respectively.

The spectrum of T is denoted by σ(T) and the spectral radius by ρ(T). The notation ∥·∥ stands for the spectral norm.

Recall that asc(T) (des(T)), the ascent (descent) of T∈B(X), is the smallest non-negative integer n such that

$$\text{N(T}{}^{\mathrm{n}}\text{)=N(T}{}^{\mathrm{n+1}}\text{)(R(T}{}^{\mathrm{n}}\text{)=R(T}{}^{\mathrm{n+1}}\text{)).}$$

If no such n exists, then asc(T)=∞ (des(T)=∞). A square matrix always has its Drazin inverse. An operator T∈B(X) has its Drazin inverse T^D if and only if it has finite ascent and descent [9], [11], [14]. In such a case,

$$\text{Ind}\text{(T)=}\text{asc}\text{(T)=}\text{des}\text{(T)=k.}$$