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

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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 ap- proximate 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.
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