Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 669–686

On Real and Complex Spectra in some Real C^{∗}-Algebras and Applications

V. Didenko and B. Silbermann

Abstract. A real extension Ã of a complex C^{∗}-algebra A by some element m which has a number of special properties is proposed. These properties allow us to introduce some suitable operations of addition, multiplication and involution on Ã. After then we are able to study Moore-Penrose invertibility in Ã. Because this notion strongly depends on the element m, we study under what conditions different elements m produce just the same involution on Ã. It is shown that the set of all additive continuous operators L_{add}(H) acting in a complex Hilbert space H possesses unique involution only (in the sense defined below). In addition, we consider some properties of the real and complex spectra of elements belonging to Ã, and show that whenever an operator sequence {Ã_{n}} ⊂ L_{add}(H) is weakly asymptotically Moore-Penrose invertible, then the real spectrum of Ã_{n}^{∗}Ã_{n} can be split in two special parts. This property has been earlier known for sequences of linear operators.

Keywords: Real C^{∗}-algebras, Moore-Penrose invertibility, singular integral equations with conjugation

AMS subject classification: 65 R 20

1. Introduction

Let A be a real C^{∗}-algebra with identity e. A usual method for introducing the notion of spectrum for elements of A is to consider the spectrum of the images of these elements in a special complex algebra which is called the complexification of A. As a result, the spectra of elements of A are some sets in the complex plane ℂ. One of the reasons for such a definition is to make impossible the situation when the spectrum of an element may turn out to be empty. Another reason for using the complexification of A is closely connected with the previous one. Namely, it is possible that the initial algebra does not have the operation of multiplication by complex scalars. However, there are real algebras which possess this operation. For such algebras the spectra defined by complexification technique may be distinguished from the usual one (see [5] or [12: Remark 1.1.11]).

V. Didenko: Odessa State University, IMEM, ul. Petra Velikogo 2, 270000 Odessa, Ukraine e-mail: didenko@post.imem.odessa.ua; supported, in part, by DFG. B. Silbermann: University of Technology, Faculty of Mathematics, D-09107 Chemnitz e-mail: Bernd.Silbermann@mathematik.tu-chemnitz.de

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

In the present paper we deal with the investigation of Moore-Penrose invertibility in special real C^{∗}-algebras. In complex algebras analogous problems were studied in [9, 10, 11, 15, 16, 18]. It turns out that the problems can be successfully treated by using spectral characteristics of some self-adjoint elements. However, self-adjoint elements of real algebras do not always possess the properties which such elements of complex algebras have. That is why we employ two different notions of the spectrum for the real C^{∗}-algebras under consideration. Namely, we consider a real and a complex spectrum and the interdependency of these two objects. It is shown that the complex spectrum can be expressed by the real parts of the spectra of some elements belonging to an auxiliary complex C^{∗}-algebra.

The notion of Moore-Penrose invertibility used in this work strongly depends on a special element m. It may happen that different elements m lead to different Moore-Penrose inverses. We give a description of a class of elements m such that they all generate just the same Moore-Penrose inverse for each a of our real algebra.

The results mentioned are used to study some problems of Moore-Penrose invertibility for additive operators, acting in a complex Hilbert space H. In particular, we investigate the asymptotic Moore-Penrose invertibility in algebras appearing in spline approximation methods for singular integral equations with conjugation. Note that the stability problem in such algebras was studied in [2, 3, 7, 8, 14]. Finally, let us remark one technical detail. Definitions, examples, and remarks on the one hand and lemmata, theorems, and propositions on the other hand are numbered separately.

2. Algebra Ã and some its properties

Let R be a ring over the real number field ℝ. We assume that R contains a complex C^{∗}- algebra A with identity e and an element m ∉ A, which are connected by the following relations:

(A_{1}) For each a ∈ A, the element mam is in A.

(A_{2}) m^{2} = e and me = m.

(A_{3}) For each λ ∈ ℂ, m(λe) = \overline{λ}m.

(A_{4}) The null element 0 of the C^{∗}-algebra A is also that of the ring R.

(A_{5}) For each a ∈ A, (mam)^{∗} = ma^{∗}m, where ”∗” means an involution on A. In the sequel, the set of all such elements m will be referred to as M(A).

Now, we consider a subset Ã ⊂ R which consists of all elements ã having the form

ã = b + cm  (1)

with b, c ∈ A. It is easily seen that the set Ã is closed with respect to the operations of addition and multiplication. Moreover, one can define the operation of multiplication of elements of A by complex scalars as

λã = λ(b + cm) = (λb) + (λc)m ∀λ ∈ ℂ.

The set Ã with the above operations becomes a real algebra (this is not a complex algebra because, in general, ã(λb̃) ≠ λ(ãb̃)).

Let us note the following result.

Lemma 2.1. Let assumptions (A_{1}) − (A_{5}) hold. Then each element ã of Ã has a unique representation in the form (1).

Proof. Indeed, if an element ã has two different representations ã = b_{1} + c_{1}m and ã = b_{2} + c_{2}m, then we immediately get

(b_{1} − b_{2}) + (c_{1} − c_{2})m = 0.  (2)

Multiplying this equality by ie from the right and by (−i) from the left we get

−(b_{1} − b_{2}) + (c_{1} − c_{2})m = 0.  (3)

Combining (2) with (3) and remembering assumption (A_{4}) we obtain our claim.

As a next step we introduce an involution on Ã. This can be done by using the previous lemma. Namely, for each ã ∈ Ã, ã = b + cm (b, c ∈ A) we put

ã^{⋆} = b^{∗} + mc^{∗}.  (4)

It is a simply matter to show that the so defined involution possesses most of the basic properties we know for involutions on complex C^{∗}-algebras. For instance, one has the following relations:

• For each ã ∈ Ã, the element ã^{⋆} belongs to Ã.

• For each ã ∈ A, (ã^{⋆})^{⋆} = ã.

• For any ã, b̃ ∈ A and α, β ∈ ℂ, (ãb̃)^{⋆} = b̃^{⋆}ã^{⋆} and (αã + βb̃)^{⋆} = ã^{⋆}\overline{α}e + b̃^{⋆}\overline{β}e.

In addition, we can see that m^{⋆} = m and, for each ã ∈ A, ã^{⋆} = ã^{∗}.

Therefore, it will cause no confusion if we will use the same symbol ”∗” for the notation of the involution on the set Ã. We should only turn the attention of the reader to the fact that the above involution depends on the element m. In the sequel, we will study a number of algebras Ã which can be defined by different elements m. In these cases we will use special notations to make precise which involution is meant.

3. Real and complex spectra

Let A be a real C^{∗}-algebra with identity e, and let b ∈ A. Recall the definition of the real spectrum of b.

Definition 3.1. The real spectrum of b in A, denoted by sp_{A}b, is the set of all real numbers λ such that b − λe is not invertible in A.

As it has been pointed out earlier real algebras often possess the operation of multiplication by complex scalars. When this is the case, one may consider the complex spectrum of b in A, too. It is defined analogously, namely as the set of all complex numbers λ such that b − λe is not invertible in A. In the sequel, the complex spectrum of b in A will be denoted by Sp_{A}b.

In this section we are going to describe some relations between the complex and the real spectra for the elements of the algebra Ã introduced earlier. To do this we have to mention some known results concerning real algebras.

Theorem 3.1 (see [13] App. 1). Let b be a self-adjoint element of a unital (real or complex) C^{∗}-algebra A, and let R(b) be the smallest closed real C^{∗}-subalgebra containing the elements b and e. Then R(b) is isometrically isomorphic to the algebra C(sp_{R(b)}b) of all continuous real functions on sp_{R(b)}b.

Lemma 3.2 (see [3]). Let b be a self-adjoint element of a complex C^{∗}-algebra A. Then sp_{R(b)}b = Sp_{A}b (= sp_{A}b).

Lemma 3.3 (see [3]). Let B be a real C^{∗}-subalgebra of a complex C^{∗}-algebra A. Then B is inverse closed in the algebra A, i.e. if an element b ∈ B is invertible in A, then it is also invertible in B.

Now we are in a position to give a description of the complex spectra for elements of Ã by real components of the spectra of some elements belonging to a known complex C^{∗}-algebra.

Let ã ∈ Ã, and let A^{2×2} be the complex C^{∗}-algebra of (2 × 2)-matrices with entries in A. By Ψ, Ψ : Ã → A^{2×2} we denote the transformation

Ψ(ã) = Ψ(b + cm) = ( \begin{matrix}b & c \\ mcm & mbm \end{matrix} )  (5)

where b, c ∈ A.

Theorem 3.4. Let ã = b + cm ∈ Ã. Then

SpÃã = ⋃_{φ∈[0,2π)} {sp_{A^{2×2}}^{+}Ψ(ã_{φ})} e^{iφ}  (6)

where

ã_{φ} = e^{−iφ}b + e^{iφ}cm  (φ ∈ [0, 2π))  (7)

and

sp_{B}^{+}b = {Sp_{B}b} ∩ {ℝ^{+} ∪ 0}.

Proof. Indeed, for each λ = |λ|e^{iφ} we have

ã − λe = (ã_{φ} − |λ|e)e^{iφ}.

Hence, the element ã − λe is invertible in Ã if and only if all elements ã_{φ} − |λ|e for φ ∈ [0, 2π) are so. We consider the subset E2×2 ; Ã of A^{2×2} which consists of all matrices of A^{2×2} having the form (5). It is easily seen that E2×2 ; Ã is a real subalgebra of A^{2×2}. In addition, we can see that an element ã is invertible in Ã if and only if Ψ(ã) is invertible in E2×2 ; Ã. Taking into account the relation

Ψ(ã_{φ} − |λ|e) = Ψ(ã_{φ}) − |λ|E  (8)

where E = ({e \atop 0} {0 \atop e}), we obtain

SpÃã = ⋃_{φ∈[0,2π)} {sp_{E2×2 ; Ã}^{+}Ψ(ã_{φ})}e^{iφ}.

To complete the proof we only need to remember Lemma 3.3.

Corollary 3.5. Let ã ∈ Ã. Then

spÃã = sp_{A^{2×2}}Ψ(ã).  (9)

It should be noted here that such an equality is not always true for the complex spectrum SpÃã of ã (see Example 5.2 below).

We again consider the transformation Ψ : Ã → A^{2×2}.

Lemma 3.6 (cf. [3: Lemma 5]). The transformation Ψ is a ^{∗}-isomorphism between the real algebras Ã and E2×2 ; Ã.

Due to this lemma we can equip the algebra Ã with the norm

||ã||Ã = ||Ψ(ã)||_{A^{2×2}}.

With this norm Ã becomes a real C^{∗}-algebra, i.e. it satisfies the condition

||ã||2 ; Ã = ||ãã^{∗}||Ã  ∀ã ∈ Ã.  (10)

Thus, throughout this paper the algebra Ã is assumed having the norm (10).

4. On the uniqueness of involution in Ã

In Section 2 we defined an involution in the algebra Ã which, generally speaking, depends on the element m of the ring R. Let us explain more precisely what this means.

Suppose we have two different elements m_{1} and m_{2} of R possessing the properties (A_{1}) − (A_{5}) and producing just the same algebra Ã. If ã ∈ Ã, then we do not know whether the corresponding involutions

\begin{aligned}(ã)_{m_{1}}^{∗} & = (b_{1} + c_{1}m_{1})_{m_{1}}^{∗} = b_{1}^{∗} + m_{1}c_{1}^{∗}  (b_{1}, c_{1} ∈ A) \\ (ã)_{m_{2}}^{∗} & = (b_{2} + c_{2}m_{2})_{m_{2}}^{∗} = b_{2}^{∗} + m_{2}c_{2}^{∗}  (b_{2}, c_{2} ∈ A) \end{aligned}

coincide. In this section we give some conditions when the equality (ã)_{m_{1}}^{∗} = (ã)_{m_{2}}^{∗} is fulfilled for each ã of Ã provided that Ã_{m_{1}} = Ã_{m_{2}} (= Ã).

Theorem 4.1. Let Ã_{m_{1}} and Ã_{m_{2}} be the real algebras generated by a complex C^{∗}- algebra A and by elements m_{1} and m_{2}, respectively. Then the following assertions are equivalent:

1. Ã_{m_{1}} = Ã_{m_{2}} (= Ã) and, for each ã ∈ Ã,

(ã)_{m_{1}}^{∗} = (ã)_{m_{2}}^{∗}.  (11)

2. The element m_{1}m_{2} belongs to the algebra A and

(m_{1}m_{2})^{∗} = m_{2}m_{1}.  (12)

Proof. Necessity part: Let the algebra Ã_{m_{1}} coincide with Ã_{m_{2}}, and let they both have just the same involution. Then there exist f, g ∈ A such that m_{1} = f + gm_{2}. Following the proof of Lemma 2.1 we get m_{1} = −f + gm_{2}. Hence, m_{1} = gm_{2}, or

m_{1}m_{2} = g,  (13)

i.e. m_{1}m_{2} ∈ A.

Let us show equality (12). To do this we compute each of the involutions for the element m_{1}. Firstly, we have

(m_{1})_{m_{1}}^{∗} = m_{1}  (14)

and, secondly,

(m_{1})_{m_{2}}^{∗} = (gm_{2})_{m_{2}}^{∗} = m_{2}g^{∗}.  (15)

Comparing (14) with (15) we obtain m_{1} = m_{2}g^{∗}. Multiplying this equality by m_{2} from the left and remembering (13) we get our claim.

Sufficiency part. Let us suppose that the element m_{1}m_{2} belongs to the algebra Ã and satisfies equality (12). First of all, we show that the algebras Ã_{m_{1}} and Ã_{m_{2}} coincide. Let ã ∈ Ã_{m_{1}}. Then there exist b, c ∈ A such that ã = b + cm_{1}. Hence,

ã = b + cm_{1} = b + cm_{1} · m_{2}^{2} = b + c · (m_{1}m_{2})m_{2} = b + c_{1}m_{2},

i.e. ã ∈ Ã_{m_{2}} because b, c_{1} ∈ A. The inclusion Ã_{m_{2}} ⊂ Ã_{m_{1}} is proved analogously. Now we take an element ã of Ã and find its involutions generated by each of the elements m_{1} and m_{2}. We have

(ã)_{m_{1}}^{∗} = (b + cm_{1})_{m_{1}}^{∗} = b^{∗} + m_{1}c^{∗}

and

\begin{aligned}(ã)_{m_{2}}^{∗} & = (b + cm_{1})_{m_{2}}^{∗} = (b + c(m_{1}m_{2})m_{2})_{m_{2}}^{∗} = b^{∗} + m_{2}(c(m_{1}m_{2}))^{∗} \\ & = b^{∗} + m_{2}(m_{1}m_{2})^{∗}c^{∗} = b^{∗} + m_{2}(m_{2}m_{1})c^{∗} = b^{∗} + m_{1}c^{∗}, \end{aligned}

i.e. (ã)_{m_{1}}^{∗} = (ã)_{m_{2}}^{∗}

Corollary 4.2. Let both the elements m_{1}, m_{2} ∈ R satisfy assumptions (A_{1})−(A_{5}). If there exists a unitary element a ∈ A such that m_{2} = m_{1}a, then m_{1} and m_{2} define just the same algebra Ã, and for each ã ∈ Ã one has (ã)_{m_{1}}^{∗} = (ã)_{m_{2}}^{∗}.

Indeed, since m_{1}m_{2} = a ∈ A we only need to prove equality (12). But this is almost apparent, because (m_{1}m_{2})^{∗} = (m_{1}m_{1}a)^{∗} = a^{∗} = a^{−1} = m_{2}m_{1}, which proves the claim.

The following two assertions give us some methods to construct elements of R which would have properties (A_{1}) − (A_{5}) and would produce an algebra Ã with just the same involution.

Corollary 4.3. Let m satisfy assumptions (A_{1}) − (A_{5}), and let q be a unitary element of A such that

q^{∗} = mqm.  (16)

Then:

1. The element m_{q} = mq satisfies all conditions (A_{1}) − (A_{5}).

2. Ã_{m} = Ã_{m_{q}} (= Ã).

3. For each ã ∈ Ã, (ã)_{m}^{∗} = (ã)_{m_{q}}^{∗}.

Proof. In view of the previous corollary we only have to show the first claim of this assertion. Note that the relations (A_{1}), (A_{3}), (A_{4}) are evident, and (A_{2}) immediately follows from (16). Moreover, for the element (m_{q}b m_{q})^{∗} (b ∈ A) we have

\begin{aligned}(m_{q}bm_{q})^{∗} & = (mqbmq)^{∗} = q^{∗}(mqbm)^{∗} = q^{∗}(mb^{∗}q^{∗}m) \\ & = m(mq^{∗}m)b^{∗}m(mq^{∗}m) = mq · b^{∗} · mq = m_{q}b^{∗}m_{q}. \end{aligned}

This yields the proof.

Corollary 4.4. Let m satisfy assumptions (A_{1}) − (A_{5}), and let q ∈ A be a selfadjoint element such that q^{2} = e. Then for the element m_{q} = qmq all assertions of Corollary 4.3 are true.

Proof. The validity of the relations (A_{1}) − (A_{5}) can be proved by straightforward computation. Here we only show the validity of equality (A_{5}). For each b ∈ A we have

(m_{q}bm_{q})^{∗} = (qmqbqmq)^{∗} = q^{∗}m(qbq)^{∗}mq^{∗} = qmqb^{∗}qmq = m_{q}b^{∗}m_{q}.

It remains to prove the second assertion of Theorem 4.1. Indeed, the element mm_{q} = (mqm)q evidently belongs to the algebra A, and

(mm_{q})^{∗} = ((mqm)q)^{∗} = q^{∗}mq^{∗}m = qmqm = m_{q}m.

This finishes the proof

Now we would like to give some examples of complex C^{∗}-algebras and its extensions generated by different elements possessing properties (A_{1}) − (A_{5}).

Example 4.1. Let Γ be the unit circle in the complex plane ℂ, i.e. Γ = {t ∈ ℂ : |t| = 1}. By L_{2} = L_{2}(Γ) we denote the set of all complex-valued Lebesgue-measurable functions φ on Γ such that ∫_{Γ} |φ(t)|^{2}|dt| < ∞. Provided with the scalar product

(φ, ψ) = \frac{1}{2π} ∫_{Γ} φ(t)\overline{ψ(t)} |dt| (φ, ψ ∈ L_{2}(Γ))

this set actually becomes a Hilbert space. By M we denote the operator of complex conjugation in L_{2}(Γ), i.e.

(Mφ)(t) = \overline{φ(t)} ∀φ ∈ L_{2}(Γ).

As is shown in [3], the extension of the C^{∗}-algebra A = L(L_{2}(Γ)) by the operator M leads to the algebra of all additive continuous operators Ã = L_{add}(L_{2}(Γ)) acting on the space L_{2}(Γ). Due to the fact that the operator M satisfies relations (A_{1}) − (A_{5}), we can introduce a ”good” involution on this set.

Just the same involution on L_{add}(L_{2}(Γ)) can be also obtained by using the operator M_{1} = SMS, where S is the singular integral operator of Cauchy, i.e.

(Sφ) = \frac{1}{πi} ∫_{Γ} \frac{φ(τ) dτ}{τ − t} (φ ∈ L_{2}(Γ))

because S^{∗} = S and S^{2} = S (cf. [6], Chapter 1 and Corollary 4.4).

Example 4.2. Let us consider the Hilbert space L_{2}(Γ) again. We take a continuously differentiable function α = α(t) (t ∈ Γ) such that

α^{2}(t) = α(α(t)) = t  ∀t ∈ Γ.  (17)

It is also supposed that

|α′(t)| = 1  ∀t ∈ Γ.  (18)

For example, one can choose the function α(t) = \frac{1}{t} (t ∈ Γ). We denote by W_{α} the operator of the Carleman shift

(W_{α}φ)(t) = φ(α(t)) (φ ∈ L_{2}(Γ), t ∈ Γ)

and by B_{a} the operator of multiplication by a continuous function a, i.e.

(B_{a}φ)(t) = a(t)φ(t) (φ ∈ L_{2}(Γ), t ∈ Γ).

Let A be the smallest closed C^{∗}-subalgebra of the C^{∗}-algebra L(L_{2}(Γ)) containing the operator W_{α} and all operators B_{a} with a ∈ C(Γ). We consider the extensions of this algebra by the elements M and M_{1} = MW_{α}.

It is easily seen that M satisfies all assumptions (A_{1})−(A_{5}). Therefore, we only have to check the conditions of Corollary 4.3 for the operator W_{α}. Indeed, it follows from (17) and (18) that W_{α}^{2} = I and W_{α}^{∗} = B_{|α|}W_{α} = W_{α}. In addition, a straightforward computation gives

MW_{α}M = W_{α} = W_{α}^{∗},

i.e. all conditions of Corollary 4.3 are fulfilled. Therefore, the elements M and M_{1} define just the same algebra Ã and just the same involution on this algebra.

Example 4.3. Let H be a Hilbert space and let L_{add}(H) be the set of all additive continuous operators acting in H. As it was established in [3] the algebra L_{add}(H) can be considered as an extension of the C^{∗}-algebra L(H) by any continuous additive operator M with the following properties:

• M^{2} = I, where I is the identical operator of L(H).

• (Mφ, ψ) = \overline{(φ, Mψ)} for all φ, ψ ∈ H.

The set of all such operators will be denoted by L_{M}(H). Now we are in a position to show that all such operators produce just the same involution on L_{add}(H). First of all, we note that each operator M satisfying the above conditions is an anti-linear one, i.e.

M(λx) = \overline{λ}Mx ∀λ ∈ ℂ and ∀x ∈ H.

Hence, for arbitrary M_{1}, M_{2} ∈ L_{M}(H), the operatorM_{1}M_{2} belongs to L(H). In addition, for any φ, ψ ∈ H one finds

(M_{1}M_{2}φ, ψ) = \overline{(M_{2}φ, M_{1}ψ)} = (φ, M_{2}M_{1}ψ).

Thus (M_{1}M_{2})^{∗} = M_{2}M_{1}, and using Theorem 4.1 finishes the proof.

We see that a wide class of anti-linear operators produces just the same involution on L_{add}(H) (in the sense of definition (4)). Moreover, in the last case it turns out that all possible involutions (4) coincide!

Below, we are going to prove this claim. However, before we have to introduce some additional notations.

Let H_{ℝ} be the same Hilbert space H considered as a real space. As a scalar product on H_{ℝ} we will use the form

⟨x, y⟩ = ⟨x, y⟩_{H_{ℝ}} = Re(x, y)  (x, y ∈ H_{ℝ}).  (19)

For each A ∈ L_{add}(H) we refer to as A_{ℝ}^{∗} such an operator B ∈ L_{add}(H) which satisfies the relation

⟨Ax, y⟩ = ⟨x, By⟩ ∀x, y ∈ H.

We list without proofs some elementary properties of these operators:

1) For each A ∈ L_{add}(H), the operator A_{ℝ}^{∗} exists and is uniquely determined;

2) For any A, B ∈ L_{add}(H), (A + B)_{ℝ}^{∗} = A_{ℝ}^{∗} + B_{ℝ}^{∗} and (AB)_{ℝ}^{∗} = B_{ℝ}^{∗}A_{ℝ}^{∗};

3) For each A ∈ L_{add}(H), (A_{ℝ}^{∗})_{ℝ}^{∗} = A;

4) For each A ∈ L(H), A_{ℝ}^{∗} = A^{∗};

5) For each M ∈ M(L(H)), the operator MM_{ℝ}^{∗} belongs to L(H).

Theorem 4.5. Let H be a complex Hilbert space. Then any operators M_{1}, M_{2} ∈ M(L(H)) produce just the same involution on L_{add}(H).

Proof. First of all, we suppose that dimH > 1 and consider an arbitrary operator M ∈ M(L(H)). Let A ∈ L(H). On account of properties 1) – 4) one can write

(MA^{∗}M)_{ℝ}^{∗} = M_{ℝ}^{∗}AM_{ℝ}^{∗}.  (20)

On the other hand,

(MA^{∗}M)_{ℝ}^{∗} = (MA^{∗}M)^{∗} = MAM.

Therefore, according to condition (A_{2}) we get

(MM_{ℝ}^{∗})A = A(MM_{ℝ}^{∗}) ∀A ∈ L(H).

Since dimH > 1 the algebra L(H) is irreducible. Hence, the operator MM_{ℝ}^{∗} ∈ L(H) is an operator of multiplication by some scalar (cf.,[1], Proposition 2.3.1) i.e. MM_{ℝ}^{∗} = λI for some λ ∈ ℂ. However, the operator MM_{ℝ}^{∗} is self-adjoint. This implies λ ∈ ℝ, and applying axiom (A_{2}) once again we obtain λ = ±1. Thus, there may exist only two situations, M_{ℝ}^{∗} = M or M_{ℝ}^{∗} = −M.

Let us now show that the second case is impossible. Indeed, if we suppose that M_{ℝ}^{∗} = −M, then I + MM_{ℝ}^{∗} = 0. Therefore, for any x ∈ H we have

||x||^{2} ≤ (x, x) + (M_{ℝ}^{∗}x, M_{ℝ}^{∗}x)⟨x, x⟩ + ⟨M_{ℝ}^{∗}x, M_{ℝ}^{∗}x⟩
= ⟨x, x⟩ + ⟨MM_{ℝ}^{∗}x, x⟩
= ⟨(I + MM_{ℝ}^{∗})x, x⟩
≤ ||(I + MM_{ℝ}^{∗})x|| ||x||

(using the Cauchy-Schwarz inequality), whence

||x|| ≤ ||(I + MM_{ℝ}^{∗})x|| = 0 ∀x ∈ H,

which is impossible. Hence, for each M ∈ M(L(H)) we obtain M_{ℝ}^{∗} = M.

It remains to use the necessary and sufficient condition (12). Namely, for any M_{1}, M_{2} ∈ M(L(H)) we can write

(M_{1}M_{2})^{∗} = (M_{1}M_{2})_{ℝ}^{∗} = (M_{2})_{ℝ}^{∗}(M_{1})_{ℝ}^{∗} = M_{2}M_{1}.

Thus if dimH > 1, then all the operators of M(L(H)) produce just the same involution on L_{add}(H). Let us now consider the situation dimH = 1. In this case the algebra L(H) consists of the operator of multiplication by complex scalars only, and it is a simple matter to check that

M(L(H)) = {e^{iφ}M : φ ∈ [0, 2π)}

where M is the operator of complex conjugation. However, all such elements produce just the same involution on L_{add}(H)

Remark 4.4. Theorem 4.5 can be generalized on further complex C^{∗}-algebras. Properties 1) - 5) show what assumptions one needs in order to guarantee that the corresponding results remain true.

All the previous considerations have dealt with the cases when different elements of M(A) produced just the same algebra Ã. However, it may also occur another situation. Now we give an example when different elements m_{1} and m_{2} generate non-coinciding algebras Ã_{m_{1}} and Ã_{m_{1}}, respectively.

Example 4.5. Let a continuous function α satisfy relation (17) (but optionally relation (18)), and let B_{a}, W_{α}, M and M_{1} be the same operators as in Example 4.2. By A we now denote the smallest closed C^{∗}-subalgebra of L(L_{2}(Γ)) containing all operators B_{a}, with function a being in C(Γ). Because

M_{1}B_{a}M_{1} = B\overline{a∘α}  and  (M_{1}B_{a}M_{1})^{∗} = B_{a∘α} = M_{1}B_{a}^{∗}M_{1}

the element M_{1} generates the algebra Ã_{M_{1}}. However, it is easily seen that Ã_{M_{1}} ≠ Ã_{M}.

5. Moore-Penrose invertibility in the algebra Ã

After introducing operations of multiplication and involution on the algebra Ã we may also consider the notion of Moore-Penrose invertibility in this algebra. To do this we suppose that the algebra Ã is generated by a complex C^{∗}-algebra A and by an element m with properties (A_{1}) − (A_{5}).

Definition 5.1. An element ã ∈ Ã is said to be m-Moore-Penrose invertible in the algebra Ã if there exists an element b̃ ∈ Ã such that the relations

ãb̃ã = ã,  b̃ãb̃ = b̃,  (ãb̃)_{m}^{∗} = ãb̃,  (b̃ã)_{m}^{∗} = b̃ã.  (21)

are true.

If such an element b̃ exists, then it is called an m-Moore-Penrose inverse of ã and denoted by ã_{m}^{+}.

Lemma 5.1. Let the elements m_{1} and m_{2} produce just the same algebra Ã. If (m_{1}m_{2})^{∗} = m_{2}m_{1}, then each element ã ∈ Ã is m_{1}-Moore-Penrose invertible if and only if it is m_{2}-Moore-Penrose invertible and if ã_{m_{1}}^{+} = ã_{m_{2}}^{+}.

Proof. If (m_{1}m_{2})^{∗} = m_{2}m_{1}, then the m_{1}-involution on Ã coincides with the m_{2}- involution. It implies the validity of the first assertion of Lemma 5.1. The calculation

ã_{m_{1}}^{+} = ã_{m_{1}}^{+} ãã_{m_{1}}^{+} = ã_{m_{1}}^{+}(ãã_{m_{1}}^{+})_{m_{1}}^{∗} = ã_{m_{1}}^{+}(ã_{m_{1}}^{+})_{m_{1}}^{∗}ã_{m_{2}}^{∗}
= ã_{m_{1}}^{+}(ã_{m_{1}}^{+})_{m_{1}}^{∗}ã_{m_{2}}^{∗}(ã_{m_{2}}^{+})_{m_{2}}^{∗}ã_{m_{2}}^{∗} = ã_{m_{1}}^{+}((ã_{m_{1}}^{+})_{m_{1}}^{∗}ã_{m_{1}}^{∗})(ã_{m_{2}}^{+})_{m_{2}}^{∗}ã_{m_{2}}^{∗}
= ã_{m_{1}}^{+} ãã_{m_{1}}^{+}(ãã_{m_{2}}^{+})_{m_{2}}^{∗} = ã_{m_{1}}^{+} ãã_{m_{2}}^{+} = (ã_{m_{1}}^{+} ã)_{m_{1}}^{∗}ã_{m_{2}}^{+} = ã_{m_{1}}^{∗}(ã_{m_{1}}^{+})_{m_{1}}^{∗}ã_{m_{2}}^{+}
= ã_{m_{2}}^{∗}(ã_{m_{2}}^{+})_{m_{2}}^{∗}ã_{m_{2}}^{∗}(ã_{m_{1}}^{+})_{m_{1}}^{∗}ã_{m_{2}}^{+} = (ã_{m_{2}}^{+} ã)_{m_{2}}^{∗}(ã_{m_{1}}^{+} ã)_{m_{1}}^{∗}ã_{m_{2}}^{+}
= ã_{m_{2}}^{+}(ãã_{m_{1}}^{+} ã)ã_{m_{2}}^{+} = ã_{m_{2}}^{+} ãã_{m_{2}}^{+} = ã_{m_{2}}^{+}

gives the second assertion

Corollary 5.2. Let H be a complex Hilbert space, and let M_{1}, M_{2} ∈ M(L(H)). Then an element A ∈ L_{add}(H) is M_{1}-Moore-Penrose invertible if and only if it is M_{2}- Moore-Penrose invertible and A_{M_{1}}^{+} = A_{M_{2}}^{+} (A ∈ L_{add}(H)).

The proof of this corollary immediately follows from Theorem 4.5.

Therefore, in what follows we will suppose that either the element m is fixed or all the elements of M(A) produce just the same involution, and we will write ã^{∗} and ã^{+} instead of ã_{m}^{∗} and of ã_{m}^{+}, respectively.

As it has been mentioned earlier, the Moore-Penrose invertibility in complex C^{∗}- algebras can be handled by using some spectral characteristics of suitable self-adjoint elements. However, considering a real algebra we may meet situations when the spectra of self-adjoint elements possess unusual properties, for instance, they can contain complex points.

Example 5.2. Let H = ℂ, A be the algebra of all operators of multiplications by a ∈ ℂ, and let M(x) = \overline{x} for all x ∈ ℂ. Then

Ã = {b + cM : b, c ∈ A}.

We consider a self-adjoint element ã of Ã. It is clear that ã = b + cM is self-adjoint if and only if \overline{b} = b. If we again exploit transformation (5), we obtain that Sp_{A}ã is the circle of the radius |c| with the center at the point b, i.e. if c ≠ 0, then the spectrum of ã contains complex points, as well.

In the proposition below there are collected some results we need to continue the investigation of the Moore-Penrose invertibility in the algebra Ã.

Proposition 5.3. Let Ã be generated by an element m and a C^{∗}-algebra A with identity e. If ã ∈ Ã, then the following assertions are equivalent:

1) ã is Moore-Penrose invertible.

2) ã^{∗}ã is invertible or 0 is an isolated point of the real spectrum of ã^{∗}ã.

3) There exists a projection p̃ in RÃ(ã^{∗}ã) such that ã^{∗}ãp̃ = 0 and ã^{∗}ã + p̃ is invertible.

4) There exists a projection q̃ in Ã such that ãq̃ = 0 and ã^{∗}ã + q̃ is invertible.

If one of these conditions is fulfilled, then q̃ is uniquely determined and ã^{+} = (ã^{∗}ã + q̃)^{−1}ã^{∗}.

The proof of this proposition follows from [3: Propositions 7 and 8, Corollaries 11 and 15] and from Corollary 3.5 of the present article.

Following [3] we will say that a C^{∗}-subalgebra B of the C^{∗}-algebra A is m-closed if for each b ∈ B the element mbm is in B again.

Proposition 5.4 (cf. [3: Corollary 13]). Let A be a complex C^{∗}-algebra with identity e and B be an m-closed C^{∗}-subalgebra of A containing e. Then the element b̃ of B̃ is Moore-Penrose invertible in B̃ if and only if it is Moore-Penrose invertible in Ã.

From now on, we are going to restrict ourselves to considering a special real C^{∗}- algebra F̃. Namely, let H be a complex Hilbert space, and let F be the set of all bounded sequences {A_{n}} of bounded linear operators A_{n} acting in H. This set equipped with the norm

||{A_{n}}||_{F} = sup_{n} ||{A_{n}}||

and with the natural operations of addition, multiplication, scalar multiplication and with the involution {A_{n}}^{∗} = {A_{n}^{∗}} becomes a C^{∗}-algebra with identity. Considering the extension of F by any element M with properties (A_{1}) − (A_{5}) we obtain a real algebra F̃ consisting of all the sequences {Ã_{n}} the members of which are additive continuous operators acting on H. Let G̃ be the set of all sequences of F̃ tending to zero in the operator norm. It is easily seen that G̃ is an ideal of F̃. In addition, it is worth noticing that the operator norm on F̃ is equivalent to the norm the set F̃ is equipped with as the extension of F by M (see [3]).

Definition 5.3. A sequence {Ã_{n}} ∈ F̃ is said to be weakly asymptotically Moore-Penrose invertible if there exists a sequence {B̃_{n}} ∈ F̃ such that the sequences

{Ã_{n}B̃_{n}Ã_{n} − Ã_{n}}, {B̃_{n}Ã_{n}B̃_{n} − B̃_{n}}, {(Ã_{n}B̃_{n})^{∗} − Ã_{n}B̃_{n}}, {(B̃_{n}Ã_{n})^{∗} − B̃_{n}Ã_{n}}

belong to the ideal G̃ or, in other words, if the coset {Ã_{n}}+G̃ is Moore-Penrose invertible in the quotient algebra F̃/G̃.

It has been shown in [17] that whenever a sequence {A_{n}} ∈ F is weakly asymptotically Moore-Penrose invertible, then the spectrum Sp(A_{n}^{∗}A_{n}) of A_{n}^{∗}A_{n} can be split in two parts one of which is bounded from zero by a positive constant (independent of n), while the other part tends to zero if n tends to infinity. We now intend to show that this property remains true for elements of the real algebra F̃ as well. However, the spectrum Sp(Ã_{n}^{∗}Ã_{n}) (which coincides with the real spectrum of Ã_{n}^{∗}Ã_{n} in that case) should be replaced by the real spectrum sp(Ã_{n}^{∗}Ã_{n}) of Ã_{n}^{∗}Ã_{n}.

Proposition 5.5. A sequence {Ã_{n}} is weakly asymptotically Moore-Penrose invertible in F̃ if and only if there exist non-negative numbers d and r_{n} with d > 0 and r_{n} → 0 as n → ∞ such that

sp(Ã_{n}^{∗}Ã_{n}) ⊆ [0, r_{n}] ∪ [d, ∞]  (22)

for all sufficiently large n.

To prove this assertion one can exploit the scheme of the proof of the corresponding result for complex C^{∗}-algebras (see [17]). However, one has to be sure in the validity of auxiliary results used in [17]. We are now going to give brief comments for some propositions we need.

In the theory of complex C^{∗}-algebras an ”almost projection” lemma is well-known (see [19: Lemma 5.1.6]). A careful consideration of its proof shows that this lemma remains also true for real C^{∗}-algebras. More precisely, we have the following lemma.

Lemma 5.6. Let B be a real C^{∗}-algebra with identity, and let b ∈ B be a self-adjoint element with ||b^{2} − b|| < \frac{1}{4}. Then there exists a self-adjoint element g ∈ B such that b + g is a projection and ||g|| ≤ 2||b^{2} − b||.

For the proof of this result in the real situation one can use the isometrical isomorphism between the real algebra R(b) and C_{ℝ}(spR(b)) (see Theorem 3.1), if one observes that the corresponding proof in the complex situation deals with real functions only.

Let {Ã_{n}} be in F̃. By {Ã_{n}}^{∘} we denote the coset of F̃/G̃ which contains the sequence {Ã_{n}}, i.e. {Ã_{n}}^{∘} := {Ã_{n}} + G̃. Suppose that {Ã_{n}} is weakly asymptotically Moore-Penrose invertible. Due to Proposition 5.4 there is a sequence {P̃_{n}} ∈ F̃ such that ({P̃_{n}}^{∘})^{2} = {P̃_{n}}^{∘} and ({P̃_{n}}^{∘})^{∗} = {P̃_{n}}^{∘}. It is possible, therefore, to pick up a sequence {P̃_{n}} ∈ {P̃_{n}}^{∘} such that P̃_{n}^{∗} = P̃_{n} and ||P̃_{n}^{2} − P̃_{n}|| → 0 as n → ∞. Together with Lemma 5.6 this implies that one can choose a sequence {Π̃_{n}} ∈ {P̃_{n}}^{∘} such that each member of it is a projection, i.e. Π̃_{n}^{2} = Π̃_{n} and Π̃_{n}^{∗} = Π̃_{n} for all n ∈ ℕ.

Combining this result with Proposition 5.3 we obtain

Lemma 5.7. A sequence {Ã_{n}} ∈ F̃ is weakly asymptotically Moore-Penrose invertible if and only if there exists a sequence {Π̃_{n}} of projections on H such that the sequence {Ã_{n}^{∗}Ã_{n} + Π̃_{n}} is stable and ||Ã_{n}Π̃_{n}|| → 0 as n → ∞. The sequence {Π̃_{n}} is unique modulo G̃ and {(Ã_{n}^{∗}Ã_{n} + Π̃_{n})^{−1}A_{n}^{∗}}^{∘} = {{Ã_{n}}^{+}}^{∘}.

It should be noted that in place of q̃ in assertion 4) of Proposition 5.3 the element ã^{⊓} = e − ã^{+}ã can be chosen. According to this designation we refer to as Π{Ã_{n}} the set of all projections in ({Ã_{n}}^{∘})^{⊓}.

Lemma 5.8. Let {Ã_{n}} ∈ F̃ be weakly asymptotically Moore-Penrose invertible. Then there is a sequence {Π̃_{n}} ∈ Π{Ã_{n}} such that Π̃_{n} ∈ R_{L_{add}(H)}(Ã_{n}^{∗}Ã_{n}) for all n ∈ ℕ. The sequence {Π̃_{n}} is unique in the following sense: If {Π̃_{n}^{(1)}}, {Π̃_{n}^{(2)}} ∈ R_{L_{add}(H)}(Ã_{n}^{∗}Ã_{n}) for all n > n_{0}, then {Π̃_{n}^{(1)}} = {Π̃_{n}^{(2)}} for all sufficiently large n.

The proof of this lemma also follows the corresponding proof of Theorem 3 in [17], and the only additional result we need here is the Gelfand-Naimark theorem for commutative real C^{∗}-algebras. However, this can be found in [5].

Proof of Proposition 5.5. After having proved Lemmas 5.7 and 5.8 we can use Theorem 3.1 to obtain relation (22). However, as is shown before, in real algebras the spectral properties of self-adjoint elements can be different from those in complex algebras. Therefore, we first show that the real spectrum of Ã_{n}^{∗}Ã_{n} does not contain negative points. Indeed, as it follows from Corollary 3.5,

sp_{L_{add}(H)}(Ã_{n}^{∗}Ã_{n}) = sp_{L^{2×2}(H)}(Ψ(Ã_{n})^{∗}Ψ(Ã_{n})).

However, the self-adjoint operator Ψ(Ã_{n})^{∗}Ψ(Ã_{n}) does not have any negative points in its spectrum since it belongs to the complex C^{∗}-algebra L^{2×2}(H). Hence

sp_{L_{add}(H)}(Ã_{n}^{∗}Ã_{n}) ⊂ [0, +∞].

Let {Ã_{n}} be weakly asymptotically Moore-Penrose invertible. Due to Lemmas 5.7 and 5.8 we can take a sequence of projections {Π̃_{n}} such that {Ã_{n}} ∈ R(Ã_{n}^{∗}Ã_{n}) (n ∈ ℕ) and ||Ã_{n}^{∗}Ã_{n}Π̃_{n}|| → 0 as n → ∞ and the sequence {Ã_{n}^{∗}Ã_{n} + Π̃_{n}} is stable. Then there exists a number d > 0 such that

||(\{Ã_{n}^{∗}Ã_{n} + Π̃_{n}\}^{∘})^{−1}||_{F̃/G} < \frac{1}{d},

and we put

r_{n} := ||Ã_{n}^{∗}Ã_{n}Π̃_{n}||.  (23)

Using the definition of the norm in F̃/G̃ we can choose a number n_{0} such that

||\{Ã_{n}^{∗}Ã_{n} + Π̃_{n}\}^{−1}||_{L_{add}(H)} < \frac{1}{d}  (n > n_{0}).  (24)

Now we fix some n > n_{0}. Due to the isometrical isomorphism between R := R(Ã_{n}^{∗}Ã_{n}) and C(sp_{R}(Ã_{n}^{∗}Ã_{n})) (see Theorem 3.1) we can identify the element Ã_{n}^{∗}Ã_{n} with the function x → x, and the projection Π̃_{n} with the function x → p_{n}(x), where p_{n} takes the values 0 and 1 only. Then it follows from (23) and (24) that

x + p_{n}(x) > d  and  xp_{n}(x) < r_{n}  (x ∈ sp_{R}(Ã_{n}^{∗}Ã_{n})).

Analyzing the last two inequalities we obtain our claim.

Now we consider the notion of asymptotical Moore-Penrose invertibility for sequences of F̃. We recall that a sequence {Ã_{n}} ∈ F̃ is said to be asymptotically Moore-Penrose invertible if there is an n_{0} such that the operators Ã_{n} are Moore-Penrose invertible for all n > n_{0} and if sup_{n>n_{0}} ||Ã_{n}^{+}|| < +∞.

Theorem 5.9. A sequence {Ã_{n}} ∈ F̃ is weakly asymptotically Moore-Penrose invertible if and only if it can be represented as a sum of an asymptotically Moore-Penrose invertible sequence and a sequence of G̃.

Proof. Necessity part: Let {Ã_{n}} ∈ F̃ be weakly asymptotically Moore-Penrose invertible, and let {Π̃_{n}} be the projection sequence defined in the proof of Proposition 5.5. By {B̃_{n}} we denote the operator

B̃_{n} = Ã_{n}(I − Π̃_{n}).  (25)

Then, as it follows from the proof of Proposition 5.5, the real spectrum of B̃_{n}^{∗}B̃_{n} + Π̃_{n} is contained in the set {1} ∪ [d, +∞). Therefore, the operator B̃_{n} is Moore-Penrose invertible, B̃_{n}^{+} = (B̃_{n}^{∗}B̃_{n} + Π̃_{n})^{−1}B̃_{n}^{∗} and ∥B̃_{n}^{+}∥ ≤ 1/\sqrt{d}. Henceforth, the sequence {B̃_{n}} is asymptotically Moore-Penrose invertible. In addition, we have ∥Ã_{n}Π̃_{n}∥ → 0 as n → ∞ (cf. Lemma 28). Finally, from (25) we obtain

Ã_{n} = B̃_{n} + G̃_{n}  with ∥G̃_{n}∥ → 0 as n → ∞.  (26)

The proof of the sufficiency is evident.

The above results are of practical interest and can be applied to the Moore-Penrose regularization of approximation methods for equations with additive operators. Below we consider this problem for spline approximation methods for singular integral equations with conjugation.

In the space L_{2}(Γ) (see Example 4.1) we consider the equation

\begin{matrix}(Ax)(t) & ≡ a(t)x(t) + \frac{b(t)}{πi} ∫_{Γ} \frac{x(τ)dτ}{τ − t} + c(t)\overline{x(t)} + \frac{d(t)}{πi} ∫_{Γ} \frac{\overline{x(τ)}dτ}{τ − t} \\ & + ∫_{Γ} k_{1}(t, τ)x(τ) dτ + ∫_{Γ} k_{2}(t, τ)\overline{x(τ)} dτ \\ & = g(t) \end{matrix}  (t ∈ Γ) (27)

where a, b, c, d are supposed to be piecewise continuous functions on Γ and k_{1}, k_{2} ∈ C(Γ × Γ). As was shown in [2] the investigation of spline-approximation methods for equation (27) can be reduced to studying an algebra of operator sequences. This algebra is generated by a special sequence of anti-linear operators {M_{n}}, by sequences tending to zero in the operator norm as well as by diagonal sequences ϕ_{n}^{ε} and by sequences of circulants ψ̂_{n} formed by given transformations of piecewise continuous functions ϕ and ψ, respectively. More precisely, the sequences which appear in spline approximation methods for the equation (27) can be represented in the form

{Ã_{n}} = {a_{n}^{ε}α̂_{n} + b_{n}^{ε}β̂_{n} + (c_{n}^{ε}γ̂_{n} + d_{n}^{ε}θ̂_{n})M_{n} + G_{n}}, ||G_{n}|| → 0 as n → ∞ (28)

where a_{n}^{ε}, b_{n}^{ε}, c_{n}^{ε}, d_{n}^{ε} are related with the coefficients a, b, c, d of (27) whereas α̂_{n}, β̂_{n}, γ̂_{n}, θ̂_{n} reflect the method used. For the form of these terms we refer the reader to [2]. In [3] the sequence (28) was studied with respect to the weak asymptotic Moore-Penrose invertibility. Now we are able to say something more about the asymptotic behaviour of this sequence. However, we need additional notations.

Thus, let P and Q stand for the operators \frac{1}{2}(I +S) and \frac{1}{2}(I −S), respectively, where S was defined in Example 4.1, and let a, b, c, d and α, β, γ, θ be piecewise continuous functions. We form the matrices

\begin{aligned}A(t) & = ( \begin{matrix}a(t) & 0 \\ 0 & \overline{a(t)} \end{matrix} ) \\ D(t) & = ( \begin{matrix}d(t) & 0 \\ 0 & \overline{d(t)} \end{matrix} ) \\ γ^{⋆}(t) & = ( \begin{matrix}γ(t) & 0 \\ 0 & \overline{γ(\overline{t})} \end{matrix} ) \end{aligned}  \begin{aligned}B(t) & = ( \begin{matrix}b(t) & 0 \\ 0 & \overline{b(t)} \end{matrix} ) \\ α^{⋆}(t) & = ( \begin{matrix}α(t) & 0 \\ 0 & \overline{α(\overline{t})} \end{matrix} ) \\ θ^{⋆}(t) & = ( \begin{matrix}θ(t) & 0 \\ 0 & \overline{θ(\overline{t})} \end{matrix} ) \end{aligned}  \begin{aligned}C(t) & = ( \begin{matrix}c(t) & 0 \\ 0 & \overline{c(t)} \end{matrix} ) \\ β^{⋆}(t) & = ( \begin{matrix}β(t) & 0 \\ 0 & \overline{β(\overline{t})} \end{matrix} ) \\ Λ & = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) . \end{aligned}

Corollary 5.10. Let a, b, c, d ∈ P C(Γ) as well as α, β, γ, θ ∈ P C(Γ). The sequence (28) can be represented in the form (26) if and only if all the operators W_{u}, W^{u} ∈ L(L_{2}^{2}(Γ)) (u ∈ Γ),

\begin{aligned}W^{u} & = [A(t)α^{⋆}(u + 0) + B(t)β^{⋆}(u + 0) + (C(t)γ^{⋆}(u + 0) + D(t)θ^{⋆}(u + 0))Λ]P \\ & + [A(t)α^{⋆}(u − 0) + B(t)β^{⋆}(u − 0) + (C(t)γ^{⋆}(u − 0) + D(t)θ^{⋆}(u − 0))Λ]Q \end{aligned}

On Real and Complex Spectra in Real Algebras 685

and

\begin{aligned}W_{u} & = P [A(u + 0)α^{⋆}(\overline{t}) + B(u + 0)β^{⋆}(\overline{t}) + (C(u + 0)γ^{⋆}(\overline{t}) + D(u + 0)θ^{⋆}(\overline{t}))Λ] \\ & + Q[A(u − 0)α^{⋆}(\overline{t}) + B(u − 0)β^{⋆}(\overline{t}) + (C(u − 0)γ^{⋆}(\overline{t}) + D(u − 0)θ^{⋆}(\overline{t}))Λ] \end{aligned}

are normally solvable, and the norms of their Moore-Penrose inverses are uniformly bounded with respect to u ∈ Γ.

The proof of this corollary follows from Theorem 5.9 and from [3: Theorem 25]. Note that the Fredholmness of the operators W_{u} and W^{u} is sufficient for the uniform boundedness of their Moore-Penrose inverses (cf. [15]).

Remark 5.4. In order to regularize the sequence {Ã_{n}} of (28) practically, we have to consider the matrix sequence ({Ψ(Ã_{n})},

Ψ(Ã_{n}) = ( \begin{matrix}a_{n}^{ε}α̂_{n} + b_{n}^{ε}β̂_{n} & c_{n}^{ε}γ̂_{n} + d_{n}^{ε}θ̂_{n} \\ M_{n}(c_{n}^{ε}γ̂_{n} + d_{n}^{ε}θ̂_{n})M_{n} & M_{n}(a_{n}^{ε}α̂_{n} + b_{n}^{ε}β̂_{n})M_{n} \end{matrix} ) .

These matrices can be regularized (in the sense of (26)) using their singular value decompositions (see [17]). After then we can apply the mapping Ψ^{−1} to regularized matrices in order to get the corresponding regularization for the initial sequence (28).

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Received 24.08.1998