convolution type operators on cones and their finite sections

Math. Nachr. 278, No. 3, 290 – 311 (2005) / DOI 10.1002/mana.200310241

Convolution type operators on cones and their finite sections

Helena Mascarenhas∗1 and Bernd Silbermann∗∗1

1 Technische Universitat Chemnitz, Fakultat fur Mathematik, 09107 Chemnitz, Germany

Received 29 July 2002, revised 6 April 2003, accepted 22 April 2003Published online 10 January 2005

Key words Convolution operators, singular values, finite sectionMSC (2000) Primary: 45E10; Secondary: 47A05, 65F20

This paper is concerned with finite sections of convolution type operators defined on cones, whose symbol isthe Fourier transform of an integrable function on R

2. The algebra of these finite sections satisfies a set of ax-ioms (standard model) that ensures some asymptotic properties like the convergence of the condition numbers,singular values, ε-pseudospectrum and also gives a relation between the singular values of an approximationsequence and the kernel dimensions of a set of associated operators. This approach furnishes a method todetermine whether a Fredholm convolution operator on a cone is invertible.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Let a ∈ L1(R2)

and C(a) be the usual convolution operator

C(a) : L2(R2) −→ L2

(R2)

g −→∫

R2a(t− s)g(s) ds . (1.1)

Given a closed cone K ⊂ R2 with vertex at the origin, CK(a) denotes the following operator on the conedefined in L2

(R2)

by

CK(a) = χK(C(a) + λI)χKI + (1 − χK) I ,

where χK is the characteristic function ofK and λ ∈ C. We will say that CK(a) is a convolution operator on thecone K although, strictly speaking, it is a natural extension to L2

(R2)

of the convolution operator on K definedin [14].

The Fredholm and invertibility properties of these operators were studied by several authors, see for instance[14], [5], [11], [1] and the notes and comments in [3], Ch. 9.

Simonenko showed in [14] that CK(a) is Fredholm if and only if the symbol F (a) + λ, where F is thefamiliar Fourier transform, does not vanish in R2 and λ is nonzero. Later, Douglas and Howe also proved that inthis case the index is zero [5]. What can be said about invertibility? If CK(a) is Fredholm and the kernel functiona satisfies some conditions then CK(a) is invertible (see [11], [1]), but whether this result remains true for anya in L1

(R2)

is not known until now (see [3]). However, we will show that the kernel dimension of a Fredholmoperator CK(a) can be computed with the help of numerical analysis.

In this paper we are not only interested in this question, but also in asymptotic properties of finite sectionsof convolution type operators on cones. Kozak studied these finite sections and used for the first time Banachalgebras techniques to establish sufficient and necessary conditions for the stability of the approximation se-quences [9]. We shall prove that the approximation sequences contain much more information. We are especiallyconcerned with properties like convergence of singular values, norms of inverses, ε-pseudospectrum, k-splittingproperty and others.

∗ e-mail: [email protected], Phone: +351 218419639, Fax: +351 218417598∗∗ Corresponding author: e-mail: [email protected], Phone: +49 371 531 4128, Fax: +49 371 531 4131


Math. Nachr. 278, No. 3 (2005) / www.mn-journal.com 291

Using a local principle and limit operators techniques, we show that a certain C*-algebra of approximationsequences containing the finite sections of convolution type operators on cones is a standard model in the senseof [7]. In particular, we recover Kozak’s results. If an algebra is standard, then the asymptotics mentioned aboveare the consequence of a general theory and in particular there is a relation between the singular values of anapproximation sequence and the kernel dimensions of a set of associated operators. It turns out that when weconsider a particular approximation sequence An of a Fredholm operator CK(a), this relation tells us that thesingular values of An split and the number of singular values of An on some interval [0, εn] for n large enoughis equal to the kernel dimension of CK(a). Since the index of CK(a) is zero, if we compute the singular values,for example by the help of methods of numerical analysis for large matrices, we can decide whether CK(a) isinvertible.

In the course of the paper, we will need some known results (published only in russian) which we present herewith full proofs, for the convenience of the reader.

We shall deal with algebras with non-trivial center. For these algebras the Allan-Douglas principle is a pow-erful tool to describe them, so we begin by stating it ([8], Ch. 1). Let A be a Banach unital algebra and C bea closed subalgebra of the center of A which contains the identity. C is a commutative algebra and by MC wedenote the space of its maximal ideals. To each maximal ideal x in MC we associate the smallest closed idealJx of A which contains x and we denote by Φx the quotient projection from A onto the quotient algebra A/Jx

(also called local algebra).

Theorem 1.1 (Allan-Douglas principle) a) An element A ∈ A is invertible if and only if the cosets Φx(A)are invertible in A/Jx for every x ∈ MC .

b) If Φx(A) is invertible in A/Jx, then Φy(A) is invertible in A/Jy for all y in some open neighborhood of x.

2 An algebra of convolution operators on cones

We begin by studying a C∗-algebra that contains convolution operators on cones and operators of multiplicationby continuous functions on R2. We shall describe necessary and sufficient conditions for an operator in thisalgebra to be Fredholm.

A continuous function f in R2 can be viewed as a function of a complex variable, and given t > 0, we defineft as the function

ft : S1 −→ C

eiθ −→ f(teiθ).

We say that f ∈ C(R2)

if the functions ft converge uniformly on S1 as t tends to infinity. With pointwiseoperations and supremum norm C(R2

)is a C∗-algebra. The multiplicative linear functionals in C(R2

)are the

evaluation functionals f → f(x) for each x ∈ R2 and the functionals f → limt→∞ f(teiθ) for every eiθ ∈ S1.So, the maximal ideal space of C(R2

)is the compactification of R2 (equipped with the usual topology) with

the “infinite” circle and we will denote it by R2 . In other words, if we consider the following homeomorphismdefined on the open ball B1(0),

ξ : B1(0) −→ R2

x −→ x

1 − |x| ,(2.1)

and its unique continuous extension ξ to the closed ball B1(0),

ξ : B1(0) −→ R2 , (2.2)

then R2 is exactly the image of ξ.From now on we denote by θ∞ the points in R2 \R

2 given by ξ(eiθ), with θ ∈ [0, 2π[ and we say that a

sequence hn ∈ R2 converges to θ∞ if ξ−1(hn) converges to eiθ. A basic neighborhood of θ∞ is an open set of


292 Mascarenhas and Silbermann: Convolution operators

the type

WR,ε (θ∞) =|x| eiθ′

: |x| > R, and x|x| = eiθ′

, θ′ ∈ ]θ − ε, θ + ε[

⋃ θ′∞ : θ′ ∈ ]θ − ε, θ + ε[ .

(2.3)

We also denote f(θ∞) = limt→∞ f(teiθ).We say that K ⊂ R2 is a cone at the origin if there exist θ1, θ2 ∈ [0, 2π[ with |θ1 − θ2| = π, such that

K = H1 ∩H2 , (2.4)

where for j = 1, 2 each

Hj =z ∈ C : Im (e−iθz) ≥ 0

(2.5)

is a rotation of the usual upper half-space by the angle θj in the positive direction.The algebra we are interested in is the smallest C∗-algebra A of L(L2

(R2))

that contains the convolutionoperators C(a) in (1.1), the multiplication operators by functions in C(R2

), χH1I and χH2I. By Aπ we denote

the image of A in the Calkin algebra by the quotient projection π : L(L2(R2)) → L(L2

(R2))/K, where K is

the ideal of compact operators of L(L2(R2)). In order to show that Aπ has a non-trivial center, we introduce the

following lemma.

Lemma 2.1 Let F1 and F2 be two closed disjoint subsets of R2. Then there exist δ > 0 such that |x−y| > Rδfor all R > 0, every x ∈ F1 ∩ R2\BR(0) and every y ∈ F2 ∩ R2\BR(0).

P r o o f. Let F1 = ξ−1(F1) and F2 = ξ−1(F2). Since ξ is a homeomorphism from B1 (0) onto R2, thesets F1 and F2 are closed and disjoint. Hence δ = dist

(F1, F2

)> 0. Let R ≥ 0, x ∈ F1 ∩ R

2\BR(0) and

y ∈ F2 ∩ R2\BR(0). Then∣∣ξ−1(x) − ξ−1(y)

∣∣ > δ, which can be written as∣∣∣∣ x

1 + |x| −y

1 + |y|∣∣∣∣ > δ .

Assuming without loss of generality that |x| ≤ |y|, it follows by a simple argument of Euclidean geometry that∣∣∣∣ x

1 + |x| −y

1 + |x|∣∣∣∣ > δ

which leads to

|x− y| > (1 + |x|) δ > Rδ ,

finishing the proof.

Proposition 2.2 ([14]) If f ∈ C(R2)

then C(a)fI − fC(a) is a compact operator.

P r o o f. The operator C(a)fI − fC(a) is compact if and only if given two arbitrary closed disjoint setsF1, F2 ⊂ R2, χF1C(a)χF2I is compact (see [2] Cor. 5.8). Note that the space considered in [2] is L2 (X) withX a compact set, but L2

(R2)

is indeed isomorphic to L2(R2, µ), where µ is the extended Lebesgue measure

such that µ (M) = 0 if M ⊂ R2 \R2. For g in L2(R2), we have:

(χF1C(a)χF2g) (s) =∫

R2χF1 (s) a(s− t)χF2 (t) g(t) dt .

Functions in L1(R2)

can be approximated by continuous functions with compact support, thus we can as-sume that a is a continuous function with support contained in the centered ball of radius M. Define a(s, t) =χF1(s)a(s − t)χF2(t). By the previous lemma, there exist δ > 0 and R > M such that |s − t| > Rδ > M,for every s ∈ F1 ∩ R2\BR(0) and t ∈ F2 ∩ R2\BR(0). Thus, if |s| > 2R or |t| > 2R then a(s, t) = 0.Since a has compact support and is bounded, we have

∫R2

∫R2 |a(s, t)|2 ds dt < ∞. Therefore, χF1C(a)χF2I is

a Hilbert-Schmidt operator and so it is compact.



Clearly, fI commutes with χH1I and χH2I, so the last proposition tells us that C(R2)

is isomorphic to aC∗-subalgebra of the center of Aπ, with maximal ideal space isomorphic to R2. So we can apply Allan-Douglasprinciple (Thm. 1.1), considering the projections Φx : Aπ → Aπ/Jx , with x running through R2. The nextproposition shows that the local algebras Aπ/Jx are very simple for x ∈ R2.

Proposition 2.3 Let x ∈ R2 and f ∈ C(R2).

a) Φx(fI) = Φx(f(x)I), Φx (C(a)) = 0 and

Φx(χHjI) =

⎧⎪⎪⎨⎪⎪⎩Φx(I) , if x ∈ int Hj ,

Φx (0) , if x /∈ Hj ,

Φx(χHj I), if x ∈ ∂Hj ,

where ∂Hj is the boundary of Hj , j = 1, 2.b) The local algebras Aπ/Jx are isomorphic to Ck (with pointwise multiplication), with k depending on the

position of x, and k ∈ 1, 2, 4.

P r o o f. a) For the local elements Φx(fI) and Φx(χHj I) the result is straightforward. Let f be a contin-uous function with compact support and f(x) = 1. We start by showing that the operator C(a)fI defined onL2(R2)

by,

[C(a)fI(g)] (x) =∫

R2a(x− t)f(t)g(t) dt , a ∈ L1

(R

2),

is a compact operator. We can suppose that a is a continuous function with compact support because thisset of functions is dense in L1

(R2). If we define a(x, t) = a(x − t)f(t), then a is a continuous function

with compact support, which means that C(a)fI is a Hilbert-Schmidt operator and therefore compact. Thus,Φx(C(a)fI) = 0. But since Φx(fI) = f(x)I = I, we have Φx(C(a)) = 0.

b) If x /∈ ∂H1 and x /∈ ∂H2, then from assertion a), the algebra Aπ/Jx is generated by the identity andthus is isomorphic to C. If x ∈ ∂H1\0 or x ∈ ∂H2\0, then Aπ/Jx is respectively generated by Φx(χH1I)or Φx(χH2I). Since Φx(χHj I) is idempotent its spectrum is contained in 0, 1, and in fact it equals the set0, 1. Suppose that the spectrum is 1; if Φx(χHj I) is invertible then due to Theorem 1.1, b), there existsa neighborhood of x such that Φy(χHj I) is invertible for every y in this neighborhood. If we choose y in theexterior of Hj , we have Φy(χHjI) = 0 and so it is not invertible. Using the same argument, we can showthat the spectrum cannot be 0. Since it is not empty it has to be 0, 1 and so Aπ/Jx is isomorphic toC (σ (Φx(χHj I)

)) C2. Let suppose now that x = 0. If θ1 = θ2 then Aπ/Jx C2. Assume that θ1 = θ2.The algebra Aπ/Jx is generated by two idempotents, Φx(χH1I) and Φx(χH2I) that commute with each other,and thus it is a linear space of dimension less or equal to four. Since by definition |θ1 − θ2| = π, it can beshown by simple computations that Φx (I), Φx(χH1I), Φx(χH2I) and Φx (χH1χH2I) are linearly independent.Therefore the local algebra at zero is isomorphic to C4, where the product in C4, given by the Gelfand transform,is the pointwise product.

If x ∈ R2 \R2 then the local algebras are more involved and in order to describe them, we will introducesome results about limit operators. Let k ∈ R2 and denote by Vk the shift operator Vk : L2

(R2) → L2

(R2),

g(t) → g(t− k).

Definition 2.4 Let h = (hn)n∈Nbe a sequence in R

2 converging to some θ∞ ∈ R2 \R2, andA an operator in

L(L2(R2))

. If the strong limit V−hnAVhn exists when n tends to infinity, then we say thatA has a limit operatorwith respect to h = (hn)n∈N

and we denote it by Ah = s-limV−hnAVhn .

Proposition 2.5 ([12]) Let A,B,Am ∈ L(L2(R2))

and ‖Am − A‖ → 0 when m tends to +∞. Let h =(hn)n∈N

be a sequence converging to some θ∞ ∈ R2 \R2.i) If Ah, , Bh exist then (AB)h and (A+B)h also exist and we have

(AB)h = AhBh , (A+B)h = Ah +Bh .

ii) If Amh exists for all m large enough, then Ah exists and Ah = limAm

h .iii) If Ah exists then ‖Ah‖ ≤ ‖A‖.



Let us analyze the limit operators of the generators of our algebra A and of the compact operators with respectto some sequences. If we choose a general sequence hn → θ∞ and consider the limit operators of C(a), fI andT ∈ K, we conclude that they always exist and are the same, independently of how hn converges to θ∞. But forthe operator χHj I the situation is different. If, for example, θ = θj the existence and form of the limit operatorof χHjI depends strongly on how the sequence hn converges to θ∞ (see [13], Prop. 6c). However, if we alsoimpose that hn/|hn| = eiθj , then (χHj I)h = χHj I as we will show. So, if hn is a sequence converging to θ∞and hn/|hn| = eiθ , the limit operators depend only on θ and so we will denote this limit operator by Aθ .

Proposition 2.6 Let θ∞ ∈ R2 \R2 and h = (hn)n∈Nbe a sequence in R2 such that hn/|hn| = eiθ and

hn → θ∞. Then:

i) (C(a))h = (C(a))θ = C(a).

ii) (fI)h = (fI)θ = f(θ∞)I .

iii) (χHjI)h = (χHj I)θ =

⎧⎪⎪⎨⎪⎪⎩χHj I , if Im (ei(θ−θj)) = 0 ,

I , if Im (ei(θ−θj)) > 0 ,

0 , if Im (ei(θ−θj)) < 0 .

j = 1 , 2 .

iv) (T )h = (T )θ = 0 for all T ∈ K.

P r o o f. i) By definition, (V−hnC(a)Vhng) (x) =∫

R2 a(x + hn − t)g(t − hn) dt =∫

R2 a(x − s)g(s) ds =(C(a)g)(x) where a ∈ L1

(R2). So C(a) is shift invariant and thus (C(a))h = C(a).

ii) Let g be a function in L2(R2)

whose support is the ball BM (0) for some M > 0. We have by definition,(V−hnfVhng)(x) = f(x + hn)g(x). We know that f is continuous in θ∞, which means given ε > 0 there is aneighborhood WR,δ (θ∞) as in (2.3) such that |f (x) − f (θ∞) | < ε for every x in WR,δ (θ∞) . For n ≥ p andp ∈ N, BM (0) + hn is contained in WR,δ (θ∞) , and so for n ≥ p

‖(f(x+ hn) − f (θ∞))g‖ ≤ supx∈BM (0)

|f(x+ hn) − f(θ∞)| ‖g‖L2 ≤ ε ‖g‖L2 .

Since the functions in L2(R2)

with compact support are dense in L2(R2), we get

‖(V−hnfVhn − f (θ∞))g‖L2 −→ 0 for every g ∈ L2(R

2).

iii) Suppose Im(ei(θ−θj)

)> 0. By definition,(

V−hnχHjVhng)(x) = χHj (x+ hn)g(x) .

Once again we consider g inL2(R2)

with compact support. We have for n large enough (χHj (x+hn)−1)g(x) =0, so we get s-limV−hnχHjVhn = I.

If Im(ei(θ−θj)

)< 0 then the proof is analogous. If θ = θj or |θ − θj | = π then V−hnχHjVhn = χHjI,

because hn/|hn| = eiθj .iv) Let g and f be functions with compact support in R2. Then for n large enough

∫R2 g(x − hn)f(x) dx

is zero, which implies that Vhn converges weakly to zero. Since Vhn

w 0 and compact operators take weakly

convergent sequences into strongly convergent sequences, TVhn goes strongly to zero and so V−hnTVhn

s→ 0for every compact operator T .

Given θ ∈ [0, 2π[ and hn ∈ R2 such that hn → θ∞ and hn/|hn| = eiθ, by Propositions 2.5 and 2.6, the stronglimit of V−hnAVhn exists and is unique for every A ∈ A, so we can define for each θ and A ∈ A , the operator

Aθ = s- limV−hnAVhn .

The following proposition shows that the local algebras at the “infinite” points are isomorphic to algebras ofoperators.



Proposition 2.7 Let θ∞ ∈ R2 \R2 and Aπθ = Φθ(Aπ). The map

symθ : Aπθ −→ L (L2

(R2))

Aπ + Jθ −→ Aθ

is well-defined and it is an isometric isomorphism onto its image.

P r o o f. Suppose θ = θ1, and let B be the following C∗-subalgebra of AB = alg(C(a), χH1I, λI, where λ ∈ C .

From the fact that (Aθ)∗ = (A∗)θ for every A ∈ A and from Proposition 2.5 it follows that,

Sθ : A −→ L (L2(R2))

A −→ Aθ(2.6)

is a *-homomorphism and by Proposition 2.6 i), iii), B = Bθ for every B ∈ B, which means that the restrictionof Sθ to B is the identity. We claim that

A = B Kθ ,

where denotes the direct sum and Kθ is the smallest closed ideal in A that contains the compact operators inA and fI, with f ∈ C(R2

)and f (θ∞) = 0. If A ∈ Kθ then by Proposition 2.6 iv) Aθ = 0 and if A ∈ B then

Aθ = A, so B ∩ Kθ = 0. The elements of the type

A =n1∑i=1

BifiI +n2∑

i=n1

BifiχH2I +n3∑

i=n2

χH2BifiI + T , (2.7)

with Bi ∈ B, fi ∈ C(R2)

and T a compact operator, form a dense set in A. The operator∑n

i=1 BifiI can bewritten as B + J, where

B =n∑

i=1

Bifi(θ∞)I and J =n∑

i=1

Bi(fi − fi(θ∞))I

are elements of B and Kθ respectively. The projection χH2I is of the form

χH2I = (χH2I)θ + χH2I − (χH2I)θ

where (χH2I)θ is I or 0 and χH2I − (χH2I)θ ∈ Kθ. Since Kθ is an ideal we can conclude that every elementof the form (2.7) is equal to some B + J , with B ∈ B and J ∈ Kθ. Thus B Kθ is dense in A. But B is aC∗-subalgebra and Kθ is a closed two sided ideal so its sum is a C∗-algebra ([4], 1.8.4), and we get the claim.

Due to Sθ (Kθ) = 0 and π (Kθ) = Jθ, the application symθ is a well defined ∗-homomorphism and Aπ =BπJ θ. Thus, every coset in Aπ can be uniquely written as Bπ + Jπ, where Bπ ∈ Bπ and Jπ ∈ Jθ . Thus, ifsymθ (Aπ + Jθ) = symθ (Bπ + Jθ) = Bθ is zero then Aπ + Jθ = 0 and so symθ is injective. Since symθ isan isomorphism between the C∗-algebras Aπ

θ and B, it is isometric.In the other cases the proof is analogous; just note that if θ = θj and |θ − θj| = π then we would have

A = B Kθ , with B = algC(a), λI, λ ∈ C, and χHj I ∈ Kθ for j = 1, 2.

Theorem 2.8 Let A = alg C(a), fI, χH1I, χH2I, where f ∈ C(R2). Then A ∈ A is Fredholm if and

only if Aθ = symθ (A) is invertible for every θ ∈ [0, 2π[ and Φx(A) is invertible in Ck (with pointwise productand k = 1, 2 or 4) for every x ∈ R2.

P r o o f. It follows from Allan-Douglas principle and Propositions 2.3 and 2.7.

Corollary 2.9 ([6] and [14]) a) The operator χH1(C(a) + fI)χH1I + (1− χH1)I is Fredholm if and only if

1. f(x) = 0, for all x ∈ H1,



2. C(a) + f(θ∞)I is invertible for every θ such that Im(ei(θ−θ1)

) ≥ 0.b) The operator χK(C(a) + fI)χKI + (1 − χK) I is Fredholm if and only if

1. f (x) = 0, for all x ∈ K,

2. C(a) + f(θ∞)I is invertible for every θ such that Im(ei(θ−θ1)

) ≥ 0 and Im(ei(θ−θ2)

) ≥ 0.

P r o o f. a) From Proposition 2.6 iii), Theorem 2.8 and the fact that χH1(C(a) + λI)χH1I + (1 − χH1)I,with λ ∈ C, is invertible if and only if C(a) + λI is invertible (see [6], Thm. 1) we get the claim.

b) The proof follows from a), just note that χK = χH1χH2 and so (χKI)θ = (χH1I)θ(χH2I)θ.

Corollary 2.10 Let A ∈ alg C(a), λI, χH1I, then A is Fredholm if and only if A is invertible.

P r o o f. From Proposition 2.6 we have Aθ1 = A for every A in the algebra A and from Theorem 2.8, we getthat if A is Fredholm then A = Aθ1 is invertible.

Note that if f (x) = λ, with λ ∈ C, we have the following equivalent statements:i) χK (C (a) + λI)χKI + (1 − χK) I is Fredholm.

ii) χHj (C(a) + λI)χHj I + (1 − χHj )I is invertible, for all j ∈ 1, 2 .iii) F(a)(x) + λ = 0, for all x ∈ R2 and λ = 0.

3 Finite sections of convolution operators on cones

3.1 The algebra of finite sections

Let U be the set of sequences of operators (An)n∈Nsuch that An ∈ L (L2

(R2))

and s- limV−nxAnVnx ands- limV−nxA

∗nVnx exist for every x ∈ R2. Considering the norm ‖(An)‖ = supn∈N

‖An‖ and the usual point-wise operations, U is a C∗-algebra.

Given ϕ∈C(R2), let (ϕnI) denote the sequence of expanded functions ϕn(t)=ϕ

(tn

)and B be the following

C∗-subalgebra of U ,

B =(An) ∈ U : lim ‖Anϕn − ϕnAn‖ = 0, for all ϕ ∈ C(R2

).

Given a subset U of R2, nU is the set

nU = nt : t ∈ U and χnU (t) = χU

(tn

). (3.1)

Let us now define a set Ω ⊂ R2 which will help us of to define the finite sections. We say Kx is a cone withvertex x if it is a set of the type K + x = x+ t : t ∈ K where K is a cone at the origin (see (2.4)). Let Ω bea closed bounded set of R2 containing the origin, otherwise it would not be interesting because the strong limitof χnΩI would be 0. We assume that for each point x ∈ ∂Ω there is a cone Kx with vertex x, neighborhoodsUand V of x and a C1-diffeomorphism : U → V such that

(x) = x , ′ (x) = I , and (U ∩ Ω) = V ∩Kx .

Note that this cone is uniquely defined for each x ∈ ∂Ω. We also require that if 0 ∈ ∂Ω, its associated is theidentity.

Given a coneKx, we denote byK0x = Kx − x, the translated cone to the origin, i.e. the only one that satisfies

χK0xI = V−xχKxVx . (3.2)

Proposition 3.1 Let T be a compact operator, x ∈ R2, ϕ ∈ C(R2) and f ∈ C(R2).

i) s-limV−nx (C (a) + fI)Vnx = C (a) + f (θ∞) I if x = 0, where θ satisfies eiθ = x|x| .

ii) s-limV−nxTVnx = 0 if x = 0.

iii) s- limV−nxχnΩVnx =

⎧⎪⎪⎨⎪⎪⎩χK0

xI if x ∈ ∂Ω ,

I if x ∈ intΩ ,

0 if x ∈ extΩ .

iv) s- lim V−nxϕnVnx = ϕ (x) I.



P r o o f. i) and ii) are obvious from Proposition 2.6 if we consider hn = nx.iii) Let us first consider x = 0. If 0 ∈ intΩ or 0 ∈ extΩ then clearly s- limχnΩI is the identity or zero,

respectively. We will show that if 0 ∈ ∂Ω then s- lim χnΩI = χK0I. By the definition of Ω, we know that thereis a neighborhood U of zero and a C1-diffeomorphism : U → U such that U ∩ Ω = U ∩K0. If U c = R2\Uthen,

χnΩ = χn(U∩Ω) + χn(Uc∩Ω).

Functions in L2(R

2)

can be approximated by functions with compact support, so let g be a function of that kind.We have for n large enough χn(Uc∩Ω)g = 0 , i.e. ‖χn(Uc∩Ω)g‖L2 → 0 and so s- limχn(Uc∩Ω) = 0. Sinceχn(U∩Ω) = χn(U∩K0), we get s- limχn(U∩Ω)I = χK0I.

Now, let x = 0 and Ωx = y − x, y ∈ Ω (i.e. χΩxI = V−xχΩVx). We have χnΩxI = V−nxχnΩVnx, andconsequently if x ∈ intΩ (resp. x ∈ extΩ) then 0 ∈ intΩx (resp. 0 ∈ extΩx), and we get s- lim V−nxχnΩVnx =I (resp. s- limV−nxχnΩVnx = 0). If x ∈ ∂Ω then 0 ∈ ∂Ωx, but in this case there exists a C1-diffeomorphism : U → V between two neighborhoods of zero such that (0) = 0, ′ (0) = I and (U ∩ Ωx) = V ∩K0

x. Toshow that the strong limit of χnΩxI is χK0

xI it is enough to prove that the norm

∥∥(χK0x− χnΩx

)g∥∥

Lp convergesto zero for every g ∈ Lp

(R2)

with compact support. For n large enough, the support of g is contained in nU∩nVand ∫

R2|χK0

xg (t) − χnΩxg (t) |p dt =

∣∣∣∣∫R2

|χK0xg (t) |p dt−

∫R2

|χnΩxg (t) |pdt∣∣∣∣

=

∣∣∣∣∣∫

K0x∩nV

|g (t) |p dt−∫

n(Ωx∩U)

|g (t) |p dt∣∣∣∣∣

≤∣∣∣∣∣∫

K0x∩nV

|g (t) |p(1 − ∣∣J−1(

tn

)∣∣) dt∣∣∣∣∣+

∣∣∣∣∣∫

K0x∩nV

(|g(t)|p − ∣∣g(n−1

(tn

))∣∣p) ∣∣J−1(

tn

)∣∣ dt∣∣∣∣∣ ,where J denotes the Jacobian. Since −1 (0) = I, it is easy to check that the last two integrals converge to zerowhen n tends to infinity.

iv) As in ii) it is enough to show ‖(V−nxϕnVnx − ϕ(x)I)g‖L2 → 0 for every continuous function g withcompact support. Let ϕn,nx be the function defined by ϕn,nx(t) = ϕ

(tn + x

).

From the continuity of ϕ, we deduce that for ε > 0 and n large enough we have

supt∈supp g

|ϕn,nx(t) − ϕ(x)| ≤ ε .

Since V−nxϕnVnx = ϕn,nxI, it follows that ‖(V−nxϕnVnx − ϕ(x)I)g‖L2 ≤ ε ‖g‖L2 and we get the claim.

Proposition 3.2 The following sequences belong to B :i) (C(a))n∈N.

ii) VnxTV−nx, where T is a compact operator.iii) χnKxI , where χnKx (t) is defined in (3.1).iv) χnΩI and ϕnI.

P r o o f. Due to the previous proposition we just need to prove that these sequences commute with (ϕnI) upto a sequence tending in the norm to zero, for every ϕ ∈ C(R2

).

i) We shall show that, given ε > 0 there exists n large enough such that

‖ϕnC(a) − C(a)ϕnI‖ < ε .

By definition,

(ϕnC(a) − C(a)ϕnI) g =∫

R2a(x− t)

(ϕ(

xn

)− ϕ(

tn

))g(t) dt ,



where g is in L2(R2). Functions in L1

(R2)

can be approximated by functions with compact support, so weassume that the support of the function a is contained in the ball BR (0) of radius R. If |x − t| > R thena (x− t) = 0. For |x− t| ≤ R we want to show that, given ε > 0 there is n0 ∈ N such that for all n ≥ n0∣∣ϕ( x

n

)− ϕ(

tn

)∣∣ ≤ ε

‖a‖ .

The idea is to use the uniform continuity on B1(0) of the function ψ = ϕ ζ, where ζ is the homeomorphism(2.2), and the relation of the distance in B1(0) and the distance in R2 given by ζ. Let u, s ∈ B1(0) and suppose|u| ≤ |s|. Then,

|u− s| ≤∣∣∣∣ u

1 − |u| −s

1 − |u|∣∣∣∣ and

∣∣∣∣ u

1 − |u|∣∣∣∣ ≤

∣∣∣∣ s

1 − |u|∣∣∣∣ ≤

∣∣∣∣ s

1 − |s|∣∣∣∣ ,

and by a simple argument of euclidean geometry we get,∣∣∣∣ u

1 − |u| −s

1 − |u|∣∣∣∣ ≤

∣∣∣∣ u

1 − |u| −s

1 − |s|∣∣∣∣ . (3.3)

If |u| > |s| then (3.3) can be shown in the same way. Thus,

|u− s| ≤ ∣∣ζ(u) − ζ(s)∣∣ (3.4)

for every u and s in B1 (0) .If |x− t| ≤ R, then given δ > 0 and n large enough,

∣∣ xn − t

n

∣∣ < δ and from (3.4) we have,∣∣∣ζ−1(

xn

)− ζ−1(

tn

)∣∣∣ ≤∣∣∣∣xn − t

n

∣∣∣∣ < δ .

From the uniform continuity of ψ it follows that given ε > 0 there is n0 such that,∣∣∣ψ (ζ−1(

xn

))− ψ(ζ−1

(tn

))∣∣∣ =∣∣ϕ( x

n

)− ϕ(

tn

)∣∣ < ε

‖a‖for every n ≥ n0. Thus,

‖(ϕnC(a) − C(a)ϕnI)g‖ ≤ ε

‖a‖ ‖C(a)g‖ ≤ ε ‖g‖ .

ii) The operators Vnx and V−nx are isometries, thus

‖VnxTV−nxϕnI − ϕnVnxTV−nx‖ = ‖TV−nxϕnVnx − V−nxϕnVnxT ‖ .

Let ψI = V−xϕVx, then ψnI = V−nxϕnVnx, and we have,

‖TψnI − ψnT ‖ ≤ ‖T (ψn − ψ(0))I‖ + ‖(ψn − ψ(0))T ‖ .

Since (ψn − ψ(0))I s→ 0 and(ψn − ψ(0)

)I

s→ 0 then ‖T (ψn − ψ(0))I‖ → 0 and ‖(ψn − ψ(0))T ‖ → 0.iii) Let us see that s- limV−nyχnKxVny exists for every y ∈ R2. If y = x then s- lim V−nxχnKxVnx = χK0

xI

because V−nxχnKxVnx = χK0xI, where K0

x is the cone defined in (3.2). If y = x then V−nyχnKxVny =V−n(y−x)χK0

xVn(y−x). Since there are two half-spaces H1 and H2 whose boundary passes through the origin

such that χK0x

= χH1χH2 , using Proposition 2.5 i) and Proposition 2.6 iii), with hn = n (y − x) and eiθ = y−x|y−x| ,

we get that s- limV−n(y−x)χnK0xVn(y−x) always exist.

Clearly χnKxI commutes with ϕnI for every ϕ ∈ C(R2).

iv) It is immediate.



By G we denote the closed ideal of sequences in B tending to zero with respect to the operator norm andq : B → B/G denotes the quotient projection. From now on, if A is a subalgebra of B, we define Aq =q (A) . It is easy to see that C =

(ϕnI) : ϕ ∈ C(R2

)is a C∗-subalgebra of the center of Bq isomorphic to

C(R2), and as we have seen the maximal ideal space of C(R2

)can be identified with R

2 ∪ S1. So, we canlocalize Bq via the Allan-Douglas principle (Thm. 1.1). To each x ∈ R2 we associate the maximal ideal in C,Ix = (ϕn) I + G : ϕ (x) = 0 and to each eiθ ∈ S1 the ideal Iθ = (ϕnI) + G : ϕ (θ∞) = 0 . By Jx wedenote the smallest closed ideal in Bq generated by Ix, with x ∈ R2. The following proposition characterizesthese ideals.

Proposition 3.3 Let x ∈ R2. The coset (An) + G belongs to Jx if and only if for every ε > 0 there isϕ ∈ C(R2

), depending on ε, with compact support and ϕ (x) = 1 such that for n large enough

‖AnϕnI‖L(L2) < ε .

P r o o f. If (An) + G ∈ Jx then (An) + G can be approximated by elements of the type∑m

k=1Bknψ

knI + G

where(Bk

n

) ∈ B and ψk ∈ C(R2)

and ψk (x) = 0. Let M =∑m

k=1 ‖Bkn‖B and ϕ ∈ C(R2

)be such that

ϕ (x) = 1 and ‖ψkϕ‖∞ < εM for every k = 1, . . . ,m. Then∥∥∥∥∥

m∑k=1

Bknψ

knϕnI

∥∥∥∥∥B

≤m∑

k=1

∥∥Bkn

∥∥B∥∥ψkϕI

∥∥∞ ≤ M

ε

M= ε .

It is easy to see that the result is also true not only for the elements in a dense set of Jx but also for everyelement in Jx. Note also that the result does not depend on the representation of the coset (An) + G.

On the other hand, for ε > 0 there is ϕ ∈ C(R2)

with compact support and ϕ (x) = 1 such that for n largeenough, ‖AnϕnI‖ < ε, i.e. ‖AnϕnI + G‖Bq < ε. Then we can construct a sequence ϕk

n ∈ C(R2)

such thatϕk(x) = 1 and∥∥An −An

(1 − ϕk

n

)I + G∥∥Bq =

∥∥AnϕknI + G∥∥Bq < εk ,

where εk → 0. Since (1−ϕkn) = (1−ϕk)n and (1−ϕk(x)) = 0, the coset An(1−ϕk

n)I +G belongs to Jx forevery k ∈ N and so An +G is the limit of a sequence in Jx, thus An +G ∈ Jx, because Jx is a closed ideal.

3.2 Local algebras

Let Bq/Jx be the quotient algebra and φx : Bq → Bq/Jx the canonical projection, with x in R2. Let us describesome local elements.

Proposition 3.4 Let x ∈ R2, y ∈ R2, f ∈ C(R2

)and T a compact operator.

i) φx(VnyTV−ny) = 0 if x = y.

ii) φx(fI) = f(θ∞)φx(I) if x ∈ R2 \ 0.

iii) φx(χnΩI) =

0 if x ∈ extΩ ∪ R2 \R2 .

1 if x ∈ intΩ ,

where θ satisfies eiθ = x|x| if x ∈ R2\ 0 and x coincides with θ∞ if x ∈ R2 \R2.

P r o o f. i) If x = y then there is ϕ ∈ C(R2)

such that ϕ (x) = 1 and ϕ (y) = 0. The operators V−ny andVny are isometries, so we obtain ‖VnyTV−nyϕnI‖ = ‖TV−nyϕnVny‖. Since s- limV−nyϕnVny = ϕ(y)I = 0(Proposition 3.1 iv)), and T is a compact operator we get ‖TV−nyϕnVny‖ → 0. Thus, from Proposition 3.3,VnyTV−ny + G belongs to Jx.

ii) Suppose first that x ∈ R2\ 0 and eiθ = x|x| . Recall that f(θ∞) = limt→∞ f

(teiθ). Since f belongs

to C(R2)

given ε > 0 there is a neighborhood WR,δ (θ∞) of θ∞ as in (2.3) such that for all t ∈ WR,δ(θ∞) :



|f(t) − f(θ∞)| < ε. If we consider ϕ ∈ C(R2)

whose support is a ball centered on x that does not contain zero,ϕ(x) = 1 and ‖ϕ‖ ≤ 1, then for n large enough, supp ϕn ⊂WR,δ (θ∞) and so we get:

‖(f − f(θ∞))ϕnI‖ < ε .

Thus, from Proposition 3.3, the coset (f − f(θ∞))I + G belongs to Jx.The proof for x = θ∞ ∈ R2 \R2 is analogous. If we choose ϕ with support contained in WR,δ(θ∞),

ϕ(θ∞) = 1 and ‖ϕ‖ ≤ 1, then supp ϕn ⊂ WR,ε(θ∞) for all n ≥ 1 and so ‖(f − f(θ∞))ϕnI‖ < ε, forevery n ≥ 1.

iii) It is immediate from Proposition 3.3.

Let us now consider the finite sections of the operators A in A0 = algC(a), f I, where f ∈ C(R2)

andC(a) is the convolution operator (1.1), this means the smallest closed subalgebra of B containing the sequencesχnΩAχnΩI + (1 − χnΩ) I and the ideal G, which we denote by

E = algχnΩAχnΩI + (1 − χnΩ) I, G ,

with G in G.We know that the stability of a sequence (An) in E is given by the invertibility in the quotient algebra Eq, and

(An) + G is invertible in Eq if and only if φx(An) is invertible for every x ∈ R2. From the last proposition, weconclude that the main problem occurs at the boundary points x ∈ ∂Ω. Our aim now is to show that the algebrasEq and Fq

x = algχnKxAχnKxI + (1 − χnKx)I + G/G, where χnKx is defined in (3.1) and A ∈ A0, arelocally isomorphic at x, i.e. φx(Eq) φx(Fq

x) for every x ∈ ∂Ω.Recall that to each x ∈ ∂Ω we associated aC1-diffeomorphism : U → V,whereU and V are neighborhoods

of x and in case 0 ∈ ∂Ω, ≡ I. Thus, if 0 ∈ ∂Ω then χnΩχnU = χnK0χnU and so φx (χnΩ) = φx (χnK0) andthe algebras φx(Eq) and φx(Fq

x) are the same. So, from now on x is a fixed point in ∂Ω\0.By LnV (R2) and LnU (R2) we denote the Hilbert spaces g ∈ L2(R2) : g (x) = 0 if x /∈ nV and

g ∈ L2(R2) : g(x) = 0 if x /∈ nU respectively, and by Rn the following operator acting on them:

Rn : LnV (R2) −→ LnU (R2)

(Rng)(x) =

g(n(

xn

))if x ∈ nU ,

0 if x /∈ nU .

Rn is a bounded linear operator and has an inverse given by,

R−1n : LnU (R2) −→ LnV (R2)(

R−1n g

)(x) =

g(n−1

(xn

))if x ∈ nV ,

0 if x /∈ nV .

Consider now the following subalgebras of B,

BU = (χnUBnχnUI) , (Bn) ∈ B and BV = (χnVBnχnV I) , (Bn) ∈ B ,

and the map,

H : BU −→ BV

(χnUBnχnUI) −→ (R−1

n χnUBnχnURnχnV I).

Proposition 3.5 a) H is an isometric isomorphism.b) If W ⊂ U then H (χnW I) = χn(W )I.

c) The image of BU ∩ G by H is BV ∩ G.



P r o o f. a) It is easy to check that H : BV → BU which associates to each (χnV BnχnV I) the element(RnχnVBnR

−1n χnUI

)is the inverse homomorphism ofH. So, the mapH is a C∗-isomorphism and therefore

it is also isometric.b) Let g be a function in L2

(R2)

and W ⊂ U. Then we have

H(χnW )g = R−1n χnUχnWRnχnV g = R−1

n χnW g(n(

xn

))= χn(W )g .

c) It is immediate.

Let BqU and Bq

V be the analogous subalgebras of Bq and H′ : Bq

U → BqV be the map analogous to H.

Lemma 3.6 H′ is a well defined isomorphism and satisfies

H′ (Bq

U ∩ Jx) = BqV ∩ Jx .

P r o o f. From the last proposition a) and c) we see that H′ is a well defined isomorphism. Suppose now

that, (χnUBnχnUI) + G ∈ Jx. Note that Proposition 3.3 is also true when we substitute ϕ by the characteristicfunction χW of a neighborhood W of x and consider (χnW I) instead of (ϕnI) . So, given ε > 0 there is aneighborhoodW ⊂ U of x such that for n large enough

‖χnUBnχnUχnW I‖ < ε .

By Proposition 3.5, H(χnUBnχnUχnW I) = H(χnUBnχnUI)χn(W )I and on the other hand we have‖H(χnUBnχnUχnW I)‖ = ‖R−1

n χnUBnχnUχnWRnχnV I‖. Since ‖Rn‖ and ‖R−1n ‖ are bounded by

supt∈V |J−1 (t) | and supt∈U |J (t) | respectively, where J denotes the Jacobian, we get for n large enough

‖H(χnUBnχnUI)χn(W )I‖ < Mε ,

where M is some constant. Since (W ) is a neighborhood of x it follows that H (χnUBnχnUI)χn(W )I + Gbelongs to Jx. Using the same argument for the inverse isomorphism we get the claim.

Every coset (Bn) + Jx in Bq/Jx is the same coset as (χnUBnχnUI) + Jx so we can define a map betweenthe local algebras by:

h : Bq/Jx −→ Bq/Jx

(χnUBnχnUI + G) + Jx −→ H ′ (χnUBnχnUI) + Jx .

(3.5)

Taking into account the previous lemma, h is well defined and is an isometric isomorphism. Note that h doesnot have to be the identity. For example hρ(χn(U∩Ω)I) is equal to the coset (χn(V ∩Kx)I + G) + Jx that is ingeneral different from the coset (χn(U∩Ω)I + G) + Jx. However, if (Bn) is the constant sequence (A), withA ∈ A0 we will see that h(χnUAχnUI) equals (χnV AχnV I + G) + Jx. The proof is based on the idea ofquasi-equivalence of two elements [9].

Proposition 3.7 Let x ∈ ∂Ω\0. Then:a) h(χnUC(a)χnUI) = (χnV C(a)χnV I + G) + Jx.

b) h maps the algebra φx(Eq) isomorphically onto φx(Fqx).

P r o o f. a) We need to show φx

(R−1

n χnUC(a)χnURnχnV I − χnVC(a)χnV I)

= 0, i.e, for every ε > 0there is a neighborhoodW of x such that for n large enough

‖χnW

(R−1

n χnUC(a)χnURnχnV I − χnV C(a)χnV I)χnW I‖ < ε . (3.6)

Since functions in L1(R

2)

can be approximated by infinitely differentiable functions with compact support, weassume that there exist R > 0 such that a(y) = 0 for |y| > R and we set M = max (‖a‖∞ , ‖a′‖∞) . Assume



without loss of generality that U and V are bounded convex neighborhoods of x satisfying,

sups∈V

∣∣∣∣J−1 (s)∣∣− 1

∣∣ < ε

2M,

sups∈V

∣∣∣(−1)′

(s) − I∣∣∣ < ε

2RM,

sups∈U

|′(s) − I| < 1 .

(3.7)

Let us denote by Qn the sequence of operators (3.6) and consider W ⊂ V . We will show that for everyg ∈ Lp(R2), ‖Qng‖ ≤ ε ‖g‖, where

Qn(g) =

⎧⎪⎨⎪⎩∫

n−1(W )

a(n−1

(yn

)− s)g(n(

sn

))ds−

∫nW

a(y − t)g(t) dt , if y ∈ nW ,

0 , if y /∈ nW .

Employing the substitution s = n−1(

tn

), we see that Qn has a kernel function given by

qn(y, t) =

[a(n−1

(yn

)− n−1(

tn

)) ∣∣J−1(

tn

)∣∣− a(y − t)]

if (y, t) ∈ WR ,

0 if (y, t) /∈ WR ,

where WR =(y, t) ∈ nW × nW : min

(∣∣n−1(

yn

)− n−1(

tn

)∣∣ , |y − t|) ≤ R.

We estimate

|qn(y, t)| ≤ M∣∣ ∣∣J−1

(tn

)∣∣− 1∣∣+ ∣∣a(n−1

(yn

)− n−1(

tn

))− a(y − t)∣∣

≤ ε

2+M

∣∣n−1(

yn

)− n−1(

tn

)− (y − t)∣∣

≤ ε

2+Mn sup

s∈V

∣∣∣(−1)′

(s) − I∣∣∣ ∣∣∣∣ yn − t

n

∣∣∣∣≤ ε

2+M

ε

4RM|y − t| .

For (y, t) ∈WR, we have |y − t| ≤ R or∣∣n−1

(yn

)− n−1(

tn

)∣∣ ≤ R. If the last inequality is satisfied then

|y − t| = n∣∣ (−1

(yn

))− (−1

(tn

))− (−1(

yn

)− −1(

tn

))∣∣+ ∣∣n−1(

yn

)− n−1(

tn

)∣∣≤ n sup

s∈U|′(s) − I| ∣∣(−1

(yn

)− −1(

tn

))∣∣+ ∣∣n−1(

yn

)− n−1(

tn

)∣∣≤ 2R .

Therefore, |qn(y, t)| ≤ ε which completes the proof.

b) From Proposition 3.5 b) we know that H (χnΩI) = χnKxI and so h(φx(χnΩI)) = φx(χnKxI). Thecoset φx(fI) is the same as the constant coset f(θ∞)φx(I), thus h (φx(fI)) = φx(fI). These results togetherwith a) prove that h maps the generators of φx(Eq) onto the generators of φx(Fq

x) and since h is an isometricisomorphism we get the assertion.

Let x ∈ R2 and Wx be the following map,

Wx : B −→ L(L2(R2))

(An) −→ s- limV−nxAnVnx .(3.8)

By the definition of B, the limits s- limV−nxAnVnx and s- limV−nxA∗nVnx exist and it is easy to check, us-

ing properties of the strong limit analogous to those in Proposition 2.5, that Wx is a *-homomorphism with‖Wx‖ ≤ 1.



Clearly Wx maps G onto zero, so we can define the analogous homomorphism W ′x : Bq → L(L2

(R2))

onthe quotient algebra Bq. From Proposition 3.1 iv) we can conclude that W ′

x also maps Jx onto zero because Jx

is generated by the sequences (ϕnI) with ϕ in C(R2)

and ϕ (x) = 0, thus

wx : Bq/Jx −→ L (L2(R2))

((An) + G)+J x −→ s- limV−nxAnVnx ,

is a well defined homomorphism.

Proposition 3.8 Let 0 ∈ ∂Ω∪0 andK0x be the cone defined in (3.2). The local algebraφx(Eq) is isomorphic

to the C∗-algebra of operators

L = algχK0

xAχK0

xI +

(1 − χK0

x

)I,

with A ∈ algC (a) , λI if x ∈ ∂Ω\ 0 and with A ∈ A0 if 0 ∈ ∂Ω.

P r o o f. Let us first consider x ∈ ∂Ω\ 0. From Proposition 3.7 b) it is enough to prove that the restric-tion of wx to φx(Fq

x) is an isomorphism onto L. The generators of φx(Fqx) are of the type (χnKxAχnKxI +

(1 − χnKx)I + G) + Jx with A ∈ algC (a) , λI and can be written as:

V−nx(χK0xAχK0

xI + (1 − χK0

xI))V−nx + G + Jx

because A is a shift invariant operator and χnKxI = VnxχK0xV−nx. So every element in φx(Fq

x) is of the type(V−nxLV−nx + G) + Jx, with L ∈ L, hence the restriction of wx to φx(Fq

x) is a one to one *-homomorphismand thus an isomorphism onto L.

If 0 ∈ ∂Ω, then φx(Fqx) is generated by the constant sequences

(χK0AχK0I + (1 − χK0)I + G) + Jx

with A ∈ A0 and so the restriction of wx to φx(Fqx) is an isomorphism.

3.3 Examples

We give some examples about stability of a sequence. The obtained conditions are based on Allan-Douglasprinciple (see also Theorem 4.2 below).

Let A, Aij be an operators in A0 = algC (a) , fI with f ∈ C(R2).

1) Let 0 ∈ ∂Ω. The sequence∑m

i=1 Πlj=1 [χnΩAijχnΩI + (1 − χnΩ) I ] is stable if and only if the following

conditions are satisfied:i)∑m

i=1 Πlj=1 [χK0AijχK0I + (1 − χK0)I ] is invertible;

ii)∑m

i=1 Πlj=1

[χK0

x(Aij)θ χK0

xI +

(1 − χK0

x

)I]

is invertible for every x ∈ ∂Ω\ 0, where (Aij)θ is thelimit operator (2.6) and eiθ = x

|x| .

2) Let Ω be the disk BR(0) and A = C(a) + fI . Then χnΩAχnΩI + (1 − χnΩ)I is stable if and only if A isinvertible.

3) Let 0 ∈ Ω and ∂Ω be a smooth set except at zero and Ω ⊂ K0 and A = C (a) + fI . ThenχnΩAχnΩI + (1 − χnΩ) I is stable if and only if χK0AχK0I + (1 − χK0)I is invertible.

4 Standard model. Singular values

If an algebra is a standard model in the sense of ([7], Section 6.1.1), then many asymptotic properties are au-tomatically fulfilled like the convergence of singular values, ε-pseudospectrum and, in particular, the splittingproperty of the singular values. The algebra E considered in the previous section is not a standard model, sowe will consider a larger one here. Let E be the smallest C∗-algebra of B that contains E and the sequences(VnxTV−nx) for every x ∈ ∂Ω∪0 and T ∈ K. We will show that the extended algebra E is a standard algebrafor the family Wx defined in (3.8) with x ∈ ∂Ω ∪ 0; this means that (An) is stable if and only if Wx(An) isinvertible for every x ∈ ∂Ω ∪ 0.



Note that due to Proposition 3.1 i), Wx satisfies the following required condition to be a standard algebra,

Wx(VnxTV−nx) =

T if x = y ,

0 if x = y .

Let Fx be the extension of the algebra Fx

Fx = algχnKxAχnKxI + (1 − χnKx)I, VnxTV−nx, G ,where A ∈ A0, and x ∈ ∂Ω\0.

Lemma 4.1 The algebras φx

(Eq)

and φx

(Fqx

)are isomorphic.

P r o o f. Taking into account the proof of Proposition 3.8 it is easy to check that the restriction of wx toφx

(Fqx

), which we will denote by wx, is an isomorphism onto the operator algebra

L = algχK0

xAχK0

xI + (1 − χK0

x)I, T

,

with A ∈ alg C (a) , λ ∈ C and T ∈ K. The algebra L coincides with the image of wx restricted to φx

(Eq),

so ϑx : φx

(Eq) → φx

(Fqx

), given by ϑx = w−1

x wx is a well defined *-homomorphism. In order to show that

ϑx is one to one, we first prove that the set Kx = φx(VnxTV−nx) : T ∈ K is a closed ideal of φx

(Eq). Clearly

Kx is a linear space. To see that it is an ideal it is enough to consider the product of any element of Kx by anygenerator of φx

(Eq). Let A1 ∈ alg C (a) , λ ∈ C , it is easy to check that:

(χnΩA1χnΩI + (1 − χnΩ) I)(VnxTV−nx) = VnxAnTV−nx ,

with

(An) = V−nxχnΩVnxA1V−nxχnΩVnx + I − V−nxχnΩVnx

(note that A1 is shift invariant). From Proposition 3.1 iii), the strong limit of An, denoted by A, is given byA = χK0

xA1χK0

xI + (1 − χK0

x)I and since T is a compact operator, ‖(An − A)T ‖ converges to zero. Hence,

‖Vnx (An −A) TV−nx‖ also converges to zero and so,

φx (VnxAnTV−nx) = φx (VnxATV−nx) ,

and AT is clearly compact. Similarly one checks for the product on the right side. Lastly, the ideal Kx is closedbecause K is closed.

The map ϑx restricted to Kx is the identity. Indeed,wx(φx(VnxTV−nx)) = T and w−1x (T ) = φx(VnxTV−nx)

implies ϑx(φx(VnxTV−nx)) = φx(VnxTV−nx), for every T ∈ K. Note also that ϑx = w−1x wx coincides with

h, (3.5), when restricted to the algebra φx(Eq). Moreover, if E ∈ φx(Eq) ∩ Kx, then h(E) = E belongs toφx(Fq

x) ∩ Kx.

Suppose now E ∈ φx

(Eq)

and ϑx(E) = 0. The sum of the closed ideal Kx with the C∗-algebra φx(Eq) is

also a C∗-algebra ([4], 1.8.4), which clearly coincides with φx

(Eq). Thus E = E1 + E2, with E1 ∈ φx(Eq)

and E2 ∈ Kx. So, E2 = ϑx (E2) = ϑx (−E1) belongs to φx(Fqx) ∩ Kx and this implies that E2 = h−1

(E2)belongs to φx(Eq)∩Kx, but if E2 ∈ φx(Eq), then E2 = −E1 because ϑx restricted to φx(Eq) is one to one, andso E = 0 which yields the assertion.

The following theorem shows that E is a standard algebra.

Theorem 4.2 A sequence (An) ∈ E is stable if and only if Wx(An) is invertible for every x ∈ ∂Ω ∪ 0.P r o o f. If (An) is stable then (An) + G is invertible in Eq but since W

′x : Eq → L(L2

(R

2))

is a homomor-phism it follows that Wx(An) = W

′x(An) is invertible for x ∈ ∂Ω ∪ 0.

Conversely, suppose Wx(An) is invertible for every x ∈ ∂Ω ∪ 0. From the proof of the previous lemmawe get that wx : φx

(Eq) → L(L2

(R2))

is indeed an isomorphism onto the image such that wx(φx(An)) =Wx(An). Thus, φx(An) is invertible for every x ∈ ∂Ω\0.



If x = 0 it is easy to check that,

φx(An) = φx(s- limAn) = φx(Wx(An)) , (4.1)

for every generator of E , and since Wx is a homomorphism, the equality (4.1) is valid for every element in E . Sothe invertibility of Wx(A) gives the invertibility of φx(Wx(A)).

Let now x ∈ intΩ and x = 0, and let y be a point in the intersection of the boundary of Ω with the half linestarting at the origin and passing through x. We claim that for every φx(An) in φx

(E),φx(An) = φx(Sθz(Wy(An))) , (4.2)

where z is some point belonging to intK0y , θz satisfies eiθz = z

|z| and Sθz(Wy(An)) is the limit operator ofWy(An) in the θz direction. Note that Sθz Wy is a homeomorphism (see (2.6)), thus we just need to show(4.2) for the generators of E . From Proposition 3.4, φx(χnΩAχnΩI + (1 − χnΩ)I) = φx (Aθ) , with eiθ = x

|x|direction, and Wy(χnΩAχnΩI + (1 − χnΩ)I) = χK0

yAθχK0

yI + (1− χK0

y)I. Thus, since z ∈ intK0

y it followsthat,

Aθ = Sθz

(χK0

yAθχK0

yI +

(1 − χK0

y

)I).

For the other generators of E the equality (4.2) is immediate. So, the invertibility of Wy(An) implies the invert-ibility of φx(An).

If x ∈ extΩ ∪ R2 \R2 then φx(χnΩI) = 0 and φx(VnzTV−nz) = 0 for every z ∈ ∂Ω ∪ 0, thus φx(An) =λφx(I) where λ does not depend on x. Now choose some y ∈ ∂Ω; by hypothesis Wy(An) is invertible and soφy(An) is also invertible. From Theorem 1.1 b) there is a neighborhood N of y such that for every t ∈ N thecoset φt(An) is invertible; but for t ∈ N ∩ extΩ, φt(An) = λφt(I), and so λ = 0. Consequently φx(An) isinvertible for every x ∈ extΩ ∪ R2 \R2.

Finally the invertibility of φx(An) for every x ∈ R2 implies, from the Allan-Douglas principle, the stabilityof (An).

The algebra E is a standard model, so from [7], Ch. 6, we know that if (An) is Fredholm, (i.e. if Wx(An)is Fredholm for every x ∈ ∂Ω ∪ 0) then (An) has the k-splitting value property. This means that there existd > 0 and εn ≥ 0, with lim εn = 0, such that the set of singular values of An is contained in [0, εn] ∪ [d,+∞] ,and for n large enough the number k of singular values in [0, εn], counted with respect to their multiplicity isindependent of n. Moreover, k satisfies:

k =∑

x∈∂Ω∪0dim KerWx(An) , (4.3)

where only a finite number of Wx(An) is not invertible.

Remark 4.3 DefiningL∞0 (R2) as the set of bounded functions on R2 that vanish at infinity, one has analogous

results by considering operators fI with f belonging to the algebraL∞0 (R2)+C(R2

)and, in particular, Theorem

4.2 and Examples 1), 2) and 3) have exactly the same form.

The following examples are applications of (4.3) together with Theorem 2.8, and Corollaries 2.9 and 2.10. LetAn be the sequence An = χnΩAχnΩ + (1 − χnΩ) I with A in the algebra generated by C(a) and fI.

Example 4.4 Let 0 ∈ ∂Ω and ∂Ω be a smooth set except at zero. If the operators χK0AχK0I + (1 − χK0)Iand χH0

xAθχH0

xI +

(1 − χH0

x

)I are Fredholm for every x ∈ ∂Ω\ 0

(eiθ = x

|x|), then

k = dim Ker (χK0AχK0I + (1 − χK0)I) .

Example 4.5 Let 0 ∈ ∂Ω, ∂Ω be a smooth set except at zero andA = C(a)+λI, with λ ∈ C. If the operatorCK(a) = χK(C(a)+λI)χKI+(1−χK)I (whereK is the cone at the origin associated to Ω) is Fredholm then

k = dim KerCK(a) .



Example 4.6 Let ∂Ω be a smooth set, 0 ∈ intΩ and A = C (a) + fI with f ∈ C(R2). If C(a) + fI is

Fredholm then

k = dim Ker (C(a) + fI) .

Note that k is determined by the sequence An.

5 Approximation by matrices

5.1 Computation of the kernel dimension of CK(a)

Concerning the question “If the convolution operator on the cone is Fredholm then is it also invertible ?”, weknow from Example (4.5) above that if we can compute the singular values of An = χnΩ (C (a) + λI)χnΩI +(1 − χnΩ) I for n large enough then we know the kernel dimension of the operator CK(a) and hence, we knowif it is or not invertible. The point is that the operators An are Riesz-Schauder operators and the computa-tion of their singular values needs a further discretization procedure. In what follows we give an idea of howto construct another approximation sequence (Bn) of the convolution operator on the cone that satisfies thek-splitting property and the condition (4.3) with the same k. We will see that the singular values of (Bn) can beobtained by computing the singular values of matrices of finite dimension.

Let pm, with m > 0, be an operator that associates to each g a step function given by:

pm : Lp(R) −→ Lp(R) , 1 ≤ p < ∞ ,

g(t) −→+∞∑

j=−∞gjχ[ j

m , j+1m ]

(5.1)

where gj is the mean value of g on the interval[

jm ,

j+1m

], i.e.

gj = m

∫ j+1m

jm

g (t) dt .

The following proposition is well-known. For convenience we give a proof.

Proposition 5.1 The projection pm satisfies the following properties:i) s- lim pm = I when m→ ∞. If g ∈ Lp(R) is differentiable and g′ ∈ Lp(R) then

‖g − pm(g)‖Lp ≤ 1m

‖g′‖Lp .

ii) If c(a) is the usual convolution operator in L2(R), where a ∈ L1(R), then

limm→∞ ‖(I − pm)c(a)‖ = lim

m→∞ ‖c(a)(I − pm)‖ = 0 .

P r o o f. i) We just need to prove the second statement because the set of differentiable functions whose deriva-tive belongs toLp (R) is dense inLp(R) and so it implies the strong convergence of pm to identity. Let g ∈ Lp(R)be differentiable with g′ ∈ Lp(R). We have

‖g − pm(g)‖p =j=+∞∑j=−∞

∫ j+1m

jm

|g (t) − gj |p dt

and

|g(t) − gj| =

∣∣∣∣∣ g(t) −m

∫ j+1m

jm

g(s) ds

∣∣∣∣∣=

∣∣∣∣∣m∫ j+1

m

jm

g(t) − g(s) ds

∣∣∣∣∣ =

∣∣∣∣∣m∫ j+1

m

jm

∫ t

s

g′(u) du ds

∣∣∣∣∣≤ m

∫ j+1m

jm

∫ j+1m

jm

|g′(u)| du ds =∫ j+1

m

jm

|g′(u)| du .



Thus,

‖g − pm(g)‖p ≤j=+∞∑j=−∞

∫ j+1m

jm

(∫ j+1m

jm

|g′(u)| du)p

dt

≤ 1m

j=+∞∑j=−∞

∫ j+1m

jm

|g′(u)|p du(∫ j+1

m

jm

1 du

)pq

=(

1m

)p

‖g′‖p .

ii) Let g ∈ L2(R) and a ∈ L1(R) be a differentiable function such that a′ ∈ L1(R). Then a ∗ g is alsodifferentiable and (a ∗ g)′ = a′ ∗ g. From this fact and i) we get,

‖a ∗ g − pm(a ∗ g)‖L2 ≤ 1m

‖(a ∗ g)′‖L2 =

1m

‖a′ ∗ g‖L2 ≤ 1

m‖a′‖L1‖g‖

L2 .

Hence, the norm ‖(I − pm)c(a)‖ converges to zero when m tends to infinity. It is easy to check that I − pm is aself-adjoint operator, thus ‖c(a)(I − pm)‖ converges also to zero. Since the set of differentiable functions on R

is dense in L1(R) we get the assertion.

Defining the operator Pm = pm ⊗ pm in L2(R

2)

we will have properties analogous to the one dimensionalcase. This operator Pm : L2

(R

2)→ L2

(R

2), assigns to each function g the step function,

∑(i,j)∈Z2

gijχ[ im , i+1

m ]×[ jm , j+1

m ] where gij = m2

∫ i+1m

im

∫ j+1m

jm

g(t, s) dt ds . (5.2)

It follows from the previous proposition and the properties of tensor product that s-limPm = I and

limm→∞ ‖(I − Pm)C(a)‖ = lim

m→∞ ‖C(a)(I − Pm)‖ = 0 . (5.3)

With the aim of defining the desired sequence (Bn) let us take Ω to be

Ω =[0, 1

2

]× [0, 12

] ∪ (x, y) ∈ R2 :(x− 1

2

)2+(y − 1

2

)2 ≤ 14

, (5.4)

i.e. Ω is the union of a square with a disk of radius 12 . We have that s-limχnΩI = χKI, where the cone K is the

quarter plane R2++.

Fig. 1



We can associate to each m > 0 a partition of R2 into squares given by

R2 =

⋃(i,j)∈Z2

[im ,

i+1m

]× [ jm ,

j+1m

],

and we define Ωmn as the union of all squares of side 1

m lying inside nΩ (see Fig. 1).It is easy to check that PmχΩm

nI = χΩm

nPm and χΩm

nχnΩ = χnΩχΩm

n= χΩm

n.

Let m = nα, for some α > 1. To simplify the notation we write χΩnα instead of χΩnαn

. By (Bn) we denotethe following sequence,

Bn = PnαχΩnα (C(a) + λI)χΩnαPnα + λ(χnΩ − χΩnαPnα)I + (1 − χnΩ)I .

Proposition 5.2 The sequence (Bn) converges strongly to CK(a) and has the k-splitting property withk = dim kerCK(a).

P r o o f. We know that

An = χnΩ (C(a) + λI)χnΩI + (1 − χnΩ) I

has the k-splitting property with k = dim KerCK(a), so it is enough to show that lim ‖An − Bn‖ = 0, [7]. Weestimate

‖An −Bn‖ = ‖χnΩC(a)χnΩI − PnαχΩnαC(a)PnαχΩnα‖≤ ‖(χnΩI − PnαχΩnα )C(a)PnαχΩnα ‖ + ‖PnαχΩnαC(a)(χnΩI − PnαχΩnα )‖

+ ‖(χnΩI − PnαχΩnα )C(a)(χnΩI − PnαχΩnα I )‖ .

Since PnαχΩnα is a bounded sequence and (χnΩ − PnαχΩnα ) I is a bounded self-adjoint sequence, we just needto show ‖(χnΩ − PnαχΩnα )C(a)‖ converges to zero as n goes to infinity. We have

‖(χnΩ − PnαχΩnα )C(a)‖ = ‖(χnΩ − χΩnα )C(a) + (χΩnα I − χΩnαPnα)C(a)‖≤ ‖(χnΩ − χΩnα )C(a)‖ + ‖(I − Pnα)C(a)‖ .

From (5.3) the second sequence converges to zero, and to prove that the first sequence converges to zero it isenough to consider kernel functions a ∈ Cc

(R2)

because the set of continuous functions with compact supportis dense in L1

(R2). Let g ∈ L2

(R2), then

‖(χnΩI − χΩnα I)C(a)g‖2L2 =

∫R2

∣∣∣∣χnΩ\Ωnα (x)∫

R2a(x− t)g(t) dt

∣∣∣∣2 dx≤∫

R2χnΩ\Ωnα (x)

(∫R2

|a(x− t)|2 dt∫

R2|g(t)|2 dt

)dx

≤ ‖a‖L2 ‖g‖L2

∫R2χnΩ\Ωnα (x) dx .

The last integral is the area of the set nΩ\Ωnα . This set is contained in a three quarter annulus with outer radiusn2 and inner radius n

2 −D, where D is the diagonal of a square of side 1nα (see Figure 1). Hence,

area (nΩ\Ωnα) ≤ 3π4

(n2

4−(n

2−

√2

nα

)2)

≤ 3π2√

21

nα−1.

Since α > 1, we have limn→∞ ‖(χnΩ − χΩnα )C(a)‖ = 0.



The last proposition tells us that we need to compute the singular values of Bn, which we denote by σ2(Bn).It is easy to check that for

Bn = PnαχΩnα (C(a) + λI)χΩnαPnα ,

viewed as an operator defined on Im (PnαχΩnα ), we have

σ2

(Bn

)\ |λ| , 1 = σ2 (Bn) \ |λ| , 1 .Thus, from the k-splitting property of (Bn) it follows for n large enough, that

σ2

(Bn

) ∩ [0, εn] = σ2(Bn) ∩ [0, εn] ,

with εn → 0, which means in particular that(Bn

)also has the k-splitting property, with the same value of k. Let

us now see that each Bn can be identified with a finite matrix, for each n. Consider the following isomorphism,

Γ : ImPnα −→ l2(Z2)

∑(i,j)∈Z2

gijχ[ inα , i+1

nα ]×[ jnα , j+1

nα ] −→ gij(i,j)∈Z2

The restriction of Γ to Im (PnαχΩnα ) is an isomorphism onto the space l2(Ωnα ∩ Z2

), which has finite dimen-

sion equal to the number of squares of area(

1nα

)2inside nΩ. Denoting by ΓR this restriction, we have that the

operator

ΓRBnΓR : l2(Ωnα ∩ Z

2) −→ l2

(Ωnα ∩ Z

2)

is identified with a matrix of size d (n)× d (n) , with d (n) = dim Im (PnαχΩnα ). Since d (n) is less or equal tothe number of squares inside [0, n] × [0, n], we estimate d (n) ≤ n2(α+1).

Remark 5.3 We know that the singular values sd(n)1 ≤ s

d(n)2 ≤ . . . ≤ s

d(n)d(n) of the matrices d (n) × d (n)

satisfy

limn→∞ s

d(n)k = 0 and lim

n→∞ inf sd(n)k+1 > 0

with k = dim KerCK(a). Some natural questions arise. For instance, one can ask how fast does the sequences

d(n)k converge to zero? How large has to be the dimension of the matrices to observe the k-splitting phenomena?

One can introduce some parameters in our sequence (Bn) that change the speed limit of sd(n)k and the size of

the matrices. If we consider a more general operator prm, that associates to each g, a function that is a polynomial

of degree r in each interval[

jm ,

j+1m

](note the operator (5.1) is of this type with r = 0), then one can show based

on [10] that the results are still true and the convergence speed of sd(n)k is higher when r is larger. We can also

introduce another parameter, substituting the sets nΩ by nβΩ, where β is some positive number. The previousresults are also true and Proposition 5.2 is in force for α > β (see proof). So, in this case d (n) is less or equal ton2(α+β).

5.2 Another way to compute the kernel dimension of CK(a)

It is clearly of interest to have more than one numerical approach to compute the kernel dimension of CK(a).Actually one can choose other sets Ω and sequences (An) satisfying the k-splitting property such that we getinformation about the kernel dimension of CK(a). Here we take for Ω the simple square,

Ω = [0, 1] × [0, 1] , (5.5)

with vertices x1 = (0, 0), x2 = (1, 0), x3 = (1, 1) and x4 = (0, 1). We consider for a fixed m ∈ N thefollowing operators:

T1 = PmχmΩI , T2 = V−x2PmχmΩVx2 ,

T3 = V−x3PmχmΩVx3 , T4 = V−x4PmχmΩVx4 ,



where Pm is the interpolation operator (5.2). Note that the operator PmχnΩI has dimension m4 and so Ti, withi = 1, . . . , 4 are compact operators. Defining the sequences,

An = χnΩ(C(a) + λI)χnΩI + (1 − χnΩ)Iand

An = An(I − Vnx2T2V−nx2 − Vnx3T3V−nx3 − Vnx4T4V−nx4) ,

we have that if CK(a) is Fredholm, where K is the first quadrant, then Wx(An) is Fredholm for every x ∈ ∂Ω.Indeed, if CK(a) is Fredholm then Wxi(An) = CK0

xi(a)(I − Ti), with i = 2, 3, 4, are Fredholm operators and

Wx(An) with x ∈ ∂Ω\ x1, . . . , x4 are invertible operators. Thus, (An) has the k-splitting property with kgiven by

k = dim KerCK(a) +4∑

i=2

dim KerCK0xi

(a)(I − Ti) .

If we know dim KerCK0xi

(a)(I−Ti), then we can compute dim KerCK(a). For that we will use a simple obser-vation made by A. Rogoshin, a Ph.D. student of P. Junghanns (private communication), that says the following:Let X be a Banach space, A ∈ L(X) be a Fredholm operator and Ym ∈ L(X) be a family of projections suchthat s-limYm = I and dim ImYm = dm (dm ∈ N). Then, there ism0 ∈ N such that dim KerA(I−Ym) = dm,for m > m0.

Setting X = L2(K0

xi

)and Ym = Ti = Ti(m), we can apply the previous result to

Ai = χK0xi

(C(a) + λI)χK0xiI

and obtain for m large enough that

dim KerCK0xi

(a)(I − Ti) = dim Ker (Ai(I − Ti(m))) = m4 . (5.6)

The point is that we want to choose a fixedm such that (5.6) is in force. With that aim, we consider the sequence

Dn = An(I − T1 − Vnx2T2V−nx2 − Vnx3T3V−nx3 − Vnx4T4V−nx4) (5.7)

that satisfies the k-splitting property with

k(m) =4∑

i=1

dim KerCK0xi

(a)(I − Ti) .

The kernel dimension of CK0xi

(a)(I −Ti) is greater than or equal tom4 for every i ∈ 1, 2, 3, 4 and form large

enough it is exactly m4, so when k (m) is equal to 4m2 a correct m is found.To know dim KerCK(a) from the equation (5.6), we need to compute k, and as before we face the problem

of computing the singular values of the infinite dimensional operators An. So we need to find a sequence Bn

such that ‖An −Bn‖ → 0 and whose singular values can be obtained by computing the singular values of finitematrices.

For n large enough we have that VnxiTiV−nxi(1 − χnΩ)I = 0 and moreover if m,n ∈ 2l : l ∈ N

then(I −Pn)VnxiTiV−nxi = VnxiTiV−nxi(I−Pn) = 0. From these facts and χnΩPn = PnχnΩI and equality (5.3)it is not difficult to show that

Bn = PnχnΩ(C(a) + λI)χnΩ(I − Vnx2T2V−nx2 − Vnx3T3V−nx3 − Vnx4T4V−nx4)χnΩPn

+λχnΩ(I − Pn) + (1 − χnΩ)I

satisfies ‖An −Bn‖ → 0 and σ2(Bn)\ |λ| , 1 = σ2

(Bn

)\ |λ| , 1 , where Bn are the operators

Bn = PnχnΩ(C(a) + λI)χnΩ (I − Vnx2T2V−nx2 − Vnx3T3V−nx3 − Vnx4T4V−nx4)χnΩPn

that can be identified with matrices of dimensionm4 ×m4.Note that to compute the singular values of Dn in (5.7) we compute the singular values of the analogous

matrices.



Remark 5.4 We want to point out that the approach to compute the kernel dimension of CK(a), consideringfor Ω a square can be also applied to the analogous operators in the discrete case.

Remark 5.5 It is well-known that the kernel dimension of a Fredholm operator is not stable under smallperturbations, however the method we present here is stable, and the main reason is, roughly speaking, that thesingular values of a matrix are stable under small perturbations. So, if we perturb an approximation sequence ofmatrices, although the singular values are not exactly the same, we can still see the splitting property with thesame k.

Remark 5.6 In order to avoid heavier notation we considered a fixed Ω, (5.4) or (5.5), such that s-limχnΩ =χKI, where K is the quarter plane R++. Nevertheless, all the previous results remain true for any other conewith an appropriate Ω.

Remark 5.7 We can apply the same method of matrices approximation considering the set (5.4) or (5.5), tocompute the kernel dimension of operators of the type χK(C(a) + fI)χKI + (1 − χK)I with f ∈ L∞

0 (R2) +C(R2

)(see Remark 1).

Remark 5.8 The developed approach works also in the block case.

Acknowledgements The authors thank the referees for a few useful remarks and suggestions, in particular for some simpli-fications in the proofs of Lemma 2.1 and Proposition 3.7 a).

The first author was supported by Fundacao para a Ciencia e Tecnologia under the grant Praxis XXI BD 19702/99 and theFCT project POCTI/MAT/34222/99.

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convolution type operators on cones and their finite sections

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