PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 83(97) (2008), 37–47 DOI: 10.2298/PIM0897037S

DRAZIN INVERSES OF OPERATORS WITH RATIONAL RESOLVENT

Christoph Schmoeger

Communicated by Stevan Pilipović

Abstract. Let A be a bounded linear operator on a Banach space such that the resolvent of A is rational. If 0 is in the spectrum of A, then it is well known that A is Drazin invertible. We investigate spectral properties of the Drazin inverse of A. For example we show that the Drazin inverse of A is a polynomial in A.

1. Introduction and terminology

In this paper X is always a complex Banach space and L(X) the Banach algebra of all bounded linear operators on X. For A ∈ L(X) we write N(A) for its kernel and A(X) for its range. We write σ(A), ρ(A) and R_{λ}(A) for the spectrum, the resolvent set and the resolvent operator (A − λ)^{−1} (λ ∉ σ(A)) of A, respectively. The ascent of A is denoted by α(A) and the descent of A is denoted by δ(A).

An operator A ∈ L(X) is Drazin invertible if there is C ∈ L(X) such that (i) CAC = C, (ii) AC = CA and (iii) A^{ν+1}C = A^{ν} for nonnegative integer ν.

In this case C is uniquely determined (see [2]) and is called the Drazin inverse of A. The smallest nonnegative integer ν such that (iii) holds is called the index i(A) of A. Observe that

0 ∈ ρ(A) ⇔ A is Drazin invertible and i(A) = 0.

The following proposition tells us exactly which operators are Drazin invertible with index > 0:

1.1. Proposition. Let A ∈ L(X) and let ν be a positive integer. Then the following assertions are equivalent:

(1) A is Drazin invertible and i(A) = ν.

(2) α(A) = δ(A) = ν.

(3) R_{λ}(A) has a pole of order ν at λ = 0.

2000 Mathematics Subject Classification: Primary 47A10.

Key words and phrases: rational resolvent, Drazin inverse.

Proof. [2, § 5.2] and [3, Satz 101.2]. □

The next result we will use frequently in our investigations.

1.2. Proposition. Suppose that A ∈ L(X) is Drazin invertible, i(A) = ν ⩾ 1, P is the spectral projection of A associated with the spectral set {0} and that C is the Drazin inverse of A. Then

P = I − AC, N(C) = N(A^{ν}) = P (X),

C(X) = N(P) = A^{ν}(X),

C is Drazin invertible, i(C) = 1, ACA is the Drazin inverse of C,

0 ∈ σ(C) and σ(C) ∖ \{0\} = \{\frac{1}{λ} : λ ∈ σ(A) ∖ \{0\}\}.

Proof. We have P = I − AC, N(A^{ν}) = P (X) and σ(C) ∖ \{0\} = \{\frac{1}{λ} : λ ∈ σ(A) ∖ {0}} by [2, § 52]. It is clear that 0 ∈ σ(C). From Proposition 1.1 and [3, Satz 101.2] we get N(P) = A^{ν}(X). If x ∈ X then Cx = 0 ⇔ P x = x, hence N(C) = P (X). From P = I − AC = I − CA it is easily seen that N(P) = C(X). Let B = ACA. Then

C^{2}B = CBC = CACAC = CAC = C,
CB = CACA = ACAC = BC
BCB = ACACACA = ACACA = ACA = B.

This shows that C is Drazin invertible, B is the Drazin inverse of C and that i(C) = 1. □

Now we introduce the class of operators which we will consider in this paper. We say that A ∈ L(X) has a rational resolvent if

R_{λ}(A) = \frac{P (λ)}{q(λ)}

where P (λ) is a polynomial with coefficients in L(X), q(λ) is polynomial with coefficients in ℂ and where P and q have no common zeros. We use the symbol F(X) to denote the subclass of L(X) consisting of those operators whose resolvent is rational. For A ∈ L(X) let H(A) be the set of all functions f : △(f) → ℂ such that △(f) is an open set in ℂ, σ(A) ⊆ △(f) and f is holomorphic on △(f). For f ∈ H(A) the operator f(A) ∈ L(X) is defined by the usual operational calculus (see [3] or [4]).

The following proposition collects some properties of operators in F(X). An operator A ∈ L(X) is called algebraic if p(A) = 0 for some nonzero polynomial p.

1.3. Proposition. Let A ∈ L(X). Then

(1) A ∈ F(X) if and only if σ(A) consists of a finite number of poles of R_{λ}(A).

(2) A ∈ F(X) if and only if A is algebraic.

(3) If dim A(X) < ∞, then A ∈ F(X).

(4) If A ∈ F(X) and f ∈ H(A), then f(A) = p(A) for some polynomial p.

(5) If A ∈ F(X), the p(A) ∈ F(X) for every polynomial p.

Proof. [4, Chapter V.11] □

1.4. Corollary. Suppose that A ∈ F(X) and 0 ∈ ρ(A). Then A^{−1} ∈ F(X) and A^{−1} is a polynomial in A.

Proof. Define the function f : ℂ ∖ {0} → ℂ by f(λ) = λ^{−1}. Then f ∈ H(A) and f(A) = A^{−1}. Now apply Proposition 1.3 (4) and (5). □

Remark. That A ∈ F(X) and 0 ∈ ρ(A) implies A^{−1} ∈ F(X) is also shown in [1, Theorem 2]. In Section 2 we will give a further proof of this fact.

2. Drazin inverses of operators in F(X)

Throughout this section A will be an operator in F(X) and σ(A) = {λ_{1}, …, λ_{k}}, where λ_{1}, …, λ_{k} are the distinct poles of R_{λ}(A) of orders m_{1}, …, m_{k} (see Proposition 1.3 (1)).

Recall that m_{j} = α(A − λ_{j}) = δ(A − λ_{j}) (j = 1, …, k). Let

(2.1)  m_{A}(λ) = (λ − λ_{1})^{m_{1}} ⋯ (λ − λ_{k})^{m_{k}}.

By [4, Theorem V.10.7],

m_{A}(A) = 0.

The polynomial m_{A} is called the minimal polynomial of A. It follows from [4, Theorem V.10.7] that m_{A} divides any other polynomial p such that p(A) = 0. In what follows we always assume that m_{A} has degree n, thus n = m_{1} + ⋯ + m_{k} and that m_{A} has the representations (2.1) and

(2.2)  m_{A}(λ) = a_{0} + a_{1}λ + a_{2}λ^{2} + ⋯ + a_{n−1}λ^{n−1} + λ^{n}.

Observe that

0 ∈ ρ(A) ⇔ a_{0} ≠ 0

and that

0 is a pole of order ν ⩾ 1 of R_{λ}(A) ⇔ a_{0} = ⋯ = a_{ν−1} = 0 and a_{ν} ≠ 0.

Now we are in a position to state our first result. Recall from Proposition 1.1 that if λ_{0} ∈ σ(A), then A − λ_{0} is Drazin invertible.

2.1. Theorem. If λ_{0} ∈ σ(A) and if C is the Drazin inverse of A − λ_{0}, then there is a scalar polynomial p such that C = p(A).

Proof. Without loss of generality we can assume that λ_{0} = λ_{1} = 0. Let ν = m_{1}. Then we have

m_{A}(λ) = a_{ν}λ^{ν} + aν + 1λ^{ν+1} + ⋯ + λ^{n−1} + λ^{n}

and that a_{ν} ≠ 0. Let

q_{1}(λ) = −\frac{1}{a_{ν}} (a_{ν+1} + a_{ν+2}λ + ⋯ + λ^{n−(ν+1)}).

Then

A^{ν+1}q_{1}(A) = −\frac{1}{a_{ν}} (a_{ν+1}A^{ν+1} + a_{ν+2}A^{ν+2} + ⋯ + A^{n})

= −\frac{1}{a_{ν}} (m_{A}(A) − a_{ν}A^{ν}) = A^{ν}.

Let B = q_{1}(A). Then A^{ν+1}B = A^{ν} and BA = AB. For the Drazin inverse C we have

A^{ν+1}C = A, CAC = C and CA = AC.

Thus

A^{ν+1}(B − C) = A^{ν+1}B − A^{ν+1}C = A^{ν} − A^{ν} = 0

This shows that (B − C)(X) ⊆ N(A^{ν+1}). By Proposition 1.1, α(A) = ν, thus (B − C)(X) ⊆ N(A^{ν}), therefore (B − C)(X) ⊆ P_{1}(X), where P_{1} denotes the spectral projection of A associated with the spectral set {0} (see Proposition 1.2). Since P_{1} = I − AC = I − CA, it follows that

B − C = P_{1}(B − C) = P_{1}B − P_{1}C = P_{1}B − (I − CA)C
= P_{1}B − C + CAC = P_{1}B,

thus C = B − P_{1}B. We have P_{1} = f(A) for some f ∈ H(A). By Proposition 1.3 (4), f(A) = q_{2}(A) for some polynomial q_{2}. Now define the polynomial p by p = q_{1} − q_{2}q_{1}. It results that

p(A) = q_{1}(A) − q_{2}(A)q_{1}(A) = B − P_{1}B = C.  □

2.2. Corollary. If λ_{0} ∈ σ(A) and if C is the Drazin inverse of A − λ_{0}, then C ∈ F(X).

Proof. Theorem 2.1 and Proposition 1.3 (5). □

2.3. Corollary. Let A be a complex square matrix and λ_{0} a characteristic value of A. Then the Drazin inverse of A − λ_{0} is a polynomial in A.

Proof. Theorem 2.1 and Proposition 1.3 (3). □

Let T ∈ L(X). An operator S ∈ L(X) is called a pseudo inverse of T provided that T ST = T. In general the set of all pseudo inverses of T is infinite and this set consists of all operators of the form ST S + U − ST UT S, where U ∈ L(X) is arbitrary (see [2, Theorem 2.3.2]). Observe that if T is Drazin invertible with i(T) = 1, then the Drazin inverse of T is a pseudo inverse of T.

2.4. Corollary. If λ_{0} ∈ σ(A), then the following assertions are equivalent:

(1) λ_{0} is a simple pole of R_{λ}(A);

(2) there is a pseudo inverse B of A − λ_{0} such that B(A − λ_{0}) = (A − λ_{0})B;

(3) there is a polynomial p such that p(A) is a pseudo inverse of A − λ_{0}.

Proof. (1) ⇔ (2): Proposition 1.1.

(1)⇒ (3): We can assume that λ_{0} = 0. Let q_{1} and B as in the proof of Theorem 2.1. Then A^{2}B = A and AB = BA, hence ABA = A.

(3) ⇒ (1): Again we can assume that λ_{0} = 0. With B = p(A) we have ABA = A and AB = BA. Set C = BAB; then ACA = A, CAC = C and AC = CA. This shows that C is the Drazin inverse of A and that i(A) = 1. By Proposition 1.1, λ_{0} = 0 is a simple pole of R_{λ}(A). □

2.5. Corollary. Let X be a complex Hilbert space and suppose that N ∈ L(X) is normal and that σ(N) is finite. We have:

(1) N ∈ F(X),

(2) If λ_{0} ∈ σ(N), then there is a polynomial p such that

(N − λ_{0})p(N)(N − λ_{0}) = N − λ_{0}.

Proof. By [3, Satz 111.2], each point in σ(N) is a simple pole of R_{λ}(N), thus N ∈ F(X). Now apply Theorem 2.4. □

Our results suggest the following.

Question. If A ∈ F(X) and if B is a pseudo inverse such that AB = BA, does there exist a polynomial p with B = p(A)?

The answer is negative:

Example. Consider the square matrix

A = (\begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}) .

It is easy to see that the minimal polynomial of A is given by m_{A}(λ) = λ^{2} − 3λ = λ(λ − 3), hence σ(A) = {0, 3} and A^{2} = 3A. Let

B = \frac{1}{3} (\begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}) .

Then AB = BA = \frac{1}{3}A, thus ABA = \frac{1}{3}A^{2} = A, hence B is a pseudo inverse of A. Since A^{2} = 3A, any polynomial in A has the form αI + βA with α, β ∈ ℂ. But there are no α and β such that B = αI + βA. An easy computation shows that the Drazin inverse of A is given by \frac{1}{9}A and that i(A) = 1.

If 0 is a simple pole of R_{λ}(A), then we have seen in Theorem 2.4 that A has a pseudo inverse. If 0 is a pole of R_{λ}(A) of order ⩾ 2, then, in general A does not have a pseudo inverse, as the following example shows.

Example. Let T ∈ L(X) be any operator with T (X) not closed (of course X must be infinite dimensional). Define the operator A ∈ L(X ⊕ X) by the matrix

A = (\begin{matrix}0 & 0 \\ T & 0 \end{matrix}) .

Then the range of A is not closed. By [2, Theorem 2.1], A has no pseudo inverse. From A^{2} = 0 it follows that A ∈ F(X ⊕X) and that 0 is a pole of order 2 of R_{λ}(A).

Now we return to the investigations of our operator A ∈ F(X). To this end we need the following propositions.

2.6. Proposition. Suppose that T ∈ L(X), 0 ∈ ρ(T), λ ∈ ℂ ∖ {0} and that k is a nonnegative integer. Then:

(1) N(T − λ)^{k}) = N((T^{−1} − \frac{1}{λ})^{k});

(2) α(T − λ) = α(T^{−1} − \frac{1}{λ}).

Proof. We only have to show that N((T − λ)^{k}) ⊆ N((T^{−1} − \frac{1}{λ})^{k}). Take x ∈ N((T − λ)^{k}). Then 0 = (T − λ)^{k}x, thus 0 = (T^{−1})^{k}(T − λ)^{k}x = (1 − λT^{−1})^{k}x, hence x ∈ N((T^{−1} − \frac{1}{λ})^{k}). □

2.7. Proposition. Suppose that T ∈ L(X), 0 ∈ σ(T), λ ∈ ℂ ∖ {0} and k is a nonnegative integer. Furthermore suppose that T is Drazin invertible and that C is the Drazin inverse of T. Then:

(1) N((T − λ)^{k}) = N((C − \frac{1}{λ})^{k});

(2) α(T − λ) = α(C − \frac{1}{λ});

Proof. (2) follows from (1).

(2) Let ν = i(T). We use induction. First we show that N(T −λ) = N(C − \frac{1}{λ}). Let x ∈ N(T − λ), then T x = λx and T^{ν}x = λ^{ν}x. We have

λC^{2}x = C^{2}T x = CT Cx = Cx,

hence C(1−λC)x = 0, thus (1−λC)x ⊆ N(C). By Proposition 1.2, N(C) = N(T^{ν}), therefore

0 = T^{ν}(1 − λC)x = (1 − λC)T^{ν}x = (1 − λC)λ^{ν}x,

therefore x ∈ N(C − \frac{1}{λ}). Now let x ∈ N(C − \frac{1}{λ}). From Cx = \frac{1}{λ}x we see that x ∈ C(X) = N(P), where P is as in Proposition 1.2. From P = I − T C we get x = T Cx = T (\frac{1}{λ}x), thus T x = λx, hence x ∈ N(T − λ). Now suppose that n is a positive integer and that

N((T − λ)^{n}) = N((C − \frac{1}{λ})^{n}).

Take x ∈ N((T − λ)^{n+1}). Then (T − λ)x ∈ N((T − λ)^{n}) = N((C − \frac{1}{λ})^{n}), thus

0 = (C − \frac{1}{λ})^{n}(T − λ)x = (T − λ)(C − \frac{1}{λ})^{n}x.

This gives

(C − \frac{1}{λ})^{n}x ∈ N(T − λ) = N(C − \frac{1}{λ}),

therefore x ∈ N((C − \frac{1}{λ})^{n+1}). Similar arguments show that N((C − \frac{1}{λ})^{n+1}) ⊆ N((T − λ)^{n+1}). □

In what follows we use the notation of the beginning of this section. Recall that we have σ(A) = {λ_{1}, …, λ_{k}}. If 0 ∈ σ(A), then we always assume that λ_{1} = 0, hence σ(A) ∖ {0} = {λ_{2}, …, λ_{k}}.

2.8. Proposition.

(1) If 0 ∈ ρ(A), then σ(A^{−1}) = \{\frac{1}{λ_{1}} , …, \frac{1}{λ_{k}} \}.

(2) If 0 ∈ σ(A) and if C is the Drazin inverse of A, then 0 ∈ σ(C) and σ(C) ∖ \{0\} = \{\frac{1}{λ_{2}} , …, \frac{1}{λ_{k}} \}.

Proof. (1) follows from the spectral mapping theorem.

(2) is a consequence of Proposition 1.2. □

For our next result recall from Corollary 1.4 that if 0 ∈ ρ(A), then A^{−1} ∈ F(X).

2.9. Theorem. Suppose that 0 ∈ ρ(A). Then

(1) If the minimal polynomial m_{A} has the representation (2.1), then the minimal polynomial m_{A^{−1}} of A^{−1} is given by

m_{A^{−1}}(λ) = (λ − \frac{1}{λ_{1}} )^{m_{1}} ⋯ (λ − \frac{1}{λ_{k}} )^{m_{k}} .

(2) If the minimal polynomial m_{A} has the representation (2.2), then m_{A^{−1}} is given by

m_{A^{−1}}(λ) = \frac{1}{a_{0}} + \frac{a_{n−1}}{a_{0}} λ + ⋯ + \frac{a_{1}}{a_{0}} λ^{n−1} + λ^{n}.

Proof. Proposition 2.6 shows that

α(A^{−1} − \frac{1}{λ_{j}}) = α(A − λ_{j}) = m_{j} (j = 1, …, k),

thus (1) is shown. Furthermore m_{A^{−1}} has degree m_{1} + ⋯ + m_{k} = n. Now define the polynomial q by

q(λ) = \frac{1}{a_{0}} + \frac{a_{n−1}}{a_{0}} λ + ⋯ + \frac{a_{1}}{a_{0}} λ^{n−1} + λ^{n}.

Then

a_{0}A^{n}q(A^{−1}) = A^{n}(a_{0}(A^{−1})^{n} + a_{1}(A^{−1})^{n−1} + ⋯ + a_{n−1}A^{−1} + 1)
= m_{A}(A) = 0.

Since a_{0} ≠ 0 and 0 ∈ ρ(A), it results that q(A^{−1}) = 0. Because of degree of q = n = degree of m_{A^{−1}}, we get q = m_{A^{−1}}. □

Remark. The proof just given shows that there is a polynomial q such that q(A^{−1}) = 0. Therefore we have a simple proof for the fact that A^{−1} ∈ F(X).

2.10. Theorem. Suppose that 0 ∈ σ(A) and that 0 is a pole of R_{λ}(A) of order ν ⩾ 1. Let C denote the Drazin inverse of A (recall from Corollary 2.2 that C ∈ F(X)).

(1) If m_{A} has the representation (2.1), then

m_{C}(λ) = λ(λ − \frac{1}{λ_{2}} )^{m_{2}} ⋯ (λ − \frac{1}{λ_{k}})^{m_{k}}.

(2) If m_{A} has the representation (2.2), then

m_{C}(λ) = \frac{1}{a_{ν}} λ + \frac{a_{n−1}}{a_{ν}} λ^{2} + ⋯ + \frac{a_{ν+1}}{a_{ν}} λ^{n+1−(ν+1)} + λ^{n+1−ν}.

Proof. Proposition 2.7 gives

α(C − \frac{1}{λ_{j}}) = α(A − λ_{j}) = m_{j} (j = 2, …, k).

By Proposition 1.1 and Proposition 1.2, α(C) = 1. Thus (1) is valid. We have

m_{A}(λ) = a_{ν}λ^{ν} + a_{ν+1}λ^{ν+1} + ⋯ + a_{n−1}λ^{n−1} + λ^{n},

hence

(2.3)  0 = m_{A}(A) = a_{ν}A^{ν} + a_{ν+1}A^{ν+1} + ⋯ + a_{n−1}A^{n−1} + A^{n}.

If ν ⩽ l ⩽ n, then

C^{n+1}A^{l} = C^{n+1}C^{l}A^{l} = C^{n+1−l}(CA)^{l}
= C^{n+1−l}CA = C^{n−l}CAC = C^{n+1−l}.

Then multiplying (2.3) by C^{n+1}, it follows that

0 = a_{ν}C^{n+1−ν} + a_{ν+1}C^{n+1−(ν+1)} + ⋯ + a_{n−1}C^{2} + C.

Now define the polynomial q by

q(λ) = \frac{1}{a_{ν}} λ + \frac{a_{n−1}}{a_{ν}} λ^{2} + ⋯ + \frac{a_{ν+1}}{a_{ν}} λ^{n+1−(ν+1)} + λ^{n+1−ν}.

Then q(C) = 0. Since degree of q = n + 1 − ν = 1 + m_{2} + ⋯ + m_{k} = degree of m_{C}, we get q = m_{C}. □

2.11. Corollary. With the notation in Theorem 2.10 we have

C(A − λ_{2})^{m_{2}} ⋯ (A − λ_{k})^{m_{k}} = 0.

Proof. Let D = (A − λ_{2})^{m_{k}} ⋯ (A − λ_{k})^{m_{k}}. From A^{ν}D = m_{A}(A) = 0 we see that D(X) ⊆ N(A^{ν}). Since N(A^{ν}) = N(C) (Proposition 1.2), CD = 0. □

Notation. X^{∗} denotes the dual space of X and we write T^{∗} for the adjoint of an operator T ∈ L(X). Recall from [4, Theorem IV. 8.4] that

(2.4)  \overline{T (X)} = N(T^{∗})^{⊥} (T ∈ L(X)).

2.12. Proposition. Suppose that T ∈ L(X), λ ∈ ℂ ∖ {0} and that j is a nonnegative integer. Then

(1) If 0 ∈ ρ(T), then (T − λ)^{j}(X) = (T^{−1} − \frac{1}{λ})^{j}(X).

(2) If 0 ∈ σ(T), if T is Drazin invertible and if C denotes the Drazin inverse of T, then \overline{(T − λ)^{j}(X)} = \overline{(C − \frac{1}{λ})^{j}(X)}.

Proof. (1) Let y = (T − λ)^{j}x ∈ (T − λ)^{j}(x) (x ∈ X). Then

(T^{−1} − \frac{1}{λ})^{j}T^{j}x = ((T^{−1} − \frac{1}{λ})T )^{j}x = (1 − \frac{T}{λ})^{j}x

= \frac{(−1)^{j}}{λ^{j}} (T − λ)^{j}x = \frac{(−1)^{j}}{λ^{j}} y,

therefore y ∈ (T^{−1} − \frac{1}{λ})^{j}(X).

(2) Let ν = i(T). Then T^{ν+1}C = T^{ν}, T C = CT and CT C = C. Hence

(T^{∗})^{ν+1}C^{∗} = (T^{∗})^{ν}, T^{∗}C^{∗} = C^{∗}T^{∗} and C^{∗}T^{∗}C^{∗} = C^{∗}.

Thus T^{∗} is Drazin invertible and C^{∗} is the Drazin inverse of T^{∗}. By Proposition 2.7,

N((T^{∗} − λ)^{j}) = N((C^{∗} − \frac{1}{λ})^{j}),

therefore the result follows in view of (2.4). □

2.13. Corollary.

(1) If 0 ∈ ρ(A), then (A − λ_{j})^{m_{j}}(X) = (A^{−1} − \frac{1}{λ_{j}})^{m_{j}}(X) (j = 1, …, k).

(2) If 0 ∈ σ(A) is a pole of order ν ⩾ 1 of R_{λ}(A) and if C is the Drazin inverse of A, then A^{ν}(X) = C(X) and

(A − λ_{J})^{m_{j}}(X) = (C − \frac{1}{λ_{j}})^{m_{j}}(X) (j = 2, …, k).

Proof. (1) is a consequence of Proposition 2.12.

(2) That A^{ν}(X) = C(X) is a consequence of Proposition 1.2. Now let j ⩽ {2, …, k}. Because of Proposition 1.1 and Theorem 2.10 we see that

α(C − \frac{1}{λ_{j}}) = δ(C − \frac{1}{λ_{j}}) = m_{j} = α(A − λ_{j}) = δ(A − λ_{j}).

By [3, Satz 101.2], the subspaces (A − λ_{j})^{m_{j}}(X) and (C − \frac{1}{λ_{j}})^{m_{j}}(X) are closed. Now apply Proposition 2.12. □

For j = 1, …, k let P_{j} denote the spectral projection of A associated with the spectral set {λ_{j}}. Observe that

P_{i}P_{j} = 0 for i ≠ j and P_{1} + ⋯ + P_{k} = 1.

If 0 ∈ ρ(A), then denote by Q_{j} the spectral projection of A^{−1} associated with the spectral set \{\frac{1}{λ_{j}} \} (j = 1, …, k). If 0 ∈ σ(A) and if C is the Drazin inverse, then denote by Q_{1} the spectral projection of C associated with the spectral set \{0\} and by Q_{j} the spectral projection of C associated with the spectral set \{\frac{1}{λ_{j}} \} (j = 2, …, k).

2.14. Corollary. P_{j} = Q_{j} (j = 1, …, k).

Proof. By [3, Satz 101.2], we have

P_{j}(X) = N((A − λ_{j})^{m_{j}}) and N(P_{j}) = (A − λ_{j})^{m_{j}}(X)

(j = 1, …, k). If 0 ∈ ρ(A), then

Q_{j}(X) = N((A^{−1} − \frac{1}{λ_{j}})^{m_{j}}) and N(Q_{j}) = (A^{−1} − \frac{1}{λ_{j}})^{m_{j}}(X)

(j = 1, …, k). Now apply Proposition 2.6 and Corollary 2.13 (1) to get

P_{j}(X) = Q_{j}(X) and N(P_{j}) = N(Q_{j}),

hence P_{j} = Q_{j} (j = 1, …, k).

Now let 0 ∈ σ(A). By Proposition 1.2, Proposition 2.7, Corollary 2.13 (2) and [3, Satz 101.2], we derive

P_{1}(X) = N(C) = Q_{1}(X), N(P_{1}) = C(X) = N(Q_{1}),
P_{j}(X) = N((C − \frac{1}{λ_{j}})^{m_{j}}) = Q_{j}(X)
N(P_{j}) = (C − \frac{1}{λ_{j}} )^{m_{j}}(X) = N(Q_{j})

(j = 2, …, k). Hence P_{j} = Q_{j} (j = 1, …, k). □

For A we have the representation A = ∑_{j=1}^{k} λ_{j}P_{j} + N, where N ∈ L(X) is nilpotent and N = ∑_{j=1}^{k}(A−λ_{j})P_{j} (see [4, Chapter V. 11]). If p = max{m_{1}, …, m_{k}}, then it is easily seen that N^{p} = 0. If A has only simple poles, then N = 0.

2.15. Corollary.

(1) If 0 ∈ ρ(A), then there is a nilpotent operator N_{1} ∈ L(X) with

A^{−1} = ∑_{j=1}^{k} \frac{1}{λ_{j}} P_{j} + N_{1}

(2) If 0 ∈ σ(A) and if C is the Drazin inverse of A, then

C = ∑_{j=2}^{k} \frac{1}{λ_{j}} P_{j} + N_{1}, where N_{1} ∈ L(X) is nilpotent.

Proof. Corollary 2.14. □

With the notation of Corollary 2.15 (2) we have AC = 1 − P_{1}, CP_{1} = 0 (see Proposition 1.2) and

ACA = (1 − P_{1})( ∑_{j=2} kλ_{j}P_{j} + N) = A − P_{1}( ∑_{j=2}^{k} λ_{j}P_{k} + N) = A − P_{1}N;

hence A = ACA + P_{1}N, P_{1}N is nilpotent and

(ACA)P_{1}N = ACP_{1}AN = 0 = NACP_{1}A = P_{1}N(ACA).

Recall that ACA is the Drazin inverse of C and that i(ACA) = 1. The following more general result holds:

2.16. Theorem. Suppose that T ∈ L(X) is Drazin invertible, i(T) = ν ⩾ 1 and that C is the Drazin inverse of T. Then there is a nilpotent N ∈ L(X) such that T = T CT + N, N(T CT) = (T CT)N = 0 and N^{ν} = 0.

This decomposition is unique in the following sense: if S, N_{1} ∈ L(X), S is Drazin invertible, i(S) = 1, N_{1} is nilpotent, N_{1}S = SN_{1} = 0 and if T = S + N_{1}, then S = T CT and N = N_{1}.

Proof. Let N = T − T CT; then

N^{ν} = (T (1 − CT))^{ν} = T^{ν}(1 − CT)^{ν} = T^{ν}(1 − CT)
= T^{ν} − T^{ν}CT = T^{ν} − T^{ν+1}C = T^{ν} − T^{ν} = 0.

For the uniqueness of the decomposition we only have to show that S = T CT. There is R ∈ L(X) such that SRS = S, RSR = R and SR = RS. Consequently,

N_{1}R = N_{1}RSR = N_{1}SR^{2} = 0 = R^{S}SN_{1} = RN_{1},

hence

T R = (S + N_{1})R = SR = RS = R(S + N_{1}) = RT.

Now let n be a nonnegative integer such that N_{1}^{n} = 0. Since SN_{1} = 0 = N_{1}S, it follows that

T^{n} = (S + N_{1})^{n} = S^{n} + N_{1}^{n} = S^{n}.

We can assume that n ⩾ ν. Thus

T^{n+1}R = S^{n+1}R = S^{n−1}SRS = S^{n} = T^{n}.

Furthermore we have T R = RT and

RT R = R(S + N_{1})R = RSR = R,

hence R = C. With S_{1} = T CT we get

S_{1}RS_{1} = T CT CT CT = T CT = S_{1},
RS_{1}R = CT CT C = CT C = RT R = R
S_{1}R = T CT C = CT CT = RS_{1}.

This shows that S = S_{1} = T CT. □

References

[1] S. R. Caradus, On meromorphic operators, I, Canad. J. Math. 19, 723–736, 1967.

[2] S. R. Caradus, Operator theory of the pseudoinverse, Queen’s Papers Pure Appl. Math. 38, Queen’s University, Kingston, Ontario, 1974.

[3] H. Heuser, Funktionalanalysis, 2nd ed., Teubner, 1986.

[4] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, 2nd ed., Wiley and Sons, 1980.

Institut für Analysis (Received 17 04 2006) Universität Karlsruhe (TH) Kaiserstrasse 89 76133 Karlsruhe Germany

christoph.schmoeger@math.uni-karlsruhe.de