SPECTRAL APPROXIMATION and INDEX for CONVOLUTION TYPE OPERATORS on CONES on Lp (R2)

SPECTRAL APPROXIMATION AND INDEX FOR CONVOLUTION TYPE OPERATORS ON CONES ON Lp (R2)

H. MASCARENHAS AND B. SILBERMANN

Abstract. We consider an algebra of operator sequences containing, among oth- ers, the approximation sequences to convolution type operators on cones acting on p 2 L (R ), with 1 < p < ∞. To each operator sequence (An) we associate a family p 2 of operators Wx(An) ∈ L(L (R )) parametrized by x in some index set. When all Wx(An) are Fredholm, the so-called approximation numbers of An have the α-splitting property with α being the sum of the kernel dimensions of Wx(An). Moreover, the sum of the indices of Wx(An) is zero. We also show that the index of some composed convolution-like operators is zero. Results on the convergence of the -pseudospectrum, norms of inverses and condition numbers are also obtained.

1. Introduction Let u ∈ L1 (R2) and λ ∈ C. We denote by C (a) the convolution operator defined by C(a): Lp(R2) → Lp(R2) R g 7→ λg (t) + 2 u(t − s)g(s)ds. R where a is given by a(x) = λ + (F (u)) (x) , with F being the Fourier transform on R2. The function a, usually called the symbol of the operator C(a), is continuous in R2 and tends to λ at infinity. It is well known that C(a) ∈ L (Lp (R2)), the Banach algebra of all bounded linear operators in Lp (R2), and its invertibility is equivalent to the invertibility of the symbol a. The set of all such symbols 2 1 2 (1) W R := a = λ + F (u): u ∈ L R , λ ∈ C ∞ 2 is a subalgebra of L (R ) closed for the norm kakW = |λ| + kukL1 and is the so- called Wiener algebra. By a cone K with vertex at the origin, we mean an angular sector in R2 , i.e (2) K := {teiθ : t ∈ [0, +∞[ , θ ∈ Γ} where Γ is a closed connected subset of unit circle T containing at least two points. We define the convolution operator on K as the restriction of C(a) to a cone K extended by the unity to the complement of K and we denote it by

CK (a) := χK C(a)χK I + (1 − χK ) I, where χK is the characteristic function of K. In 1967, Simonenko proved that CK (a) is Fredholm if and only if a is invertible and if CK (a) is Fredholm then its index is zero [Si1] . It remains an open question to know if every Fredholm operator CK (a), with a in the Wiener algebra, is invertible.

1991 Mathematics Subject Classiﬁcation. 45E10, 47A53, 47L80, 15A29. Key words and phrases. Convolution operators, pseudospectrum, Fredholm sequences, index. 1 Spectral Approximation of Convolution Type Operators 2

When K is a half-space Gohberg and Goldstein showed that this result is true [GG]. For some special symbols the answer is also true [B1, M, O]. We give a contribution to this problem by presenting an asymptotic formula for the kernel dimension of CK (a) providing it is Fredholm. We consider an algebra E of approximate sequences containing in particular approximate sequences to operators of the type χK AχK + (1 − χK )I with A in the Banach algebra

2 2 (3) A0 := alg{C (a) , fI : f ∈ C(R ), a ∈ W (R )}, where C(R2) is set of the continuous functions in R2 which have finite limit in every direction (see precise definition in section 2). Our main aim is to describe the so called Fredholm sequences in E and their properties. We mean by a Fredholm sequence a sequence which is regularizable by some special ideal D consisting of sequences of compact type. The interest of Fredholm sequences has at least three aspects: It is a generalization of the stability property, they can be used to construct generalized inverses to some sequences of interest [S2], and last but not least Fredholm sequences can be used in some cases to obtain the index and an asymptotic formula for the kernel dimension of some Fredholm operators. Using C∗ algebras techniques, it was proved in [MS], for p = 2, that Fredholm sequences in E possess some splitting property of the singular values (that is, the ∗ 1 points in the spectrum of (AnAn) 2 ) as n → ∞. Here we prove, for 1 < p < ∞, the analogous result by defining approximation numbers in the context of infinite dimensional Banach spaces and using them instead of singular values . We further prove an index formula for the operators associated to each Fredholm sequence. In [RoS] results concerning some theory of Banach algebras consisting of structured matrix sequences (in which the structure is hidden in the sequence, not in the entries separately) are obtained. Here we adapt this model, although the approximations which occur in this paper are not finite matrices. To achieve our aim we start in section 2 by introducing a Banach algebra B containing sequences with special structure and which contains E. We first study the stability problem for sequences in E by help of some localization procedure and limit operators techniques. A correspondence of a sequence (An) ∈ E and a set p 2 of operators Wx(An) ∈ L(L (R )), with x in some index set X is established. We reprove, in particular, earlier stability results of A. Kozak [K1] and add some new features which lead to the convergence of condition numbers and -pseudospectrum. These results are given in section 3. The method we use goes back to S. Roch [R], who used it in a different setting, and to our earlier paper [MS]. Notice that some results of the recent paper of [Ma] are also contained in ours, as will be explained at the end of subsection 3.4.1 We start section 4 by defining Fredholm sequences and approximations numbers. In subsection 4.3 we show that if a sequence (An) is Fredholm then the sum of the indices of Wx(An) is zero and the approximation numbers of (An) have the α-splitting property with α being the sum of the kernel dimensions of Wx(An). A second localization level is applied in subsection 4.4 to describe all Fredholm sequences belonging to E. It is shown that Fredholm sequences are those for which Wx(An) is Fredholm for every x ∈ X. In the last subsection we use the main results Spectral Approximation of Convolution Type Operators 3 to deduce that for some Fredholm convolution type operators its index is zero and an asymptotic formula for the kernel dimension is computed. Notice that we consider the above mentioned operators on Lp (R2) for simplicity. Although, one can extend the results without difficulties to related operators acting on Lp (Rn). Moreover, the system case can also be considered by the methods of this paper.

2. Localization ∞ p 2 Let F be the set of all sequences (An)n=0 of operators in L (L (R )) such that sup kAnk < ∞. Endowed with the usual pointwise operations and the norm k(An)k = sup kAnk, F is a Banach algebra. The set G of the sequences in F tending in norm to zero is a closed ideal of F and plays an important role due to Kozak’s Theorem ([BS], Proposition 2.20). This theorem states that a sequence (An) in F is stable (i.e. An is invertible for n large enough and the norms of their inverses are uniformly bounded) if and only if the G coset (An)+G is invertible in F = F/G. Thus, stability is an invertibility problem. There are no effective tools to study invertibility in F G because of its generality. The way out is to introduce a suitable subalgebra of F G which serves our aim and which is subject to localization via Allan-Douglas local principle. 2 2 x Let D be the open unit disc in R . It is easy to see that ξ : D → R , x 7→ 1−|x| is a homeomorphism. We denote by C(R2) the set of all continuous functions f on R2 for which f ◦ ξ admits a continuous extension onto the closed disc D. Provided with pointwise operations and the supremum norm C(R2) forms a commutative C∗-algebra isomorphic to C(D). Its maximal ideal space R2 can be viewed as the compactification (with the Gelfand topology) of R2 with the circle T of infinitely 2 2 2 −1 distant points; and a sequence hn ∈ R converges to θ∞ ∈ R \R if ξ (hn) converges iθ iθ to some e with θ ∈ [0, 2π[. We write f(θ∞) := f]◦ ξ(e ), where the second function is the extension of f ◦ ξ . 2 For each function ϕ ∈ C(R ), we associate the sequence (ϕnI) given by the ex- t panded functions ϕn (t) = ϕ n . Clearly, this sequence belongs to F and k(ϕnI)k = 2 p 2 p 2 kϕk∞. For x ∈ R , let Vx : L (R ) → L (R ), f(t) 7→ f(t − x) be the usual shift operator in the x-direction. We denote by B the set of all sequences (An) of F for which both limits ∗ s- lim V−nxAnVnx and s- lim V−nxAnVnx exist for every x ∈ R2, and 2 lim kAnϕnI − ϕnAnk = 0, for every ϕ ∈ C( ). n→∞ R It is not hard to check that B is a Banach subalgebra of F. Observe that the 2 commutator (An)(ϕnI) − (ϕnI)(An) belongs to G for every ϕ ∈ C(R ) which shows that we can apply localization to study B/G. In order to characterize the algebra B, let us consider the following map, for x ∈ R2, p 2 Wx : B → L (L (R )) (An) 7→ s- lim V−nxAnVnx. Taking into account the definition of B and the algebraic properties of the strong limits, the map Wx is well-defined and is a homomorphism. Besides, since Vnx is Spectral Approximation of Convolution Type Operators 4 an isometry and Wx(I) = I we deduce from the Banach-Steinhaus Theorem that kWxk = 1. We note that the expression Wx(An) will always refer to Wx applied to the sequence (An) ∈ B and Wx(A) will denote Wx applied to the constant sequence. Throughout this work we deal with several quotient algebras and homomorphisms defined on them. In order to simplify notation, if A, B are Banach algebras, J ⊂ A a closed ideal of A and H : A/J → B an homomorphism, we write H(a) in place of H(a + J).

Proposition 1. ([MS], Proposition 3.1) Let x ∈ R2, f, ϕ ∈ C(R2), a ∈ W (R2) and (Gn) ∈ G. Then the following holds i) s- lim V−nxC(a)Vnx = C(a) ii) s- lim V−nxfVnx = f(x∞)I if x 6= 0 where f(x∞) = limn→∞ f(nx) iii) s- lim V−nxϕnVnx = ϕ(x) iv) s- lim V−nxGnVnx = 0. Theorem 1. The algebra BG = B/G is an inverse closed subalgebra of F G. G Proof. Let (An) + G ∈ B be invertible. Then, there exists a bounded sequence of p 2 operators Bn in L (R ), such that

(4) BnAn = I + Gn 0 (5) AnBn = I + Gn 0 where (Gn) , (Gn) ∈ G. We ﬁrst prove that Wx(An) = s- lim V−nxAnVnx is an invertible operator. If (An) is stable then (V−nxAnVnx) is also stable, i.e. V−nxAnVnx is invertible for n large enough and the norms of the inverses are uniformly bounded. Therefore, there exists C > 0 such that: p 2 kV−nxAnVnxhk ≥ Ckhk for every h ∈ L R , and consequently, Wx(An) has closed image and its kernel dimension is zero. Similar ∗ ∗ arguments for (An) show that Wx (An) has kernel dimension zero and thus Wx(An) is invertible. −1 ∗ Let us now prove that s- lim V−nxBnVnx = Wx (An) and s- lim V−nxBnVnx = −1 ∗ 2 (Wx (An)) for every x ∈ R . From (4), we have

V−nxBnVnxV−nxAnVnx = I + V−nxGnVnx which is equivalent to −1 −1 V−nxBnVnx (Wx(An) + (V−nxAnVnx − Wx(An))) Wx (An) = (I+V−nxGnVnx)Wx (An).

Since V−nxAnVnx −Wx(An) and V−nxGnVnx converge strongly to zero, it follows that −1 V−nxBnVnx is equal to Wx (An) plus a sequence strongly convergent to zero. Thus −1 the strong limit of V−nxBnVnx exists and equals Wx (An). Similar arguments show ∗ −1 ∗ that the strong limit of V−nxBnVnx exists and equals (Wx (An)) . Finally, let us prove that

(BnϕnI − ϕnBn) ∈ G. 0 From (4) and (5), we can write I = AnBn − Gn = BnAn − Gn and then obtain 0 BnϕnI − ϕnBn = Bnϕn (AnBn − Gn) − (BnAn − Gn) ϕnBn 0 = BnϕnAnBn − BnϕnGn − BnAnϕnBn + GnϕnBn 0 = Bn(ϕnAn − Anϕn)Bn − BnϕnGn + GnϕnBn, Spectral Approximation of Convolution Type Operators 5

0 where the two last sequences (BnϕnGn) and (GnϕnBn) belong to G. Once (An) belongs to B, then (ϕnAn − Anϕn) in also in G and this completes the proof.

From Theorem (1), we see that a sequence (An) ∈ B is stable if and only if n o G G 2 An + G is invertible in B . By construction, the set C = (ϕnI) + G : ϕ ∈ C(R ) is contained in the center of BG and it can be easily seen that it is a C∗-algebra. Hence, we can use localization to study invertibility in the algebra BG.

Proposition 2. The algebra CG is isometric isomorphic to C(R2) Proof. It is clear that for every ϕ ∈ C(R2), we have

kϕk∞ = k(ϕnI)kB ≥ k(ϕnI) + GkBG . To prove the statement we just need to show the reverse inequality. From Propo- sition 1-iv), the homomorphism Wx maps G onto zero. Let us then deﬁne the analogous homomorphism

G p 2 (6) Wfx : B → L(L (R )), (An) + G 7→ Wx(An) G 2 on the quotient algebra B . For every x ∈ R , we have kWfxk = 1 and due to Proposition 1-iii) Wfx((ϕnI) + G) = ϕ(x)I . Thus,

kϕk∞ = sup |ϕ(x)| ≤ k(ϕnI) + GkBG . 2 x∈R Since CG 'C(R2) we identify the maximal ideal space of CG with R2. We associate 2 G to each x ∈ R the maximal ideal in C given by Ix = {(ϕn) I + G : ϕ (x) = 0}, and 2 to each θ ∈ [0, 2π[ the maximal ideal Iθ = {(ϕnI) + G : ϕ (θ∞) = 0}. For x ∈ R , G G G let Jx be the smallest closed ideal in B generated by Ix and let φx : B → B /Jx be the canonical quotient map. Due to the Allan-Douglas Principle we know that G G (An) + G ∈ B is invertible if and only if φx((An) + G) is invertible in B /Jx for every x in R2 .

3. Finite sections of convolution type operators on cones. Let us now introduce a set Ω ⊂ R2 which will be needed to define the finite sections of convolution type operators on cones. We say Kx is a cone with vertex x if it is a set of the type K + x = {x + t : t ∈ K} where K is a cone at the origin (see Equation (2)). We will use the notation nX, where n is a positive integer and X ⊂ R2 is a set, to denote the expanded set {nt : t ∈ X}. Let Ω be a closed bounded 2 set of R containing the origin, so that the strong limit of χnΩI is not zero. We also assume that for each point x ∈ ∂Ω there exists a cone Kx, neighborhoods U and V of x and a C1-diffeomorphism ρ : U → V, such that 0 ρ(x) = x, ρ (x) = I ρ(U ∩ Ω) = V ∩ Kx. If 0 ∈ ∂Ω we require moreover that the associated diffeomorphism ρ is the identity. Observe that the cone Kx is uniquely defined for each x ∈ ∂Ω. The next proposition states some convergence results that will be useful in the sequel Spectral Approximation of Convolution Type Operators 6

Proposition 3. ( [MS], Proposition 3.1 and Proposition 3.2) Let x ∈ R2 , H be a half-space with 0 ∈ ∂H and T ∈ K(Lp(R2)). Then,  χ 0 I, if x ∈ ∂Ω  Kx i) s- lim V−nxχnΩVnx = I, if x ∈ intΩ ,  0, if x ∈ R2\Ω 0 where, for x ∈ ∂Ω , Kx := {t − x : t ∈ Kx} is the cone shifted to the origin. 0 ii) s- lim V−nxχnKx Vnx = χKx I, for x ∈ ∂Ω.  χ I, if x ∈ ∂  H H iii) s- lim V−nxχHVnx = I, if x ∈ intH  0, if x ∈ R2\H iv) s- lim V−nxTVnx = 0 for x 6= 0. Let E be the smallest closed subalgebra of F generated by -(χnΩAχnΩI + (1 − χnΩ) I) with A in A0 -(Gn) with (Gn) ∈ G p 2 -(VnxTV−nx) with T ∈ K(L (R )) and x ∈ ∂Ω ∪ {0}, p 2 where A0 is the Banach algebra defined in (3) and K(L (R )) denotes the set of all compact operators on Lp(R2). Notice that E contains the finite sections of the operators χK AχK I + (1 − χK ) I, with A in A0, relative to the sequence of projections (χnΩI). We do not know whether E G is inverse closed. However from Proposition 1, Proposition 3 and com- mutations relations ([MS], Proposition 3.2), we deduce that the generators of E are in B. That is, E G = E/G is a closed subalgebra of the inverse closed algebra BG. G G Now, our main task is to describe the local subalgebras φx(E ) of B /Jx. 3.1. Local algebras for points not at the boundary. G For x∈ / ∂Ω, the local algebras φx(E ) are simple algebras. Proposition 4. ([MS], Proposition 3.4) Let x ∈ R2, y ∈ R2, f ∈ C(R2) and T ∈ K(Lp(R2)). Then: ( f(x )φ (I), if x ∈ 2 \{0} i) φ (fI) = ∞ x R x 2 2 f(x)φx(I), if x ∈ R \R , where f(x∞) = limn→∞ f(nx). 2 ii) φx(χnΩI) = 0 if x ∈ R \Ω and φx(χnΩI) = φx(I) if x ∈ intΩ. iii) φx(VnyTV−ny) = 0 if x 6= y . G It is convenient to define the homomorphisms Wx in the local-algebras B /Jx . From proposition 1- iii) it follows Wfx(Jx) = 0, where Wfx is defined in (6). Then, the map G p 2 wx : B /Jx → L (L (R )) ((An) + G) + Jx 7→ s- lim V−nxAnVnx

is a well deﬁned homomorphism with kwxk = 1. G For x∈ / ∂Ω, the local algebras φx(E ) can be completely described by the homomorphisms wx. Let A00 be the closure of all convolution operators with symbol in the Wiener algebra, i.e the Banach algebra 2 (7) A00 = alg{C(a): a ∈ W R }

2 Proposition 5. Let x ∈ R , A00 be the algebra deﬁned in (7) and A1 be the Banach algebra generated by all C(a) with a ∈ W (R2), all fI with f ∈ C(R2) and all Spectral Approximation of Convolution Type Operators 7

p 2 G compact operators in L (R ). The local algebra φx(E ) is isometrically isomorphic to the Banach algebra of operators A00 if x ∈ intΩ and x 6= 0 i) Lx := A1 if x = 0 and 0 ∈ intΩ 2 ii) Lx := CI if x ∈ R \Ω. Proof. i) Let x ∈ intΩ. From Proposition 3 and Proposition 4, we deduce that

(8) φx(An) = φx(Wx(An)) G G for the generators of φx(E ) and so it is also true for every element of E . Since

kφx(Wx(An))k ≤ kWx(An)kBG ≤ kWx(An)kL(Lp) and kwxk ≤ 1, then wx is an isometry. To show that its image is closed, note that for every L ∈ Lx, we have L = wx(φx(L)). 2 ii) If x ∈ R \Ω then, due to Proposition 4, φx (χnΩAχnΩI + (1 − χnΩ) I) = φx(I) G and φx(VnyTV−ny) = 0, y ∈ ∂Ω ∪ {0}. Thus φx(E ) is generated by φx(I) and therefore is isomorphic to C. 3.2. Local algebras at the boundary points. When x ∈ ∂Ω, the situation is more involved. Let us deﬁne, for each x ∈ ∂Ω, the Banach algebra Fx given by

Fx := alg{χnKx AχnKx I + (1 − χnKx )I, (Gn) , (VnxTV−nx)}, p 2 where A belongs to A0 deﬁned in (3) , (Gn) ∈ G and T ∈ K(L (R )). G G We will construct an isometry hρ from B /Jx onto itself which maps φx(E ) onto G φx(Fx ) and we will see that the second one is easier to describe. Recall that to each x ∈ ∂Ω we associated a C1-diﬀeomorphism ρ : U → V, where U and V are neighborhoods of x and in case 0 ∈ ∂Ω, ρ ≡ I.

Thus, if x = 0 and 0 ∈ ∂Ω then χnΩχnU I = χK0 χnU I which means φx (χnΩI) = G G φx (χK0 I) and the algebras φx(E ) and φx(Fx ) are the same. So, from now on x is a ﬁxed point in ∂Ω\{0}. (−1) Let Tn and Tn be the following operators, which will help us to construct the isometry hρ . p 2 p 2 Tn : L (R ) → ImχnU I ⊂ L (R ) g nρ t if t ∈ nU (T g)(x) = n n 0 if t∈ / nU. and (−1) p 2 p 2 Tn : L (R ) → ImχnV I ⊂ L (R ) g nρ−1 t if t ∈ nV T (−1)g (x) = n n 0 if t∈ / nV. Proposition 6. Let Jρ(t) denote the Jacobian matrix of ρ at t and similarly for −1 (−1) ρ . The operators Tn and Tn are bounded linear operators satisfying: −1 (−1) a) kTnk ≤ supt∈V |Jρ (t) | and kTn k ≤ supt∈U |Jρ (t) | (−1) (−1) b) TnTn = χnU I and Tn Tn = χnV I (locally invertible) (−1) (−1) c) χnU TnχnV I = Tn and χnV Tn χnU I = Tn ∗ (−1) d) The sequences V−nyTnVny, and V−ny Tn Vny converge strongly for y ∈ U ∗ (−1) and the sequences V−nyTn Vny and V−nyTn Vny converge strongly for y ∈ V . Spectral Approximation of Convolution Type Operators 8

Proof. a) Let g ∈ Lp(R2), then Z p t p kTngk = |g(nρ( ))| dt nU n Z s = |g(s)|p|Jρ−1( )|pds nV n Z ≤ sup |Jρ−1(s)|p |g(s)|pds s∈V nV So, −1 kTngk ≤ sup |Jρ (s)|kgkLp s∈V (−1) Analogously, one shows the bound for the operator Tn . b) and c) are immediate. p 2 p 2 d) Let y ∈ U and Fρ0(y) : L (R ) → L (R ) be the linear bounded operator given 0 by g(t) 7→ g(ρ (y)t). We claim that s- lim V−nyTnVny = Fρ0(y) if ρ(y) = y (note that if y = x it gives the identity map) and s- lim V−nyTnVny = 0 if ρ(y) 6= y . Let g ∈ Lp(R2) be a continuous function with compact support (it is enough to consider functions from a dense set of Lp(R2)). We have Z p t 0 p kV−nyTnVnyg − Fρ0(y)gkLp( 2) = |χnU−ny(t)g(nρ( + y) − ny) − g(ρ (y)t)| dt R 2 n R where χnU−ny is the characteristic function of nU − ny = {nt − ny, t ∈ U}. t 0 Let hn(t) = χnU−ny(t)g nρ( n + y) − ny − g(ρ (y)t). Since the open set U − y 2 contains the origin, then for a ﬁxed t ∈ R , there is n0 ∈ N such that for n ≥ n0, t ∈ nU − ny and thus, t h (t) = g(nρ( + y) − ny) − g(ρ0(y)t), if n ≥ n . n n 0 The function ρ is diﬀerentiable in U, i.e, ρ(t + y) = ρ(y) + ρ0(y)(t) + δ(t)ktk, where limt→0 δ(t) = 0. Since we assumed ρ(y) = y then

t t t t ρ( + y) = ρ(y) + ρ0(y) + δ( )k k n n n n t t t (9) = y + ρ0(y) + δ( )k k n n n and it follows that t h (t) = g ρ0(y)t + δ( )ktk − g(ρ0(y)t), if n ≥ n . n n 0 2 Once g is continuous, we have lim hn(t) = 0 for each t ∈ R , thus by the Lebesgue p dominated convergence theorem lim kV−nyTnVnyg − Fρ0(y)gk p 2 = 0. L (R ) Suppose now ρ(y) 6= y. We have Z p t p kV−nyTnVnygkLp( 2) = |χnU−ny(t)g(nρ( + y) − ny)| dt R 2 n R t 2 Deﬁning hn(t) = χnU−ny(t)g(nρ( n + y) − ny), it follows for ﬁxed t ∈ R that there t is n0 ∈ N such that for n ≥ n0, hn(t) = g(nρ( n + y) − ny). t Since limn→+∞ |n(ρ( n + y) − y)| = +∞, then hn(t) → 0 when n → +∞ because g has compact support. Thus, s- lim V−nyTnVny = 0. Spectral Approximation of Convolution Type Operators 9

(−1) One shows analogously that V−nyTn Vny also converges strongly for every y ∈ V . Concerning the adjoint sequences, we start by noting that ∗ q 2 q 2 Tn : L (R ) → ImχnV I ⊂ L (R ) g nρ−1 t |Jρ−1( t )| if t ∈ nV (T ∗g)(x) = n n n 0 if t∈ / nV.

1 1 is the adjoint operator of Tn, with p + q = 1. Now, deﬁne for y ∈ V the linear p 2 p 2 0 −1 bounded operator Feρ0(y) : L (R ) → L (R ), g(t) 7→ g(ρ (y)t)|Jρ (y)|. One can ∗ use the same arguments to prove that s- lim V−nyTn Vny = Feρ0(y) if ρ(y) = y (in case ∗ y = x it gives the identity map) and s- lim V−nyTn Vny = 0 if ρ(y) 6= y . Just note that when ρ(y) = y , the functions hn are, for n large enough, given by

t −1 t 0 −1 hn(t) = g(nρ( + y) − ny) Jρ ( + x) − g(ρ (y)t)|Jρ (y)| n n 2 and so applying (9) one gets lim hn(t) = 0 for each t ∈ R . Analogously, we have that ∗ (−1) q 2 q 2 Tn : L (R ) → ImχnU I ⊂ L (R ) g nρ−1 t |Jρ−1( t )| if t ∈ nU (T ∗g)(x) = n n n 0 if t∈ / nU. ∗ (−1) (−1) is the adjoint operator of Tn and that the strong limits of V−ny Tn Vny exit for every y ∈ U. Let Z ⊂ U ∩ V and W be an open ball centered in x such that W ⊂ Z ∩ ρ−1(Z). In order to construct the isometry hρ we deﬁne the Banach subalgebras of B given by BW = {χnW AnχnW I :(An) ∈ B} and Bρ(W ) = {χρ(W )Anχρ(W )I :(An) ∈ B} . Proposition 7. The map,

Hρ : BW → Bρ(W ) (−1) (An) 7→ (Tn AnTn) is a well defined isomorphism and satisfies: a) If S ⊂ W then Hρ (χnSI) = χnρ(S)I. b) The ideal G is invariant under Hρ, i.e Hρ(G) ⊂ G. c) Let (An) ∈ BW . If the coset An + G belongs to Jx then Hρ(An) + G also belongs to Jx . (−1) (−1) Proof. Simple computations show Tn AnTn = χρ(W )Tn AnTnχρ(W )I for (An) ∈ (−1) BW , it remains to prove that Tn AnTn is in B to have a well defined map. For y∈ / ρ(W ), we have s- lim V−nyχρ(W )Vny = 0 and so s- lim V−nyHρ(An)Vny = 0. If y ∈ ρ(W ) then, by construction of W , y ∈ U ∩ V and from Proposition 6-d) we (−1) conclude s- lim V−nyTn Vny and s- lim V−nyTnVny exist, as well as its adjoints and so it also exits s- lim V−nyHρ(An)Vny and its adjoint. 2 Suppose now that ϕ ∈ C(R ), we have to show that Hρ(An) commutes with ϕnI up to a sequence tending in norm to zero, i.e there exists (Gn) ∈ G such that (−1) (−1) (10) Tn AnTnϕn = ϕnTn AnTn + Gn. Spectral Approximation of Convolution Type Operators 10

2 Observe first that ifρ ˜ is a continuous extension of ρ to R , then TnϕnI = (ϕ ◦ρ˜)nTn. Indeed, for g ∈ Lp(R2) t t t (T ϕ g)(t) = T (ϕ( )g(t)) = χ (t)ϕ(ρ( ))g(nρ( )) n n n n nU n n = [χnU (ϕ ◦ ρ˜)nTng](t) = [(ϕ ◦ ρ˜)nTng](t). (−1) −1 (−1) −1 Analogously, the equality Tn ϕnI = (ϕ◦ρg)nTn holds, where ρg is a continuous extension of ρ−1 to R2. Thus, (−1) (−1) Tn AnTnϕnI = Tn An(ϕ ◦ ρ˜)nTn. Using the fact that there exists (Fn) ∈ G such that An(ϕ ◦ ρ˜)n = (ϕ ◦ ρ˜)nAn + Fn, it follows: (−1) (−1) Tn An(ϕ ◦ ρ˜)nTn = Tn [(ϕ ◦ ρ˜)nAn + Fn] Tn −1 (−1) (−1) = (ϕ ◦ ρ˜ ◦ ρg)nTn + Tn FnTn (−1) (−1) = ϕnTn + Tn FnTn. (−1) Now, let Gn = Tn FnTn and we obtain equality (10) finishing the proof that Hρ is well defined. It is clear that Hρ is linear. Since for (An), (Bn) ∈ BW , (−1) Hρ(AnBn) = Hρ(AnχnW Bn) = Tn AnχnW BnTn = (−1) (−1) = Tn AnTnTn χnW BnTn = Hρ(An)Hρ(Bn), −1 Hρ is a homomorphism. It is not hard to check that its inverse Hρ : Bρ(W ) → BW (−1) is the map that sends An to TnAnTn . a) Let g be a function in Lp (Rp) and S ⊂ W. Then we have: t (H (χ ) g)(t) = T −1χ T g (t) = T −1χ (t)g nρ ρ nS n nS n n nS n

= χnρ(S)(t)g(t). b) It is immediate. 2 c) Since Hρ is an isomorphism, it is enough to show, for ϕ ∈ C(R ) with 2 ϕ(x) = 0 there exist φ ∈ C(R ) with φ(x) = 0 such that Hρ (χnW ϕnχnW I) = −1 −1 χρ(nW )φnχρ(nW )I. In fact, Hρ (χnW ϕnχnW I) = χρ(nW )(ϕ ◦ ρg)nχρ(nW )I, where ρg is a continuous extension of ρ−1 to R2, thus choose φ = ϕ ◦ ρg−1and we get the result. G G G Let BW and Bρ(W ) be the subalgebras of B analogous to BW and Bρ(W ), and G G (11) Heρ : BW → Bρ(W ) G G be the map analogous to Hρ. From Proposition 7c), one has Heρ(BW ∩ Jx) = Bρ(W ) ∩ Jx. Every coset ((An) + G) + Jx is the same coset as ((χnW AnχnW ) + G) + Jx, thus we can now deﬁne a map between the local algebras by:

h : BG/J → BG/J (12) ρ x x ((An) + G) + Jx 7→ ((Hρ (An)) + G) + Jx . Spectral Approximation of Convolution Type Operators 11

Taking into account Proposition 7, hρ is an isomorphism, having −1 G G hρ : B /Jx → B /Jx (−1) ((An) + G) + Jx 7→ ( TnAnTn + G) + Jx . as its inverse homomorphism. Note that hρ does not 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 in general diﬀerent from the coset ( χn(U∩Ω)I + G) + Jx.

G G Proposition 8. The algebras φx(E ) and φx(Fx ) are isometrically isomorphic under the isomorphism hρ.

Proof. Since hρ is an isomorphism of B/Jx onto itself, one has only to check that hρ G G maps the generators of φx(E ) onto the generators of φx(Fx ) . Notice that the cosets φx(χnΩI), φx(χnΩχnW I) and φx(χn(Ω∩W )I) are the same and from Proposition 7-a), hρ(φx(χn(Ω∩W )I)) = φx(χnρ(Ω∩W )I). From the facts φx(χnV I) = φx(χnU I) = φx(I), W ⊂ U, and ρ is a diﬀeomorphism such that ρ(Ω ∩ U) = Kx ∩ V we get

φx(χnρ(Ω∩W )I) = φx(χnρ(Ω)χnρ(W )I) = φx(χnρ(Ω)χnρ(U)I)

= φx(χnρ(Ω∩U)I) = φx(χn(Kx∩V )I) = φx(χnKx I).

Since hρ maps C(a) onto itself, (see [MS], Proposition 3.18), we have

hρ (φx(χnΩAχnΩI + (1 − χnΩ)I)) = φx(χnKx AχnKx I + (1 − χnKx )I), for A ∈ A0. We are left to prove

hρ(φx(VnxKV−nx)) = φx(VnxKV−nx), where K is a compact operator. Actually, it is enough if we prove

(−1) (13) kVnxKV−nx − Tn VnxKV−nxTnk → 0.

Multiplying both sides by the isometries V−nx and Vnx , one has: (−1) (−1) kVnxKV−nx − Tn VnxKV−nxTnk = kK − V−nxTn VnxKV−nxTnVnxk

(−1) = kK − KV−nxTnVnx + KV−nxTnVnx − V−nxTn VnxKV−nxTnVnxk (−1) ≤ kK(I − V−nxTnVnx)k + k I − V−nxTn Vnx KkkV−nxTnVnxk

Since V−nxTnVnx is a bounded sequence and from Proposition 6-d) both sequences ∗ (−1) V−nxTn Vnx and V−nxTn Vnx converge strongly to identity, it follows the convergence (−1) of the sequences K(I − V−nxTnVnx) and I − V−nxTn Vnx K in norm to zero.

To prove that hρ is isometric, we know from Proposition 2-a) that −1 khρk ≤ sup Jρ (s) sup |Jρ (s)| . s∈V s∈U Given ε > 0, we may assume without loss of generality that U and V are chosen 2 so that khρk ≤ (1 + ε) . Since ε is arbitrarily, we have khρk = 1. Using the same −1 arguments one also has hρ = 1. Thus, hρ is an isometry. Spectral Approximation of Convolution Type Operators 12

p 2 Proposition 9. Let x ∈ ∂Ω, T ∈ K(L (R )) and A0 and A00 be the algebras deﬁned G in (3) and (7), respectively. The local algebra φx(E ) is isometrically isomorphic to the Banach algebra of operators 0 0 0 Lx := alg χKx AχKx I + 1 − χKx I,T with A ∈ A00 if x ∈ ∂Ω\{0} and A ∈ A0 if x = 0 and 0 ∈ ∂Ω.

Proof. Suppose x ∈ ∂Ω\{0}. From Proposition 8 it is enough to prove that wx G G restricted to φx(Fx ) is an isometric isomorphism onto Lx. The generators of φx(Fx ) are of the type φx (χnKx AχnKx I + (1 − χnKx )I) with A ∈ A00 or of the type φx (VnxTV−nx). The ﬁrst ones can be rewritten as: 0 0 0 (14) φx Vnx(χKx AχKx I + (1 − χKx )I)V−nx ,

0 because A is a shift invariant operator and χnKx I = VnxχKx V−nx, thus

(15) φx(An) = φx(VnxWx(An)V−nx) G for the generators of Fx . Since φx and Wx are continuous homomorphisms the G equality (15) is true for every element of Fx . Since kwxk = 1 and

kφx(VnxWx(An)V−nx)k ≤ kVnxWx(An)V−nxk ≤ kWx(An)k, we conclude that wx is an isometry. Moreover, the image of wx is closed, because G for every L ∈ Lx, we have L = wx(φx(VnxLV−nx)) with φx(VnxLV−nx) in φx(Fx ). G G If x = 0 and 0 ∈ ∂Ω, then φ0(E ) = φ0(Fx ) is generated by the constant sequences

((χK0 AχK0 I + (1 − χK0 )I) + G) + Jx and ((T ) + G) + Jx p 2 with A ∈ A0 and T ∈ K(L (R )). Now, using the same arguments as in the case x ∈ ∂Ω\{0} we have φ0(An) = φ0(W0(An)) for every (An) ∈ E and therefore w0 G maps φ0(E ) onto L0. 3.3. Stability for sequences in E. The following theorem describes stability of (An) ∈ E in terms of the operators Wx(An).

Theorem 2. Let (An) ∈ E. The sequence (An) is stable if and only if Wx(An) is invertible in L(Lp(R2)) for every x ∈ ∂Ω ∪ {0} .

Proof. If (An) is stable then it is clear that all operators Wx(An) are invertible p 2 in L(L (R )) with x ∈ ∂Ω ∪ {0} . Suppose now, Wx(An) is invertible for every x ∈ ∂Ω ∪ {0} . (i) If x ∈ ∂Ω\{0}, it follows, from the proof of the Proposition 9, that −1 (16) φx(An) = hρ (φx(VnxWx(An)V−nx)).

Since for every (An) ∈ E the sequence (VnxWx(An)V−nx) belong to the inverse closed G algebra B and, hρ deﬁned in (12) maps B /Jx isomorphically onto itself, it follows p 2 G that if Wx(An) is invertible in L(L (R )) then φx(An) is invertible in B /Jx . (ii) For x = 0 the equality φ0(An) = φ0(W0(An)) holds, and since (W0(An)) belongs to the inverse closed algebra B, we have that φ0(An) is invertible providing W0(An) is invertible. (iii) Suppose x ∈ intΩ\{0}. Let y be a point on the intersection of the boundary of Ω with the half line that starts at the origin and passes through x and choose z 0 to be any point in the interior of Ky . For the generating sequences (An) of E, we have Wx(An) = Wz(Wy(An)) and φx(An) = φx(Wx(An)), therefore the equalities Spectral Approximation of Convolution Type Operators 13 hold for every sequence in E. Thus from the invertibility of Wy(An) it follows the G invertibility of Wx(An) and consequently that of φx(An) in B /Jx. 2 (iv) If x ∈ R \Ω then φx (An) = λφx(I) where λ does not depend on x. Now choose some y ∈ ∂Ω; by hypothesis Wy (An) is invertible and so φy (An) is also invertible. From Allan-Douglas principle (see [RRS], Proposition 2.3.17) there exits a neighborhood N of y such that for every t ∈ N the coset φt (An) is invertible; but for t ∈ N ∩ extΩ, φt (An) = λφt(I), and so λ 6= 0. Now putting together these four cases we have that φx (An) is invertible for every x ∈ R2 and applying the Allan Douglas principle we ﬁnish the proof. We now give examples of equivalent conditions to the stability of a sequence, using the last theorem and previous results.

Example 1. Let 0 ∈ ∂Ω and Aij ∈ A0, where A0 is the algebra deﬁned in (3). The sequence m l X Y [χnΩAijχnΩI + (1 − χnΩ) I] i=1 j=1 is stable if and only if the following conditions are satisﬁed: Pm Ql i) i=1 j=1 [χK0 AijχK0 I + (1 − χK0 )I] is invertible m l P 0 0 0 ii) i=1 Πj=1 χKx Wx (Aij) χKx I + 1 − χKx I is invertible for every x ∈ ∂Ω\{0}.

Example 2. Let Ω be the closed unit disc. Assume A = C (a)+fI, with a ∈ W (R2) and f ∈ C(R2). The sequence (χnΩAχnΩI + (1 − χnΩ) I) is stable if and only if A is invertible.

Example 3. Let 0 ∈ ∂Ω, ∂Ω be a smooth set except at zero and Ω ⊂ K0. Let A = C (a) + fI. The sequence χnΩAχnΩI + (1 − χnΩ) I is stable if and only if

χK0 AχK0 I + (1 − χK0 )I is invertible. Notice that the result in example 3 was already proved in [K1] when f is a constant. 3.4. A symbol map for E. In Theorem 2 necessary and sufficient conditions for the stability of a sequence (An) ∈ E are given. What can we also say about other asymptotic properties of (An) such as the convergence of the norms of An, condition numbers or the behavior of the spectrum and ε-pseudospectrum? Having these questions in mind, we shall construct an isometric isomorphism from E G onto an algebra of operator valued functions and we shall describe asymptotic properties of (An). Let S be the set of all bounded functions defined on ∂Ω ∪ {0} with values in L (Lp(R2)). Endowed with pointwise sum and product, and with the supremum norm k(Ax)k = supx∈∂Ω∪{0} kAxk , S is a Banach algebra. It is easy to check that the map sym : B/G → S (An) + G 7→ (s- lim V−nxAnVnx)x∈∂Ω∪{0} Spectral Approximation of Convolution Type Operators 14 is a homomorphism. Noting that the norm of (An) + G in B/G is also given by k(An)kB/G = lim sup kAnk, it follows from the Banach-Steinhaus Theorem, that ksym(An)k ≤ lim inf kAnk ≤ lim sup kAnk which implies that ksymk ≤ 1. We will now define a topological property of the algebras that have central C∗ -subalgebras which will be useful to prove that sym is isometric when restricted to E G. So, let A be a Banach algebra with identity and C a closed C∗-subalgebra of the center of A which contains the identity. By the Gelfand-Naimark Theorem, C is isomorphic to the algebra of continuous functions on the maximal ideal space of C; therefore an element of C will be called a function. Definition 1. We say that A is a KMS-algebra with respect to C if for every A ∈ A and ϕ, ψ ∈ C with disjoint supports kA(ϕ + ψ)k ≤ max (kAϕk , kAψk) . The importance of KMS-algebras is the relation between the norm of an element of A with the norms of the corresponding local elements. More precisely A is KMS if and only if

(17) kAk = max kφx(A)k , x∈MC where φx(A) is the local element associated to the Allan-Douglas localization. For a proof see ([BKS], Theorem 5.3). Proposition 10. The algebra BG= B/G is a KMS-algebra with respect to CG.

Proof. We need to show that if ϕ and ψ are functions in C(R2) whose supports are disjoint, then

(18) k(An(ϕn + ψn)I)kBG ≤ max(k(AnϕnI)kBG , k(AnψnI)kBG ), G for every (An) ∈ B. The norm in B is given by inf(Gn)∈G sup kAn + Gnk and it is not diﬃcult to see that it coincides with lim sup kAnk . If we deﬁne Yn = Anϕn, Zn = Anψn,N = suppϕ and M = suppψ then the inequality (18) can be written as

(19) lim sup kYnχnN + ZnχnM k ≤ max (lim sup kYnk , lim sup kZnk) . Given g ∈ Lp(R2), we have p p p k(χnN YnχnN + χnM ZnχnM ) gkLp = kχnN YnχnN gk + kχnM ZnχnM gk p p ≤ max (kYnk , kZnk) kχnN gk + p p + max (kYnk , kZnk) kχnM gk p p ≤ max (kYnk , kZnk) kgk . Thus,

kχnN YnχnN + χnM ZnχnM kL(Lp) ≤ max (lim sup kYnk , lim sup kZnk) . Since the last inequality is true for every n ∈ N then

lim sup kχnN YnχnN + χnM ZnχnM kL(Lp) ≤ max (lim sup kYnk , lim sup kZnk) .

Now observe that (Anϕn − ϕnAn) belongs to the ideal G, ϕnχnN = ϕn and ψnχnM = ψn. So, YnχnN − χnN YnχnN and ZnχnM − χnM ZnχnM also belong to this ideal. Thus,

lim sup kYnχnN + ZnχnM k = lim sup kχnN YnχnN + χnM ZnχnM k Spectral Approximation of Convolution Type Operators 15 and we get the assertion (19). The following theorem shows that algebraic and topological properties of (An) can be translated in terms of the analogous properties of (Wx(An))x∈∂Ω∪{0} .

Theorem 3. The restriction of sym to E G is an isometric isomorphism onto the image. Moreover, for every (An) ∈ E, the sequence kAnk converges and

lim kAnk = max kWx(An)k . x∈∂Ω∪{0}

Proof. Let (An) be a sequence in E then, from Proposition 10, we know that:

k(An)kB/G = lim sup kAnk = max kφx(An)k 2 x∈R and we claim that

max kφx(An)k ≤ sup kWx(An)k , 2 x∈R x∈∂Ω∪{0} which implies that sym is an isometry. 2 If x ∈ R \Ω then φx(An) = λφx(I) and λ do not depend on x. In particular if 2 0 x ∈ R \Ω, kφx (An)k = kWx (An)k. Let y ∈ ∂Ω ∩ [0, x] and z in the exterior of Ky . Then it is easy to check that Wx(An) = Wz(Wy(An)), for every (An) of E. Thus,

kφx(An)k = kWx(An)k ≤ kWy (An)k If x ∈ intΩ\{0} and y is a point on the intersection of the boundary of Ω with the half line that starts at the origin and passes through x, from the proof of Theorem 2 iii), we have that

kφx(An)k ≤ kWx(An)k = kWy(Wz(An))k ≤ kWy (An)k .

For x = 0 the inequality kφx(An)k ≤ kWx(An)k holds (see the proof of Theorem 2 ii)). If x ∈ ∂Ω\{0}, then from equality (16) and khρk = 1 we deduce that kφx(An)k ≤ kWx(An)k. All these inequalities together lead to

(20) max kφx(An)k ≤ sup kWx (An)k 2 x∈R x∈∂Ω∪{0} which proves that sym is an isometry and so, an isomorphism onto its image. Thus, we can write

max kφx(An)k = sup kWx(An)k . 2 x∈R x∈∂Ω∪{0} Moreover from the inequalities above we deduce that for each x ∈ R2 there exists y ∈ ∂Ω ∪ {0} such that kφx(An)k ≤ kWy(An)k, therefore it is not possible to have kWx(An)k < max 2 kφx(An)k for every x ∈ ∂Ω ∪ {0}, which means that x∈R the supremum on the second side of (20) is attained. Now, observe that W (An) =

W (Ank ) for every subsequence (Ank ) of (An), which means lim sup kAnk = lim kAnk, and therefore

lim kAnk = max kWx(An)k . x∈∂Ω∪{0} Spectral Approximation of Convolution Type Operators 16

For an invertible operator A in a Banach algebra, its condition number is deﬁned as the real number condA = kAkkA−1k. It is clear that condA is always greater than or equal to one. As a consequence of Theorem 3 we obtain

Corollary 1. If (An) ∈ E is stable then the condition numbers of An converge and −1 −1 lim kAnk kAn k = max kWx(An)k max kWx (An)k. x∈∂Ω∪0 x∈∂Ω∪0

Proof. Suppose (An) ∈ E is stable . From Theorem 3

lim kAnk = max kWx(An)k. x∈∂Ω∪0

We claim the formula is also true for the inverse of (An) , i.e −1 −1 (21) lim kAn k = max kWx (An)k. x∈∂Ω∪0 From equalities (8) and (16), it is easy to check that, −1 −1 φx(Wx (An)), if x ∈ ∂Ω φx (An) = −1 −1 . hρ (φx(VnxWx (An)V−nx)) , if x∈ / ∂Ω Now, following the same reasoning as in the proof of Theorem 3, together with the result that says if An is stable then Wx(An) = s- lim V−nxAnVnx is an invertible op- −1 −1 −1 −1 erator with s- lim V−nxAn Vnx = Wx (An) and kWx (An)k ≤ lim inf kV−nxAn Vnxk (see proof of Theorem 1), we obtain the claim. 2 Corollary 2. Let An = χnΩC(a)χnΩI + (1 − χnΩ)I, with a ∈ W (R ). Then

lim kAnk = max{1, kC(a)k} ≤ max{1, kakW } 0 0 0 Proof. We have Wx(An) = χKx C(a)χKx I + 1 − χKx I, and therefore

lim kAnk = max χK0 C(a)χK0 I + (1 − χK0 )I x∈∂Ω∪{0} x x x

0 0 = max{1, χKx C(a)χKx I }. 0 Since for any interior point y of K , we have C(a) = Wy(χK0 C(a)χK0 I) and kWyk = x x x 0 0 1, then kC(a)k ≤ χKx C(a)χKx I . The reverse inequality holds obviously, proving the corollary. 3.4.1. Convergence of the pseudospectrum. As we saw, Theorem 3 and its corollaries established some relations between properties of a sequence (An) in E for large n, and its associated set of “limit operators” Wx(An). We will now consider the spectrum of An for n large enough. In particular, what is the relation between the spectrum of the sequence (An) and the one of Wx(An)? To be precise, let us first define the limit of a sequence of subsets of the complex plane. Given sets Mn ⊂ C, we define its limit set, denoted limn→∞ Mn, to be the set of all complex numbers λ for which there are integers 0 < n1 < n2 < ... and complex numbers λn, n ∈ N, such that

λn ∈ Mn and lim λn = λ. k k k→∞ k Spectral Approximation of Convolution Type Operators 17

From Theorem 2, we easily deduce that [ (22) lim σ(An) ⊂ σ(Wx(An)). n→∞ x∈∂Ω∪{0} But in general these two sets do not coincide, (see for instance, the example given in [BGS], Thm. 1.3, in the context of Wiener-Hopf operators). However, if we consider the ε-pseudospectrum, which we now define, instead of the spectrum, we will see that the results are much more satisfactory. Definition 2. For ε > 0, the ε-pseudospectrum of an operator A ∈ Lp(R2) is defined to be the set 1 σ (A) = {λ ∈ C : A − λI is not invertible or k(A − λI)−1k ≥ }. ε ε It is clear from the definition, that the spectrum of an operator is contained in its ε-pseudospectrum.

Theorem 4. Let (An) ∈ E and ε > 0. The ε-pseudospectrum of (An) converges and [ lim σε(An) = σε(Wx(An)). x∈∂Ω∪{0} The proof of theorem uses a non-trivial result due to Daniluk in the context of operators deﬁned in a Hilbert space, and which was later proved also for operators in Lp(X, dµ), where (X, dµ) is a measure space, and 1 < p < ∞ ([BGS], Theorem 5.1). This interesting result, which can be viewed as a maximum principle, states that when A is a bounded operator in Lp(X, dµ) such that A − λI is invertible for all λ in some open subset W of the complex plane, and k(A − λI)−1k ≤ M, then actually k(A − λI)−1k < M, for all λ ∈ W. We would also like to add that it is not known whether this result is true for every Banach algebra. Using this maximum principle and Theorem 3, the proof of Theorem 4 proceeds in analogy to that of Theorem 5.2 in [BGS]. Suppose now we consider the algebra E0 ⊂ E generated by (χnΩAχnΩI + (1 − χnΩ)I) with A ∈ A00 (see (7)) and Ω to be a polygon with vertex at the origin. If y is any point in one of the edges of the polygon Ω, which is not a vertex, and x is a vertex of that edge, then Wy(An) = Wy−x(Wx(An)) for every (An) ∈ E0. Thus, kWy(An)k ≤ kWx(An)k. Furthermore, if Wx(An) is invertible then Wy(An) is also invertible and −1 −1 −1 kWy (An)k = kWy−x(Wx (An))k ≤ kWx (An)k. This implies, using Corollary 1 and Theorem 4, the following result which was proved, with other methods, by Maximenko [Ma].

Corollary 3. Let (An) ∈ E0. Suppose Ω is a polygon with vertex at the origin and X is the set of its vertices. Then, −1 −1 (i) lim kAnk kA k = maxx∈X kWx(An)k maxx∈X kW (An)k if (An) is stable. n S x (ii) lim σε(An) = x∈X σε(Wx(An)). Remark 1. The results in Theorem 4 and its Corollary can be sharpened as follows. Given a sequence of sets Mn ⊂ C, let LimMn stand for the set of all complex numbers λ which are the limit of a sequence (mn) of points mn ∈ Mn. Then the limes superior (also called the partial limiting set) lim Mn appearing in the formulation of Spectral Approximation of Convolution Type Operators 18 the mentioned Theorem and Corollary can be replaced by the corresponding limes inferior (also called uniform limiting set) LimMn. This can be seen by invoking the fractality of the algebra E (see [HRS], subsection 3.3.3).

4. Fredholm sequences, Approximation numbers and Splitting property 4.1. Fredholm sequences. In order to deﬁne Fredholm sequences we start to introduce an ideal related with some sequences of compact operators. Let D ⊂ B be the smallest closed ideal of B containing the sequences converging to zero in norm and the sequences (VnxTV−nx) for every compact operator T on Lp(R2) and every x ∈ ∂Ω ∪ {0}. Proposition 11. The ideal D is the closure of the following set ( m ) X (23) ( Vnxi TiV−nxi + Gn): Ti ∈ K, (Gn) ∈ G, xi ∈ ∂Ω ∪ {0} i=1 Proof. It is clear that the sum of elements of the set (23) are of the same type. Let us now prove that the set is a right ideal. Let (An) be any sequence in B and consider a sequence of the type (VnxTV−nx) for some x ∈ ∂Ω ∪ {0} and T a compact operator. We have:

VnxTV−nxAn = VnxT (V−nxAnVnx − Wx(An))V−nx + VnxTWx(An)V−nx.

Since T is compact and s-lim [V−nxAnVnx − Wx(An)] = 0, the sequence

Gn = VnxT (V−nxAnVnx − Wx(An))V−nx converges to zero in norm. Take now R = TWx(An) which is a compact operator and it follows that

(24) VnxTV−nxAn = VnxRV−nx + Gn. Following the same ideas we obtain that (23) is also a left ideal. We know that a linear bounded operator is Fredholm if it is invertible up to an operator in the ideal of compact operators. Similarly, we say that (An) ∈ B is a Fredholm sequence if

(An) + D is invertible in B/D. Notice that this deﬁnition depends on B and D. Actually, there is also a more general notion of Fredholm sequence in the large algebra F. A sequence (An) belonging to F is called Fredholm if it is regularizable with respect to the ideal of all compact sequences. This ideal is deﬁned as follows: it is the smallest closed two-sided ideal in F which contains all sequences (Kn) of almost uniformly bounded rank, that is lim sup rankKn < ∞. It can be proved that for sequences in E these two notions coincide. This proof involves new ideas which will be part of a forthcoming paper. An application of the special lifting Theorem of S.Roch and B.Silbermann (HRS, Theorem 5.37b)) leads to the following proposition which we will give the proof for the convenience of the reader.

Proposition 12. If (An) ∈ B is a Fredholm sequence then the operator Wx(An) is Fredholm for every x ∈ ∂Ω ∪ {0} and only a ﬁnite number of these operators are non-invertible. Spectral Approximation of Convolution Type Operators 19

Proof. Suppose (An) ∈ B is a Fredholm sequence, then there are (Bn) ∈ B and (Jn) ∈ D such that AnBn = I + Jn. Pm From Proposition 11 there exists Jen ∈ D such that Jn − Jen = i=1 Vnxi TiV−nxi + Gn 1 −1 and kJenk < 2 . Now, deﬁning Ben = Bn(I + Jen) , which is in B (see Theorem 1), one has, m −1 X −1 −1 AnBen = (I + Jn)(I + Jen) = I + Vnxi TiV−nxi (I + Jen) + Gn(I + Jen) i=1 which due to (24), can be rewritten as m X (25) AnBen = I + Vnxi RiV−nxi + Gen i=1 with Ri ∈ K and Gen ∈ G. Applying Wx to both sides of (25) we have

Wx(An)Wx(Ben) = I + Ri or Wx(An)Wx(Ben) = I depending on x belong or not belong to {x1, ..., xn}, respectively. Thus, Wx(An) is left invertible for every x, except for x ∈ {x1, ..., xn} which is left Fredholm. Analogously, we show the right invertibility. 4.2. Approximation numbers. Let X be a Banach space and L(X) the algebra of linear bounded operators on X. We make the convention that if codimF is not finite, then codimF ≥ α for any integer α. 1 Definition 3. (Approximate numbers) Let X be a Banach space and α ∈ N or α ∈ {1, ..., dim X}, depending on X has non-finite or finite dimension, respectively. The αth-approximation number of an operator A in L(X) is given by:

sα(A) := inf {kA − F k : F ∈ L(X) and codimF ≥ α} , where codimF = dim X/ImF . Clearly, for every α one has

0 ≤ sα(A) ≤ sα+1(A) ≤ kAk Notice that if X is a Hilbert space of finite dimension n, then the approximation numbers coincide with the singular values of the matrix A (see [A]), i.e. √ ∗ {sα(A), α = 1, ..., n} = { λ : λ ∈ σ(A A)} Although if X is a Hilbert space of infinite dimension this is not true. Take for instance the Toeplitz operator T (eiθ) defined on the Hardy space H2(T), where T iθ is the unit circle. It has codimension one and thus s1(T (e )) = 0 and zero is not a singular value since T ∗(eiθ)T (eiθ) = I .

Proposition 13. Let α ∈ N or α ∈ {1, ..., dim X} and A, B ∈ L(X) then,

(1) sα(AB) ≤ kBk sα(A) for every α. 1 (2) If A is invertible then s1(A) ≥ kA−1k .

1Notice that this name is used in the literature for other (related) numbers Spectral Approximation of Convolution Type Operators 20

(3) If A is Fredholm of index zero, then s1(A) = ··· = sα(A) = 0 and sα+1(A) > 0 where α = dim ker A. Proof. (1) Since ImFB ⊂ ImF then codimFB ≥ codimF , thus

sα(AB) ≤ inf {kAB − FBk : codimF ≥ α} ≤ kBk sα(A). (2) If A is invertible and F is such that codimF ≥ 1, then kI − A−1F k can not be less than 1, otherwise F would be invertible. Thus, kI − A−1F k ≥ 1 , i.e. kA−1(A−F )k ≥ 1 and so kA−1kk(A−F )k ≥ 1 for every F with codimension greater than or equal to 1. Therefore 1 s (A) ≥ . 1 kA−1k (3) Suppose that A is a Fredholm operator of index zero with kernel dimension α. Then codimA = α and from the deﬁnition of approximate numbers s1(A) = ··· = sα(A) = 0. Suppose now that sα+1(A) = 0. Then there exists a sequence of operators Fn ∈ L(X) such that codimFn ≥ α + 1 and

lim kA − Fnk = 0. n→∞

Thus, for n large enough, Fn are Fredholm operators with the same index as A, i.e, for n large enough dim ker Fn = codimFn ≥ α + 1. But due to the upper semicontinuity of the kernel dimension, there is a neighborhood of A where all operators have kernel dimension less than or equal to that of A, which contradicts sα+1(A) = 0. 4.3. Splitting property of Fredholm sequences.

Deﬁnition 4. We say that (An) ∈ B has the α-splitting property if the approximation numbers of An satisfy the following conditions

lim sα(An) = 0 and lim inf sα+1(An) > 0.

Theorem 5. Let (An) ∈ B be a Fredholm sequence such that ind(An) = 0 for n large enough. Then (An) has the α-splitting property, with X α = dim ker Wx (An) . x∈∂Ω∪{0} Moreover, X ind Wx (An) = 0. x∈∂Ω∪{0} Proof. We will prove that P a) sα(An) → 0, with α = dim ker Wx(An) P x∈∂Ω b) Let β = x∈∂Ω dim coker Wx(An). There exists c > 0 such that for n large enough sβ+1(An) ≥ c. P c) α = β, which means that ind Wx(An) = 0. a) If (An) is Fredholm, then from Propositions (11) and (12) there exists (Bn) ∈ B such that m X p (26) BnAn = I + Vnxi TiV−nxi + Gn,Ti ∈ K(L ) and k(Gn)k → 0. i=1 Spectral Approximation of Convolution Type Operators 21

αi Let αi = dim ker Wxi (An), so that α = α1 + ··· + αm. Let {ei,l}l=1 be a basis of ker Wxi (An) consisting of norm one vectors. For each (i, l) with 1 ≤ i ≤ m and n p 2 n 1 ≤ l ≤ αi, define the sequence ei,l ∈ L (R ) by ei,l = Vnxi (ei,l). For each n ∈ N, define the vector space n n n V = span e1,1, ..., em,αm and for each (i, l) with 1 ≤ i ≤ m and 1 ≤ l ≤ αi, we define the functional n n fi,l : V → C given by m αi ! n X X n fi,l xj,vej,v = xi,l. j=1 v=1

Due to Lemma 6.1 in [RoS], there exists n0 ∈ N and C > 0 such that, for n ≥ n0,

m αi m αi X X X X n |xi,l| ≤ Ck xi,lei,lk. i=1 v=1 i=1 l=1 n Thus, for n ≥ n0, we have by the Hahn-Banach Theorem that fi,l can be extended to a linear continuous functional to the whole space Lp (R2) in such a way that n kfi,lk ≤ C. p 2 p 2 The operator Sn : L (R ) → L (R ) given by

m αi X X n n Sn(x) = fi,l(x)ei,l, i=1 l=1 n is a projection on the space V of dimension α, for n ≥ n0. We claim that kAnSnk → 0 when n → +∞. Let x ∈ Lp (R2). Then,

m αi X X n n p 2 kAnSnxkL (R ) = kAn fi,l(x)ei,lk i=1 l=1 m αi X X n n ≤ |fi,l(x)| kAnei,lk i=1 l=1 m αi X X n p 2 ≤ CkxkL (R ) kAnei,lk i=1 l=1 m α X Xi p 2 ≤ CkxkL (R ) kV−nxi AnVnxi ei,lk. i=1 l=1

Due to s-limV−nxi AnVnxi = Wxi (An) and ei,l ∈ ker Wxi (An), we then obtain that kV−nxi AnVnxi ei,lk → 0 for all 1 ≤ i ≤ m and 1 ≤ l ≤ αi proving the claim. Since indAn = 0 for n large enough, by assumption, and codim (I − Sn) = α then codim An(I − Sn) ≥ α. From the deﬁnition, sα(An) is the distance of An to the set of all operators whose codimension is greater than or equal to α. Thus, for n large enough

(27) sα(An) ≤ kAn − An(I − Sn)k

= kAnSnk → 0. Spectral Approximation of Convolution Type Operators 22 b) If (An) is Fredholm then it is not diﬃcult to check that (Cn) ∈ B and (Gn) ∈ G can be chosen such that

l X A C = I − V I − P V + G . n n nxi Im Wxi (An) −nxi n i=1

Notice that deﬁning βi = codim Wxi (An), we have that dim Im V I − P V = codim P = β nxi Im Wxi (An) −nxi Im Wxi (An) i and l l X X dim Im V I − P V ≤ β = β. nxi Im Wxi (An) −nxi i i=1 i=1

Since (I + Gn) is stable there exists d > 0 such that 1 k(I + G )−1k ≤ n d for n large enough, i.e by Proposition 13, we have

−1 −1 d ≤ k(I + Gn) k ≤ s1(I + Gn) = inf {kI + Gn − F k : codim F ≥ 1} ( l ) X ≤ inf kI + G − V I − P V − F k : codim F ≥ β + 1 n nxi Im Wxi (An) −nxi i=1

≤ inf {kAnCn − F k : codim F ≥ β + 1} = sβ+1(AnCn)

≤ kCnksβ+1(An) ≤ Csβ+1(An), with C = sup kCnk. c) To simplify the notation deﬁne

m m X X α(An) := dim ker Wxi (An) and β(An) := dim coker Wxi (An) . i=1 i=1

From equation (26), we have, for i = 1, ..., m, that Wxi (Bn)Wxi (An) = I + Ti and therefore ind (Wxi (Bn) Wxi (An)) = 0. Thus, m X 0 = ind (Wxi (Bn) Wxi (An))(28) i=1 m X = (ind (Wxi (Bn)) + ind (Wxi (An))) i=1

= α(Bn) − β(Bn) + α(An) − β(An)

Since An is Fredholm of index zero, for n large enough, we deduce from Equation (26) that Bn has also index zero for n large enough. Thus from a) and b) it follows that

β(An) ≥ α(An) and β(Bn) ≥ α(Bn).

But, this together with Equation (28) implies that α(An) = β(An) and α(Bn) = β(Bn). Spectral Approximation of Convolution Type Operators 23

4.4. Description of Fredholm sequences in E. In this section we characterize the Fredholm sequences (An) ∈ E, and prove that the main theorem 5 can be applied provided Wx(An) is Fredholm for every x ∈ ∂Ω ∪ {0}.

Lemma 1. Let A0 be the algebra deﬁned in (3) and M be the smallest closed ideal 1 2 of A0 generated by all operators C(a), with a = F (u) and u ∈ L (R ). The algebra A0 decomposes as 2 A0 = alg{C (a) , fI : f ∈ C(R )} = fI+˙ M

Proof. Clearly the second algebra is dense in A0, thus it remains to check that it is closed. We claim that kfIk ≤ kfI + Mk for every M ∈ M and f ∈ C(R2). Given > 0, there exists a bounded set U ⊂ R2 such that

(29) kfIkL(Lp) − ≤ kfχU IkL(Lp) = kfχU I + KkL(Lp)/K. The last equality is a standard result valid for every bounded function on a compact 1 2 set. Since C(a)χU I, with a = F (u) and u ∈ L (R ) is a compact operator for every bounded set U, ([RRS], Theorem 3.2.2), it follows that MχU I is a compact operator for every M ∈ M. Thus,

(30) kfχU I + KkL(Lp)/K ≤ k(f + M)χU IkL(Lp) ≤ kf + MkL(Lp) According to inequalities (29) and (30), and assuming that is arbitrary, we get the claim. Now, if (fnI + Mn) is a Cauchy sequence, then (fnI) is also a Cauchy 2 and therefore (fnI) converges to some f ∈ C(R ). We, then also deduce that (Mn) is Cauchy and, due to the closedness of M, it converges to some M ∈ M. Thus lim (fnI + Mn) = fI + M. The next proposition shows, in conjugation with Theorem 5, that every Fredholm sequence from E has the splitting property.

Proposition 14. If (An) ∈ E is a Fredholm sequence then, for n large enough, the operators An are Fredholm with index zero .

Proof. Observe that if (An) is a Fredholm sequence then for n large enough the operators An are Fredholm. Let us ﬁrst prove the statement for the non-closed (but dense) subalgebra E0 of E generated by the same generators of E. Due to lemma 1 a sequence in E0 is the product of sequences of the type m X (31) An = χnΩ(f + M)χnΩI + λ(1 − χnΩ)I + Gn + Vnxi TiV−nxi i=1 p with M ∈ M,(Gn) ∈ G and Ti ∈ K(L ). Suppose (An) is a Fredholm sequence, then (Bn) = (χnΩ(f + M)χnΩI + λ(1 − χnΩ)I) is also a Fredholm sequence and therefore for n large enough Bn is a Fredholm operator. Once χnΩMχnΩI is a compact operator (see [RRS], Theorem 3.2.2) , it is clear that χnΩfχnΩI + λ(1 − χnΩ)I is a Fredholm operator too for n large enough. Moreover, since it is a multiplication by a piecewise continuous function, it is also invertible. We can look to An as the sum of χnΩfχnΩI + λ(1 − χnΩ)I + Gn with the compact operators χnΩMχnΩI + Pm i=1 Vnxi TiV−nxi . Since (Gn) ∈ G, there is n0 such that 1 1 kGnk < max , , n > n0. λ infx∈K0 |f(x)| Spectral Approximation of Convolution Type Operators 24

Thus, χnΩfI + λ(1 − χnΩ)I + Gn is invertible for every n > n0 , because 1 kGnk < −1 . k((χnΩfI + λ(1 − χnΩ)I) )k

Therefore An have index zero for n large enough. Clearly, the product of sequences of the type (31) (whose elements have index zero for n large enough), has index zero for n large enough and form a dense set in E. It remains to prove the statement for the whole algebra E. If (An) is a Fredholm sequence then there exists (Bn) ∈ B such that m X p AnBn = I + Vnxi TiV−nxi + Gn,Ti ∈ K(L ) and k(Gn)k → 0. i=1

It is clear (Bn) is a Fredholm sequence. Since the set of all Fredholm sequences is an 1 open set, there exists a Fredholm sequence (Cn) ∈ E0 such that k(Bn)−(Cn)k < 2M , where M = k(An)k. Hence, m X (32) AnCn = AnBn − An(Bn − Cn) = I + Vnxi TiV−nxi + Gn − An(Bn − Cn) i=1 1 1 with k(An)((Bn) − (Cn))k < 2 . Choosing n large enough, so that kGnk < 2 , this implies, for n large enough, I +Gn −An(Bn −Cn) is invertible and therefore the right hand side of (32) is Fredholm of index zero. Since Cn is also a Fredholm operator of index zero for n large enough, because (Cn) ∈ E0, the same is true for An. In order to characterize the Fredholm sequences in terms of the limit operators, D Wx(An), we will use localization on the algebra B = B/D (observe that (An) is D Fredholm sequence if (An)+D is invertible in B ). The procedure is in parts similar to the study of the invertibility on BG. 2 π p 2 π Let, for x ∈ R , Wx : B → L(L (R ))/K be the homomorphism Wx (An) = p 2 p 2 π(Wx(An)) where π : L(L (R )) → L(L (R ))/K is the canonical quotient map. π Wx maps D onto zero, thus the related quotient map π D p 2 Wfx : B → L(L (R ))/K, is a well deﬁned homomorphism. n o D 2 ∗ Proposition 15. The set C = (ϕnI) + D : ϕ ∈ C(R ) is a C central subalgebra of BD isometrically isomorphic to C(R2). Proof. The statement follows by applying similar arguments, as in the proof of Proposition 2, to the map: π D p 2 Wfx : C → L(L (R ))/K (ϕnI) + D 7→ ϕ(x)I + K. 2 D D For x ∈ R , let Jx be the smallest closed ideal of B containing the maximal D D D D ideal {(ϕnI) + D : ϕ(x) = 0} of C , and let ψx : B → B /Jx the the canonical π D quotient map. It is easy to check that Wfx (Jx ) = 0 and so π D D p 2 wx : B /Jx → L(L (R ))/K D (An + D) + Jx 7→ (s-limV−nxAnVnx) + K. Spectral Approximation of Convolution Type Operators 25 is a well deﬁned homomorphism.

2 Proposition 16. Let a ∈ W (R ) and A0 and A00 be the algebras deﬁned in (3) and in (7), respectively. D π p 2 i) The local algebra ψx(E ) is isomorphic to Lx, the Banach subalgebra of L(L (R ))/K given by π {A + K : A ∈ A00} if x ∈ intΩ and x 6= 0 Lx := {A + K : A ∈ A0} if x = 0 and 0 ∈ intΩ D 2 ii) The local algebra ψx(E ) is isomorphic to CI, if x ∈ R \ Ω. π Proof. Using the same arguments as in the proof of Proposition 5 we show that, wx D π D restricted to ψx E is an isomorphism onto the algebra Lx if x ∈ intΩ and ψx(E ) is isomorphic to CI, if x ∈ R2 \ Ω. For the boundary points the related considerations are more complicated. They are based on the following well known Lemma. Lemma 2. Let A be a Banach algebra and I ⊂ J two closed ideals of A. Then the Banach algebra A/J is isomorphic to (A/I)/(J/I).

Proposition 17. Consider x ∈ ∂Ω. Let A0 and A00 be the algebras deﬁned in (3) and in (7), respectively. D π p 2 (i) The local algebra ψx(E ) is isomorphic to Lx, the Banach subalgebra of L(L (R ))/K given by π 0 0 0 Lx := B + K : B ∈ alg χKx AχKx I + 1 − χKx I with A ∈ A00 if x ∈ ∂Ω\{0} and A ∈ A0 if x = 0 and 0 ∈ ∂Ω. (ii) Let (An) ∈ E. The coset ψx(An) is invertible if and only if Wx(An) is Fred- holm.

G G Proof. (i) Let D , Dx be the image of the ideals D and Dx = {(VnxTV−nx) + (Gn)) : G T ∈ K, (Gn) ∈ G} in the algebra B , respectively. Let further, Jx be the ideal associated to the localization in BG and J G be the sum of the last two. More Dx precisely:

G nX o D = cl (Vnxi TiV−nxi ) + G : Ti ∈ K G Dx = {(VnxTV−nx) + G : T ∈ K} ( m ) X j j Jx = cl AnϕnI + G : ϕj(x) = 0 j=1 J G = J + DG Dx x x G where cl denotes the closure in B . The homomorphism Wfx deﬁned in (6) satisﬁes,

G Wfx (Jx) = 0 and Wfx Dx = K, thus J G is a closed ideal. Moreover, DG ⊂ J G , because DG ⊂ J for every y 6= x Dx Dx y x (see Proposition 4-iii)). G G Recalling the homomorphism Heρ : BW → Bρ(W ) (deﬁned in 11), we know that G G G G G G Heρ(Jx ∩ BW ) = Jx ∩ Bρ(W ) and Heρ(Dx ∩ BW ) = Dx ∩ Bρ(W ) (due to (13)). Thus, Spectral Approximation of Convolution Type Operators 26

H (J G ∩ BG ) = J G ∩ BG . So, eρ Dx W Dx ρ(W )

h : BG/J G → BG/J G eρ Dx Dx is a well deﬁned isomorphism which maps (A + G) + J G :(A ) ∈ E n Dx n onto (A + G) + J G :(A ) ∈ F . n Dx n x We claim BG/J G ∼ BD/J D. From Lemma (2), we have Dx = x BG/J G ∼ (BG/DG)/ J G /DG . Dx = Dx But,

J G /DG = (A + G) + D :(A + G) ∈ J + DG Dx n n x x = {(An + G) + D :(An + G) ∈ Jx} ( m ) ∼ X j j D = cl AnϕnI + D : ϕj(x) = 0 = Jx j=1

G G ∼ D and B /D = B because G ⊂ D, thus we get the claim. So one has, via ehρ, that the D D algebras ψx E and ψx Fx are isomorphic. Now, following the same arguments π D as in the proof of Proposition 9, we have that wx maps ψx Fx isomorphically to p 2 π {A + K : A ∈ Lx} ⊂ L(L (R ))/K , which is the algebra Lx. (ii) Let x ∈ ∂Ω\{0}. Suppose ψx(An) is invertible. It is easily seen, using the same arguments to obtain equation (16), that we have the analogous equation

ehρ(ψx(An)) = ψx(VnxWx(An)V−nx), for every (An) ∈ E. From the invertibility of ψx(An), we get that ψx(VnxWx(An)V−nx) π is invertible, and so wx (ψx(VnxWx(An)V−nx)) = Wx(An)+K is also invertible , which means that Wx(An) is Fredholm. Suppose now Wx(An) is Fredholm, i.e there exist p 2 B ∈ L(L (R )) and T1,T2 compact operators such that Wx(An)B = I + T1 and BWx(An) = I + T2. Applying the shifts operators to both sides of the equations, we obtain VnxWx(An)V−nxVnxBV−nx = I + VnxT1V−nx and VnxBV−nxVnxWx(An)V−nx = I + VnxT2V−nx . The sequence VnxWx(An)V−nx is in Fx and using similar arguments as in Theorem 1 we prove that (VnxBV−nx) is in B. Thus, (VnxWx(An)V−nx) is a −1 Fredholm sequence in B and therefore the coset ψx(An) = ehρ (ψx(VnxWx(An)V−nx)) D D is invertible in B /Jx . If x = 0 and 0 ∈ ∂Ω, the proof can be carried out similarly, and it is even simpler because in this case ehρ is the identity. Now applying Allan-Douglas principle together with Propositions 16 and 17 and proceeding as in the proof of Theorem 2 we get the main result of this subsection.

Theorem 6. A sequence (An) ∈ E is Fredholm if and only if Wx(An) is Fredholm for every x ∈ ∂Ω ∪ {0} . Spectral Approximation of Convolution Type Operators 27

4.5. Index and kernel dimension formula for convolution type operators. Putting together Theorem 5, Proposition 14 and Theorem 6, we obtain the last of our main results

Theorem 7. Let (An) ∈ E. Suppose Wx (An) is Fredholm for every x ∈ ∂Ω ∪ {0}. Then the number of non-invertible operators among the Wx (An) is ﬁnite, and (An) has the α-splitting property, with X α = dim ker Wx (An) , x∈∂Ω∪{0} and X ind Wx (An) = 0. x∈∂Ω∪{0} The following Corollaries show that, for special choices of the set Ω and the sequence (An), Theorem 7 allows us to obtain the index and an asymptotic kernel dimension formula for some Fredholm convolution type operators. The next Corol- lary generalizes the Simonenko’s result mentioned in the introduction.

Pm 2 2 Corollary 4. Let A = i=1 fiC(ai), with ai ∈ W (R ) and fi ∈ C(R ), i = 1...m. Suppose Ω ⊂ K0, 0 ∈ Ω and ∂Ω is a smooth set except at zero. Let, further

Ae = χK0 AχK0 I + (1 − χK0 ) I and An = χnΩAχnΩ + (1 − χnΩ)I. The operator Ae is Fredholm if and only if (An) is a Fredholm sequence. In this case we have, ind(Ae) = 0 and (An) has the α-splitting property with α = dim ker Ae.

Proof. Assume that Ae is Fredholm. Then it is clear that the limit operators Wx(Ae) 2 are invertible for every x ∈ R . Now, let x ∈ ∂Ω\{0}. Due to Ω ⊂ K0 either x ∈ ∂K0 or x ∈ intK0. If x ∈ ∂K0 then Wx(An) = Wx(Ae) and therefore Wx(An) is invertible. If x ∈ intK0 then, from Propositions 1 and 3, m ! X Wx(A) = Wx(Ae) = C fi(x∞)ai i=1 0 0 0 and Wx(An) = χKx Wx(Ae)χKx I + 1 − χKx I. This last one is a convolution op- 0 2 erator on the half-space Kx with symbol in the Wiener algebra W (R ) . Thus, it is invertible if Wx(Ae) is invertible [GG]. The statement now follows from the application of Theorems 6 and 7. Pm 2 2 Corollary 5. Let A = i=1 fiC(ai), with ai ∈ W (R ) and fi ∈ C(R ), i = 1...m. Suppose Ω is the closed unit disc and An = χnΩAχnΩ + (1 − χnΩ)I. The operator A is Fredholm is and only if (An) is a Fredholm sequence. In this case we have, indA = 0 and (An) has the α-splitting property with α = dim ker A. Proof. Using the same arguments as in the proof of Corollary 4, we obtain that Wx(An) is invertible for every x ∈ ∂Ω assuming that A is Fredholm. As a consequence of Corollaries 4 and 5 we have the following index result. Spectral Approximation of Convolution Type Operators 28

Corollary 6. Let K be a cone at the origin and A ∈ A0, where A0 is the algebra deﬁned in (3). (i) If χK AχK I + (1 − χK ) I is Fredholm then its index is zero. (ii) If A is Fredholm then its index is zero.

Proof. The operator Ae = χK AχK I + (1 − χK ) I can be approximated by operators of the form m ! X (33) χK fiC(ai) χK I + (1 − χK )I, i=1 2 2 with ai ∈ W (R ) and fi ∈ C(R ), i = 1, ..., m. Suppose Ae = lim Ak with Ak of the type (33). Since Ae is Fredholm then for k large enough Ak is also Fredholm and has index zero due to Corollary 4. Notice that we can apply this corollary, because for every cone K at the origin there exits a set Ω with the desired conditions such that K0 = K. Finally, from the stability of the index we obtain indAe = 0. The statement (ii) is proved analogous to (i). Notice that for Fredholm operators belonging to the higher dimensional analogue of A0, V. Semenjuta [Se] obtained a formula for the index in terms of the degree of a function associated to A ∈ A0 , which is by no means trivial to compute in the general case. When the dimension is two the second claim of Corollary 6 shows that the degree of this function is zero. Acknowledgement. This work was partially supported by FCT (Fundao para a Cin- cia e a Tecnologia) through the project FEDER/POCTI/MAT/59972/2004. The authors thank the referee for carefully reading the manuscript and useful remarks.

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