Collocation methods for systems of Cauchy singular integral equations on an interval

Peter Junghanns, Steffen Roch, Bernd Silbermann

Dedicated to Erhard Meister on the occasion of his seventieth birthday.

Abstract

Necessary and sufficient conditions for the stability of collocation methods with respect to Chebyshev nodes of first and second kind for Cauchy singular integral equations with piecewise continuous coefficients, given on an interval, in the scalar and in the system case are established. Moreover, the behaviour of the singular values of the matrices of the discrete equations is considered.

Keywords: Cauchy singular integral equation; collocation method; stability; Banach algebra techniques; singular value distribution

AMS classification: 45E05; 45F15; 45G05; 45L05; 45L10; 65R20

1 Introduction

Recently a collocation method, which is based on the Chebyshev nodes of second kind as collocation points and on approximating the solution by polynomials multiplied with the Chebyshev weight of second kind, was studied for both linear and nonlinear Cauchy singular integral equations (CSIE’s) on the interval [−1, 1] (see [11, 12, 22] for the linear case and [9] for the nonlinear case). There are several reasons for choosing Chebyshev nodes as collocation points independently from the asymptotic of the solution of the CSIE. At first we get a very cheap preprocessing for the construction of the matrix of the discretized equation, which is especially important in case of approximating the solution of a nonlinear CSIE by a sequence of solutions of linear equations (cf. [9]). A second reason is the possibility to apply such collocation methods to systems of CSIE’s, which is, in some sense, the main topic of the present paper. Indeed, in [11] there are only given necessary and sufficient conditions for the stability of the mentioned collocation method in the case of scalar CSIE’s of the form

a(x)u(x) + \frac{b(x)}{πi} ∫_{−1}^{1} \frac{u(y)}{y − x} dy = f(x) , −1 < x < 1 , (1.1)

where a and b are given piecewise continuous functions and the equation is considered in an appropriate weighted L^{2}-space L_{σ}^{2} . In [12, 22] the system case could only be investigated under additional conditions on the coefficients of the singular integral operator. In the present paper we study a more general situation, namely we give necessary and sufficient conditions for the stability of operator sequences {A_{n}} belonging to a C^{∗}-algebra A , which is generated by the sequences of the collocation method for equations of type (1.1). These stability conditions can be formulated in the following way. There exist *-homomorphisms W : A ⟶ L(L_{σ}^{2}) , W̃ : A ⟶ L(L_{σ}^{2}) and η_{±} : A ⟶ L(ℓ^{2}) such that, in case of collocation w.r.t. Chebyshev nodes of second kind, a sequence {A_{n}} ∈ A is stable if and only if the operators W {A_{n}} , W̃ {A_{n}} , and η_{±} {A_{n}} are invertible. In case of collocation w.r.t. Chebyshev nodes of first kind the invertibility of W {A_{n}} and W̃ {A_{n}} is necessary and sufficient for the stability of {A_{n}} ∈ A . It is important that such stability results for sequences belonging to an algebra A can be extended to the case of systems of CSIE’s.

The paper is organized as follows. In Section 2 the collocation method is described, where we also consider Chebyshev nodes of first kind as collocation points. In Section 3 some basic facts are collected and the existence of several strong limits of the involved operator sequences is established. The main result is proved in Section 4 using localization principles in C^{∗}-algebras. In the opinion of the authors it seems to be surprising that the stability conditions in the two cases of Chebyshev nodes of first and second kind are very different. The results of Section 4 are used in Section 5 to describe the behaviour of the smallest singular values of the operator sequences of the collocation method w.r.t. the Chebyshev nodes of second kind. The last Section 6 is dedicated to the very technical proof of a lemma on the local spectrum of the sequence of the collocation method in case of the Chebyshev nodes of first kind.

2 A polynomial collocation method

Let σ(x) = (1 − x^{2})^{− \frac{1}{2}} and φ(x) = (1 − x^{2})^{\frac{1}{2}} denote the Chebyshev weights of first and second kind on the interval (−1, 1), respectively, and let L_{σ}^{2} refer to the Hilbert space of all w.r.t. σ on (−1, 1) square integrable functions, equipped with inner product and norm

⟨u, v⟩_{σ} = ∫_{−1}^{1} u(x)\overline{v(x)}σ(x) dx and ∥u∥_{σ} := \sqrt{⟨u, u⟩_{σ}}.

For ω ∈ {σ, φ} and n ≥ 0, let p_{n}^{ω} stand for the w.r.t. ω orthonormal polynomial of degree n and abbreviate p_{n}^{σ} and p_{n}^{φ} to T_{n} and U_{n}, respectively. It is well known that

T_{0}(x) = \frac{1}{\sqrt{π}} , T_{n}(cos s) = \sqrt{\frac{2}{π}} cos ns , n ≥ 1, s ∈ (0, π) ,

and

U_{n}(cos s) = \sqrt{\frac{2}{π}} \frac{sin(n + 1)s}{sin s} , n ≥ 0, s ∈ (0, π) .

Further define weighted polynomials ũ_{n} := φU_{n}. Both {T_{n}}_{n=0}^{∞} and {ũ_{n}}_{n=0}^{∞} form an orthonormal basis in L_{σ}^{2} . The zeros of p_{n}^{ω} are known to be

x_{jn}^{σ} = cos \frac{2j − 1}{2n}π and x_{jn}^{φ} = cos \frac{jπ}{n + 1} where j = 1, …, n.

Further, the Lagrange interpolation operator L_{n}^{ω} acts on a function f : (−1, 1) ⟶ ℂ by

L_{n}^{ω}f = ∑_{j=1}^{n} f(x_{jn}^{ω})ℓ_{jn}^{ω} , ℓ_{jn}^{ω}(x) = ∏_{k=1,k≠j}^{n} \frac{x − x_{kn}^{ω}}{x_{jn}^{ω} − x_{kn}^{ω}} = \frac{p_{n}^{ω}(x)}{(x − x_{jn}^{ω})(p_{n}^{ω})′(x_{jn}^{ω})} .

A function a : [−1, 1] ⟶ ℂ is called piecewise continuous if it is continuous at ±1 and if the one-sided limits a(x ± 0) exist and satisfy a(x − 0) = a(x) for all x ∈ (−1, 1) . The set of all piecewise continuous functions on [−1, 1] is denoted by PC = PC[−1, 1] . For given functions a, b ∈ PC and f ∈ L_{σ}^{2}, consider the Cauchy singular integral equation

a(x)u(x) + \frac{b(x)}{πi} ∫_{−1}^{1} \frac{u(y)}{y − x} dy = f(x) , −1 < x < 1 . (2.1)

Both the Cauchy singular integral operator

S : L_{σ}^{2} ⟶ L_{σ}^{2} , u ↦ \frac{1}{πi} ∫_{−1}^{1} \frac{u(y)}{y − ·} dy

and the multiplication operators aI : L_{σ}^{2} ⟶ L_{σ}^{2}, u ↦ au, belong to the algebra L(L_{σ}^{2}) of all linear and bounded operators in L_{σ}^{2} which justifies to consider equation (2.1) on this space. For the approximate solution of (2.1), we look for a function u ∈ L_{σ}^{2} of the form

u_{n} = ∑_{k=1}^{n} ξ_{kn}ũ_{k} , ξ_{n} = [ξ_{kn}]_{k=1}^{n} ∈ ℂ^{n}

which satisfies the collocation system

a(x_{jn}^{ω})u_{n}(x_{jn}^{ω}) + \frac{b(x_{jn}^{ω})}{πi} ∫_{−1}^{1} \frac{u(y)}{y − x_{jn}^{ω}} dy = f(x_{jn}^{ω}) , j = 1, …, n . (2.2)

If we introduce Fourier projections

P_{n} : L_{σ}^{2} ⟶ L_{σ}^{2} , u ↦ ∑_{k=0}^{n−1}⟨u, ũ_{k}⟩_{σ}ũ_{k}

and weighted interpolation operators M_{n}^{ω} := φL_{n}^{ω}φ^{−1}, then the collocation system (2.2) can be rewritten as an operator equation

M_{n}^{ω}(aI + bS)P_{n}u_{n} = M_{n}^{ω}f , u_{n} ∈ im P_{n} . (2.3)

The reason for using M_{n}^{ω} instead of L_{n}^{ω} is that the range of M_{n}^{ω} coincides with that one of the Fourier projection P_{n}. Our main concern is the stability of the sequence {A_{n}} with A_{n} = M_{n}^{ω}AP_{n} and A = aI + bS. Recall that a sequence {A_{n}} is stable if there exists an n_{0} such that the operators A_{n} : im P_{n} ⟶ im P_{n} are invertible for n ≥ n_{0} and that their inverses A_{n}^{−1} are uniformly bounded:

sup_{n≥n_{0}} ∥A_{n}^{−1}P_{n}∥_{L_{σ}^{2}→L_{σ}^{2}} < ∞ .

If the sequence {A_{n}} is stable and if u^{∗} ∈ L_{σ}^{2} and u_{n}^{∗} ∈ im P_{n} are the solutions of (2.1) and (2.3), respectively, then the estimate

∥P_{n}u^{∗} − u_{n}^{∗}∥_{σ} ≤ ∥A_{n}^{−1}P_{n}∥_{L_{σ}^{2}→L_{σ}^{2}} ( ∥A_{n}P_{n}u^{∗} − Au^{∗}∥_{σ} + ∥f − M_{n}^{ω}f∥_{σ} )

shows that u_{n}^{∗} converges to u^{∗} in the norm of L_{σ}^{2} if the method (2.3) is consistent, i.e. if A_{n}P_{n} ⟶ A (strong convergence) and if M_{n}^{ω}f ⟶ f (convergence in L_{σ}^{2}). The stability result for the collocation method (2.3), which is a conclusion of Theorem 4.8, reads as follows.

Theorem 2.1 Let a, b ∈ PC. Then the sequence {M_{n}^{σ}(aI + bS)P_{n}} is stable if and only if the operator aI + bS is invertible on L_{σ}^{2}, and the sequence {M_{n}^{φ}(aI + bS)P_{n}} is stable if and only if the operators aI + bS and aI − bS are invertible on L_{σ}^{2}

Our main tools for studying the stability of an approximation sequence are the translation of the stability problem into an invertibility problem in a suitable C^{∗}-algebra and the application of local principles (see, for example, [7, Chapter 3] and [16, Chapter 7]).

For the algebraization of the stability problem, let F denote the C^{∗}-algebra of all bounded sequences {A_{n}} of linear operators A_{n} : im P_{n} ⟶ im P_{n}, provided with the supremum norm ∥{A_{n}}∥_{F} := sup_{n≥1} ∥A_{n}∥_{L_{σ}^{2}→L_{σ}^{2}} and with operations {A_{n}} + {B_{n}} := {A_{n} + B_{n}} , {A_{n}}{B_{n}} := {A_{n}B_{n}} , and {A_{n}}^{∗} := {A_{n}^{∗}} . Further, let N be the two-sided closed ideal of F consisting of all sequences {C_{n}} ∈ F such that lim_{n→∞} ∥C_{n}P_{n}∥_{L_{σ}^{2}→L_{σ}^{2}} = 0 . Then a simple Neumann series argument shows that the sequence {A_{n}} ∈ F is stable if and only if the coset {A_{n}} + N is invertible in the quotient algebra F/N . In the case at hand, it is more convenient to work in a subalgebra of F rather than in F itself (the main point being that the ideal N proves to be too small for the sake of localization). To introduce this subalgebra, define operators

W_{n} : L_{σ}^{2} ⟶ L_{σ}^{2}, u ↦ ∑_{j=1}^{n}⟨u, ũ_{n−1−j}⟩_{σ}ũ_{j},

and consider the set F^{W} of all sequences {A_{n}} ∈ F , for which the strong limits

W {A_{n}} := s-lim A_{n}P_{n} , (W {A_{n}})^{∗} = s-lim A_{n}^{∗}P_{n} , (2.4)

and

W̃{A_{n}} := s-lim W_{n}A_{n}W_{n} , (W̃{A_{n}})^{∗} = s-lim (W_{n}A_{n}W_{n})^{∗}P_{n} (2.5)

exist. Furthermore, let J refer to the collection of all sequences {A_{n}} of the form

A_{n} = P_{n}K_{1}P_{n} + W_{n}K_{2}W_{n} + C_{n} with K_{j} ∈ K(L_{σ}^{2}) , {C_{n}} ∈ N ,

where K(L_{σ}^{2}) ⊂ L(L_{σ}^{2}) stands for the ideal of all compact operators.

Lemma 2.2 (a) F^{W} is a C^{∗}-subalgebra of F, and J is a closed two-sided ideal of F^{W}.

(b) A sequence {A_{n}} ∈ F^{W} is stable if and only if the operators W {A_{n}}, W̃{A_{n}} : L_{σ}^{2} ⟶ L_{σ}^{2} and the coset {A_{n}} + J ∈ F^{W} /J are invertible.

The proof is not hard and can be found in [21], Prop. 3, or in [16], Theorem 7.7. It rests essentially on the weak convergence of the sequence {W_{n}} to zero.

3 Consistency of the method

The goal of this section is to show that the method (2.3) is consistent with equation (2.1) in the sense that M_{n}^{ω}f ⟶ f in L_{σ}^{2} under suitable restrictions for f and that A_{n}P_{n} ⟶ A strongly. The proof of the approximation properties of the interpolation operators M_{n}^{ω} is based on the following auxiliary results.

Lemma 3.1 ([14], Theor. 9.25) Let μ, ν be classical Jacobi weights with μν ∈ L^{1}(−1, 1) and let j ∈ ℕ be fixed. Then, for each polynomial q with deg q ≤ jn ,

∑_{k=1}^{n} λ_{kn}^{μ}|q(x_{kn}^{μ})|ν(x_{kn}^{μ}) ≤ const ∫_{−1}^{1} |q(x)|μ(x)ν(x) dx ,

where the constant does not depend on n and q and where x_{kn}^{μ} and λ_{kn}^{μ} = ∫_{−1}^{1} ℓ_{kn}^{μ}(x)μ(x) dx are the nodes and the Christoffel numbers of the Gaussian rule w.r.t. the weight μ , respectively.

Let Q_{n}^{μ} denote the Gaussian quadrature rule w.r.t. the weight μ,

Q_{n}^{μ}f = ∑_{k=1}^{n} λ_{kn}^{μ}f(x_{kn}^{μ}),

and write R = R(−1, 1) for the set of all functions f : (−1, 1) ⟶ ℂ , which are bounded and Riemann integrable on each interval [α, β] ⊂ (−1, 1) .

Lemma 3.2 ([5], Satz III.1.6b and Satz III.2.1) Let μ(x) = (1 − x)^{γ}(1 + x)^{δ} with γ, δ > −1. If f ∈ R satisfies

|f(x)| ≤ const (1 − x)^{ε−1−γ}(1 + x)^{ε−1−δ} , −1 < x < 1 ,

for some ε > 0 , then lim_{n→∞} Q_{n}^{μ}f = ∫_{−1}^{1} f(x)μ(x) dx . If even

|f(x)| ≤ const (1 − x)^{ε− \frac{1+γ}{2}} (1 + x)^{ε− \frac{1+δ}{2}} , −1 < x < 1 ,

then lim_{n→∞} ∥f − L_{n}^{μ}f∥_{μ} = 0 .

Corollary 3.3 Let f ∈ R and |f(x)| ≤ const (1 − x^{2})^{ε− \frac{1}{4}} , −1 < x < 1 , for some ε > 0. Then M_{n}^{ω}f ⟶ f in L_{σ}^{2} for ω = φ and ω = σ.

Proof. Since ∥f − M_{n}^{ω}f∥_{σ} = ∥φ^{−1}f − L_{n}^{ω}φ^{−1}f∥_{φ}, we can immediately apply the second assertion of Lemma 3.2 to get the assertion in case ω = φ. To consider the case ω = σ, introduce the quadrature rule

Q_{n}f = ∫_{−1}^{1}(L_{n}^{σ}f)(x)φ(x) dx = ∑_{k=1}^{n} σ_{kn}f(x_{kn}^{σ}) ,

where

σ_{kn} = ∫_{−1}^{1} \frac{T_{n}(x)}{x − x_{kn}^{σ}} \frac{φ(x)}{T′_{n}(x_{kn}^{σ})} dx = ∫_{−1}^{1} \frac{T_{n}(x)(1 − x^{2})σ(x)}{(x − x_{kn}^{σ})T′_{n}(x_{kn}^{σ})} dx = \frac{π [1 − (x_{kn}^{σ})^{2}]}{n}

for n > 2. Consequently,

Q_{n}f = \frac{π}{n} ∑_{k=1}^{n} [1 − (x_{kn}^{σ})^{2}] f(x_{kn}^{σ}) .

Since the nodes x_{kn}^{σ} of the quadrature rule Q_{n} are the zeros of 2T_{n} = U_{n} − U_{n−2}, the estimate

∫_{−1}^{1} |(L_{n}^{σ}f)(x)|^{2} φ(x) dx ≤ 2 Q_{n}|f|^{2}

holds true (see [5, Hilfssatz 2.4, §III.2]). As an immediate consequence we obtain

∥M_{n}^{σ}f∥_{σ}^{2} = ∥L_{n}^{σ}φ^{−1}f∥_{φ}^{2} ≤ \frac{2 π}{n} ∑_{k=1}^{n} |f(x_{kn}^{σ})|^{2} = 2 Q_{n}^{σ}|f|^{2} . (3.1)

Now let δ > 0 be arbitrary and p be a polynomial such that ∥φ p − f∥_{σ} < δ . For n > deg p we have ∥M_{n}^{σ}f − f∥_{σ}^{2} ≤ 2 (∥M_{n}^{σ}(φ p − f)∥_{σ}^{2} + ∥φ p − f∥_{σ}^{2}) . Since, in view of Lemma 3.2,

lim_{n→∞} Q_{n}^{σ}|φ p − f|^{2} = ∥φ p − f∥_{σ}^{2} , we get via (3.1) that lim sup_{n→∞} ∥M_{n}^{σ}f − f∥_{σ}^{2} < 6 δ^{2} , which

proves the assertion in second case, too.

The strong convergence of the sequence {M_{n}^{ω}(aI+bS)P_{n}} is part of the assertion of the following theorem. For the description of the occuring strong limits we need two further operators: the isometry

J_{σ} : L_{σ}^{2} ⟶ L_{σ}^{2} , u ⟶ ∑_{n=0}^{∞} γ_{n}⟨u, ũ_{n}⟩_{σ}T_{n} , (3.2)

where γ_{0} = \sqrt{2} and γ_{n} = 1 for n ≥ 1, and the shift operator

V : L_{σ}^{2} ⟶ L_{σ}^{2} , u ↦ ∑_{n=0}^{∞}⟨u, ũ_{n}⟩_{σ}ũ_{n+1} (3.3)

with its adjoint V^{∗} : L_{σ}^{2} ⟶ L_{σ}^{2}, u ↦ ∑_{n=0}^{∞}⟨u, ũ_{n+1}⟩_{σ}ũ_{n}.

Theorem 3.4 For a, b ∈ PC, the sequence {A_{n}^{ω}} := {M_{n}^{ω}(aI + bS)P_{n}} belongs to the algebra F^{W}. In particular, W {A_{n}^{ω}} = aI + bS and

W̃\{A_{n}^{ω}\} = \{ \begin{matrix}aI − bS & , & ω = φ , \\ J_{σ}^{−1}(aJ_{σ} + ibV^{∗}) & , & ω = σ . \end{matrix}

Proof. Step 1: Uniform boundedness. Let a, b ∈ PC and ω ∈ {φ, σ}. We have to verify the strong convergence of each of the sequences {M_{n}^{ω}(aI + bS)P_{n}}, {(M_{n}^{ω}(aI + bS)P_{n})^{∗}}, {W_{n}M_{n}^{ω}(aI + bS)W_{n}} and {(W_{n}M_{n}^{ω}(aI + bS)W_{n})^{∗}}. Let us start with showing the uniform boundedness of these sequences. Since ∥W_{n}∥ = 1, it is sufficient to prove the uniform boundedness of the sequences {M_{n}^{ω}(aI + bS)P_{n}}.

We write M_{n}^{ω}bSP_{n} as M_{n}^{ω}bP_{n}M_{n}^{ω}SP_{n} and consider first sequences of the form M_{n}^{ω}aP_{n} where a is an arbitrary function with ∥a∥_{∞} := sup {|a(x)| : x ∈ [−1, 1]} < ∞. Let u_{n} = φv_{n} ∈ im P_{n}. Then, using the algebraic accuracy of a Gaussian rule, we get the estimate

∥M_{n}^{φ}au_{n}∥_{σ}^{2} = ∥L_{n}^{φ}av_{n}∥_{φ}^{2} = Q_{n}^{φ}|av_{n}|^{2} ≤ ∥a∥_{∞}^{2} ∥v_{n}∥_{φ}^{2} = ∥a∥_{∞}^{2} ∥u_{n}∥_{σ}^{2} . (3.4)

In case ω = σ we apply Relation (3.1) and Lemma 3.1 with μ = σ and ν(x) = 1 − x^{2} to obtain

∥M_{n}^{σ}au_{n}∥_{σ}^{2} ≤ 2 ∥a∥_{∞} \frac{π}{n} ∑_{k=1}^{n} [1 − (x_{kn}^{φ})^{2}] |v_{n}(x_{kn}^{φ})|^{2} ≤ const ∥a∥_{∞}^{2} ∥v_{n}∥_{φ}^{2}

and thus

∥M_{n}^{σ}au_{n}∥_{σ} ≤ const ∥a∥_{∞} ∥u_{n}∥_{σ} , u_{n} ∈ im P_{n} , (3.5) where the constant does not depend on a , n , and u_{n} . For the uniform boundedness of the sequences {M_{n}^{ω}SP_{n}} we observe that

SφU_{n} = iT_{n+1} , n = 0, 1, 2, …, (3.6)

which shows that, for u_{n} ∈ im P_{n}, the function q_{n} := Su_{n} is a polynomial of degree not greater than n . Thus, Relation (3.1), Lemma 3.1, and the boundedness of S : L_{σ}^{2} ⟶ L_{σ}^{2} yield

∥M_{n}^{σ}Su_{n}∥_{σ}^{2} ≤ 2Q_{n}^{σ}|q_{n}|^{2} ≤ const ∥q_{n}∥_{σ}^{2} ≤ const ∥u_{n}∥_{σ}^{2} .

In case ω = φ we also use Lemma 3.1 to obtain

∥M_{n}^{φ}Su_{n}∥_{σ}^{2} = ∥L_{n}^{φ}φ^{−1}q_{n}∥_{φ}^{2} = ∑_{k=1}^{n} λ_{kn}^{φ}|q_{n}(x_{kn}^{φ})|^{2} [1 − (x_{kn}^{φ})^{2}]^{−1} ≤ const ∥q_{n}∥_{σ}^{2} .

This verifies the uniform boundedness of all sequences under consideration. Their strong convergence on L_{σ}^{2} follows once we have shown their convergence on all basis functions ũ_{n} of L_{σ}^{2}.

Step 2: Convergence of {M_{n}^{ω}(aI +bS)P_{n}}. It is an immediate consequence of Corollary 3.3 and of (3.6) that

lim_{n→∞} M_{n}^{ω}(aI + bS)P_{n}ũ_{m} = (aI + bS)ũ_{m} in L_{σ}^{2} for all m = 0, 1, 2, …

Step 3: Convergence of {(M_{n}^{ω}(aI + bS)P_{n})^{∗}}. The determination of the adjoint sequence is based upon a formula for the Fourier coefficients of the interpolating function M_{n}^{ω}f . For this goal, we write

M_{n}^{ω}f = ∑_{j=0}^{n−1} α_{jn}^{ω}(f)ũ_{j}

and get in case of ω = φ

α_{jn}^{φ}(f) = ⟨M_{n}^{φ}f, ũ_{j}⟩_{σ} = ⟨L_{n}^{φ}φ^{−1}f, U_{j}⟩_{φ} = \frac{π}{n + 1} ∑_{k=1}^{n} f(x_{kn}^{φ})ũ_{j}(x_{kn}^{φ}) , (3.7)

j = 0, …, n − 1 . In case of ω = σ we have for j = 0, …, n − 2

α_{jn}^{σ}(f) = ⟨M_{n}^{σ}f, ũ_{j}⟩_{σ} = ⟨L_{n}^{σ}φ^{−1}f, φ^{2}U_{j}⟩_{σ} = \frac{π}{n} ∑_{k=1}^{n} f(x_{kn}^{σ})ũ_{j}(x_{kn}^{σ}) .

For j = n − 1 we use the three-term recurrence relation

U_{k+1}(x) = 2xU_{k}(x) − U_{k−1}(x) , k = 1, 2, …, (3.8)

as well as the relation

T_{n+1}(x) = \frac{1}{2}[U_{n+1}(x) − U_{n−1}(x)] , n = 0, 1, 2, …, U_{−1} ≡ 0 , (3.9)

to obtain

(1 − x^{2})U_{n−1}(x) = \frac{1}{2} [T_{n−1}(x) − T_{n+1}(x)] . (3.10)

Consequently, with ψ(x) := x ,

α_{n−1,n}^{σ}(f) = ⟨L_{n}^{σ}φ^{−1}f, φ^{2}U_{j}⟩_{σ} = \frac{1}{2}⟨L_{n}^{σ}φ^{−1}f, T_{n−1}⟩_{σ}

= \frac{π}{2n} ∑_{k=1}^{n} \frac{f(x_{kn}^{σ})}{φ(x_{kn}^{σ})} T_{n−1}(x_{kn}^{σ}) = \frac{π}{2n} ∑_{k=1}^{n} f(x_{kn}^{σ})ũ_{n−1}(x_{kn}^{σ}) .

Thus,

α_{jn}^{σ}(f) = ε_{jn} \frac{π}{n} ∑_{k=1}^{n} f(x_{kn}^{σ})ũ_{j}(x_{kn}^{σ}) , j = 0, …, n − 1 , (3.11)

where ε_{jn} = 1 for j = 0, …, n − 2 and ε_{jn} = 1/2 for j = n − 1. As an immediate consequence of (3.7) we deduce for u, v ∈ L_{σ}^{2}

⟨M_{n}^{φ}aP_{n}u, v⟩_{σ} = ∑_{j=0}^{n−1} \frac{π}{n + 1} ∑_{k=1}^{n} a(x_{kn}^{φ}) ∑_{ℓ=0}^{n−1}⟨u, ũ_{ℓ}⟩_{σ}ũ_{ℓ}(x_{kn}^{φ})ũ_{j}(x_{kn}^{φ})\overline{⟨v, ũ_{j}⟩_{σ}}

= ∑_{ℓ=0}^{n−1} \overline{\frac{π}{n + 1} ∑_{k=1}^{n} \overline{a(x_{kn}^{φ})} ∑_{j=0}^{n−1}⟨v, ũ_{j}⟩_{σ}ũ_{j}(x_{kn}^{φ})ũ_{ℓ}(x_{kn}^{φ})}⟨u, ũ_{ℓ}⟩_{σ}

= ⟨u, M_{n}^{φ}\overline{a}P_{n}v⟩_{σ} .

Hence, the adjoint of M_{n}^{φ}aP_{n} : L_{σ}^{2} ⟶ L_{σ}^{2} is M_{n}^{φ}\overline{a}P_{n} : L_{σ}^{2} ⟶ L_{σ}^{2} . In case ω = σ, (3.11) implies

⟨M_{n}^{σ}aP_{n}u, v⟩_{σ} = ∑_{j=0}^{n−1} ε_{jn} \frac{π}{n} ∑_{k=1}^{n} a(x_{kn}^{σ}) ∑_{ℓ=0}^{n−1}⟨u, ũ_{ℓ}⟩_{σ}ũ_{ℓ}(x_{kn}^{σ})ũ_{j}(x_{kn}^{σ})\overline{⟨v, ũ_{j}⟩_{σ}}

= ∑_{ℓ=0}^{n−1} \overline{\frac{π}{n} ∑_{k=1}^{n} \overline{a(x_{kn}^{σ})} ∑_{j=0}^{n−1} ε_{jn}⟨v, ũ_{j}⟩_{σ}ũ_{j}(x_{kn}^{σ})ũ_{ℓ}(x_{kn}^{σ})}⟨u, ũ_{ℓ}⟩_{σ}

= ⟨u, (2P_{n} − P_{n−1})M_{n}^{φ}\overline{a}\frac{1}{2}(P_{n−1} + P_{n})v⟩_{σ} .

Thus,

(M_{n}^{φ}aP_{n})^{∗} = M_{n}^{φ}\overline{a}P_{n} and (M_{n}^{σ}aP_{n})^{∗} = (P_{n} − \frac{1}{2}P_{n−1})M_{n}^{σ}\overline{a}(P_{n−1} + P_{n}) , (3.12)

whence in both cases the strong convergence on L_{σ}^{2} of (M_{n}^{ω}aP_{n})^{∗} to \overline{a}I. For the determination of the adjoint operator of M_{n}^{ω}SP_{n} , we recall the Poincaré-Bertrand commutation formula (see [13, Chapter II, Theorem 4.4]): If ρ(x) = (1−x)^{α}(1+x)^{β} is a Jacobi weight with α, β ∈ (−1, 1) then, for u ∈ L_{ρ}^{2} and v ∈ L_{ρ^{−1}}^{2} ,

⟨Su, v⟩ = ⟨u, Sv⟩ , (3.13)

where ⟨., .⟩ refers to the unweighted L^{2}(−1, 1) inner product. Thus, the adjoint operator of S : L_{σ}^{2} ⟶ L_{σ}^{2} is φSφ^{−1}I : L_{σ}^{2} ⟶ L_{σ}^{2} . Taking into account that SP_{n}u is a polynomial of degree at most n due to (3.6), we conclude that, for all u, v ∈ L_{σ}^{2} ,

⟨M_{n}^{φ}SP_{n}u, v⟩_{σ} = ⟨L_{n}^{φ}φ^{−1}SP_{n}u, φ^{−1}P_{n}v⟩_{φ} = \frac{π}{n + 1} ∑_{k=1}^{n}(SP_{n}u)(x_{kn}^{φ})(P_{n}v)(x_{kn}^{φ})

= ⟨SP_{n}u, L_{n}^{φ}φ^{−2}P_{n}v⟩_{φ} = ⟨SP_{n}u, φM_{n}^{φ}φ^{−1}P_{n}v⟩_{σ}
= ⟨u, P_{n}φSM_{n}^{φ}φ^{−1}P_{n}v⟩_{σ} .

Analogously we get, for j = 0, …, n − 2 and u ∈ L_{σ}^{2} ,

⟨M_{n}^{σ}SP_{n}u, ũ_{j}⟩_{σ} = ⟨L_{n}^{σ}φ^{−1}SP_{n}u, φ^{2}U_{j}⟩_{σ} = \frac{π}{n} ∑_{k=1}^{n}(SP_{n}u)(x_{kn}^{σ})ũ_{j}(x_{kn}^{σ}) = ⟨SP_{n}u, L_{n}^{σ}ũ_{j}⟩_{σ}

= ⟨u, P_{n}φSφ^{−1}L_{n}^{σ}ũ_{j}⟩_{σ}

and, again using Relation (3.10),

⟨M_{n}^{σ}SP_{n}u, ũ_{n−1}⟩_{σ} = \frac{1}{2}⟨L_{n}^{σ}φ^{−1}SP_{n}u, ψU_{n−2} − U_{n−3}⟩_{σ} = \frac{π}{2n} ∑_{k=1}^{n}(SP_{n}u)(x_{kn}^{σ})ũ_{n−1}(x_{kn}^{σ})

= \frac{1}{2}⟨u, P_{n}φSφ^{−1}L_{n}^{σ}ũ_{n−1}⟩_{σ} .

Hence,

(M_{n}^{φ}SP_{n})^{∗} = P_{n}φSM_{n}^{φ}φ^{−1}P_{n} = M_{n}^{φ}φSM_{n}^{φ}φ^{−1}P_{n} (3.14)

and

(M_{n}^{σ}SP_{n})^{∗} = \frac{1}{2}P_{n}φSφ^{−1}L_{n}^{σ}(P_{n−1} + P_{n}) = \frac{1}{2}φSφ^{−1}L_{n}^{σ}(P_{n−1} + P_{n}) , (3.15)

where in (3.14) we took into account (3.6) and (3.9) and in (3.15) the relations

Sφ^{−1}T_{n} = −iU_{n−1} , n = 0, 1, 2, …, U_{−1} ≡ 0 . (3.16)

In combination with Lemma 3.2 and Corollary 3.3 it is clear now that the sequence {(M_{n}^{ω}(aI + bS)P_{n})^{∗}P_{n}\} converges strongly on L_{σ}^{2} to \overline{a}I + φSφ^{−1}\overline{b}I in both cases ω = σ and ω = φ.

Step 4: Convergence of {W_{n}M_{n}^{ω}(aI + bS)W_{n}}. We are going to verify the convergence of W_{n}M_{n}^{ω}aW_{n}ũ_{m} and W_{n}M_{n}^{ω}SW_{n}ũ_{m} for each fixed m ≥ 0. Let n > m. With the help of (3.7), the identity

ũ_{n−1−m}(x_{kn}^{φ}) = \sqrt{\frac{2}{π}} sin \frac{(n − m)kπ}{n + 1} = (−1)^{k+1}ũ_{m}(x_{kn}^{φ}) ,

and Corollary 3.3 we get

W_{n}M_{n}^{φ}aW_{n}ũ_{m} = ∑_{j=0}^{n−1} α_{n−1−j,n}^{φ}(aũ_{n−1−m})ũ_{j}

= ∑_{j=0}^{n−1} \frac{π}{n + 1} ∑_{k=1}^{n} a(x_{kn}^{φ})ũ_{n−1−m}(x_{kn}^{φ})ũ_{n−1−j}(x_{kn}^{φ})ũ_{j} = M_{n}^{φ}aũ_{m} .

10 Junghanns, Roch, Silbermann Consequently,

W_{n}M_{n}^{φ}aW_{n} = M_{n}^{φ}aP_{n} ⟶ aI in L_{σ}^{2} . (3.17) To describe the strong limit of W_{n}M_{n}^{σ}aW_{n}, we take into account that

ũ_{n−1−m}(x_{kn}^{σ}) = \sqrt{\frac{2}{π}} sin \frac{(n − m)(2k − 1)π}{2n} = \sqrt{\frac{2}{π}} sin \frac{2k − 1}{2}π cos \frac{m(2k − 1)π}{2n}

i.e.

ũ_{n−1−m}(x_{kn}^{σ}) = (−1)^{k+1}γ_{m}T_{m}(x_{kn}^{σ}) (3.18) and

L_{n}^{σ}f = ∑_{j=0}^{n−1} α̃_{jn}^{σ}(f)T_{j} with α̃_{jn}^{σ} = \frac{π}{n} ∑_{k=1}^{n} f(x_{kn}^{σ})T_{j}(x_{kn}^{σ}) .

Then, from (3.11) and Lemma 3.2, we conclude

W_{n}M_{n}^{σ}aW_{n}ũ_{m} = ∑_{j=0}^{n−1} α_{n−1−j,n}^{σ}(aũ_{n−1−m})ũ_{j}
= ∑_{j=0}^{n−1} ε_{n−1−j,n} \frac{π}{n} ∑_{k=1}^{n} a(x_{kn}^{σ})ũ_{n−1−m}(x_{kn}^{σ})ũ_{n−1−j}(x_{kn}^{σ})ũ_{j}
= ∑_{j=0}^{n−1} ε_{n−1−j,n} \frac{π}{n} ∑_{k=1}^{n} a(x_{kn}^{σ})γ_{m}T_{m}(x_{kn}^{σ})γ_{j}T_{j}(x_{kn}^{σ})ũ_{j}
= ∑_{j=0}^{n−1} \frac{π}{n} ∑_{k=1}^{n} a(x_{kn}^{σ})(J_{σ}ũ_{m})(x_{kn}^{σ})T_{j}(x_{kn}^{σ})J_{σ}^{−1}T_{j}
= J_{σ}^{−1}L_{n}^{σ}aJ_{σ}ũ_{m} ⟶ J_{σ}^{−1}aJ_{σ}ũ_{m} in L_{σ}^{2} ,

where J_{σ} is the isometry introduced in (3.2). Thus,

W_{n}M_{n}^{σ}aW_{n} = J_{σ}^{−1}L_{n}^{σ}aJ_{σ}P_{n} ⟶ J_{σ}^{−1}aJ_{σ} in L_{σ}^{2} . (3.19) For the strong convergence of the sequences related with the singular integral observe that, due to (3.6), for all n > max{m, k} ,

⟨W_{n}M_{n}^{φ}SW_{n}ũ_{m}, ũ_{k}⟩_{σ} = ⟨M_{n}^{φ}Sũ_{n−1−m}, ũ_{n−1−k}⟩_{σ} = i⟨L_{n}^{φ}φ^{−1}T_{n−m}, U_{n−1−k}⟩_{φ}
= \frac{2i}{n + 1} ∑_{j=1}^{n} cos \frac{(n − m)jπ}{n + 1} sin \frac{(n − k)jπ}{n + 1}
= −i⟨L_{n}^{φ}φ^{−1}T_{m+1}, U_{k}⟩_{φ} = −⟨M_{n}^{φ}SP_{n}ũ_{m}, ũ_{k}⟩_{σ} .

Hence, W_{n}M_{n}^{φ}SW_{n} = −M_{n}^{φ}SP_{n} ⟶ −S in L_{σ}^{2} . (3.20)

Further, the identities

J_{σ}(ũ_{n+2} − ũ_{n}) = γ_{n+2}T_{n+2} − γ_{n}T_{n} = −2φũ_{n} , n ≥ 0 , (3.21)

(3.6), (3.9), and (3.19) imply for n > m ≥ 1 , (in case n = m − 1 , note that L_{n}^{σ}T_{n} = 0)

W_{n}M_{n}^{σ}SW_{n}ũ_{m} = \frac{i}{2} W_{n}M_{n}^{σ}φ^{−1}W_{n}(ũ_{m−1} − ũ_{m+1}) = −\frac{i}{2}J_{σ}^{−1}L_{n}^{σ}φ^{−1}J_{σ}(ũ_{m+1} − ũ_{m−1})
= iJ_{σ}^{−1}L_{n}^{σ}ũ_{m−1} ⟶ iJ_{σ}^{−1}ũ_{m−1} in L_{σ}^{2} .

Obviously, W_{n}M_{n}^{σ}SW_{n}ũ_{0} = iW_{n}M_{n}^{σ}T_{n} = 0 . Hence, by means of the shift operator V introduced in (3.3) we can formulate the derived convergence result as follows:

W_{n}M_{n}^{σ}SW_{n} = iJ_{σ}^{−1}L_{n}^{σ}V^{∗}P_{n} ⟶ iJ_{σ}^{−1}V^{∗} in L_{σ}^{2} . (3.22)

Step 5: Convergence of {(W_{n}M_{n}^{ω}(aI + bS)W_{n})^{∗}}. In case ω = φ, the strong convergence of this sequence follows from (3.17), (3.20), (3.12) and (3.14), together with the outcome of step 3. In case ω = σ we have, in view of (3.19),

⟨W_{n}M_{n}^{σ}aW_{n}u, v⟩_{σ} = ⟨L_{n}^{σ}aJ_{σ}P_{n}u, J_{σ}^{−∗}P_{n}v⟩_{σ} = \frac{π}{n} ∑_{j=1}^{n} a(x_{jn}^{σ})(J_{σ}P_{n}u)(x_{jn}^{σ})\overline{(J_{σ}^{−∗}P_{n}v)(x_{jn}^{σ})}

= ⟨u, J_{σ}^{∗}L_{n}^{σ}\overline{a}J_{σ}^{−∗}P_{n}v⟩_{σ}

i.e. (W_{n}M_{n}^{σ}aW_{n})^{∗} = J_{σ}^{∗}L_{n}^{σ}\overline{a}J_{σ}^{−∗}P_{n} ⟶ J_{σ}^{∗}\overline{a}J_{σ}^{−∗} in L_{σ}^{2} . Using (3.22), we get in the same manner

⟨W_{n}M_{n}^{σ}SW_{n}u, v⟩_{σ} = i⟨L_{n}^{σ}V^{∗}P_{n}u, J_{σ}P_{n}v⟩_{σ} = \frac{πi}{n} ∑_{j=1}^{n}(V^{∗}P_{n}u)(x_{jn}^{σ})\overline{(J_{σ}^{−∗}P_{n}v)(x_{jn}^{σ})}

= i⟨φV^{∗}P_{n}u, L_{n}^{σ}φ^{−1}J_{σ}^{−∗}P_{n}⟩_{σ} = i⟨u, V M_{n}^{σ}J_{σ}^{−∗}P_{n}⟩_{σ} ,

whence the strong convergence of (W_{n}M_{n}^{σ}SW_{n})^{∗} .

For further considerations we need Fredholm and invertibility conditions for the operator aI + bS : L_{σ}^{2} ⟶ L_{σ}^{2} , if a, b ∈ PC . For this goal, we define c := (a + b)/(a − b) on [−1, 1] and f(μ) := exp(iπ(μ − 1)/2) sin(πμ/2) on [0, 1], and we associate to this operator the function

c(x, μ) := \{ \begin{matrix}c(x)(1 − μ) + c(x + 0)μ & , & μ ∈ [0, 1] , x ∈ (−1, 1) , \\ c(1) + [1 − c(1)]f(μ) & , & μ ∈ [0, 1] , x = 1 , \\ 1 + [c(−1) − 1]f(μ) & , & μ ∈ [0, 1] , x = −1 . \end{matrix}

Note that, for z_{1}, z_{2} ∈ ℂ , z_{1} + (z_{2} − z_{1})f(μ) , μ ∈ [0, 1] , describes the half circle from z_{1} to z_{2} that lies to the right of the straight line from z_{1} to z_{2} . Thus, if c(x ± 0) is finite for all x ∈ [−1, 1] , the image of c(x, μ) is a closed curve in the complex plane which possesses a natural orientation, and by wind c(x, μ) we denote the winding number of this curve w.r.t. the origin 0 .

Lemma 3.5 ([6], Theorem IX.4.1) Let a, b ∈ PC . The operator A = aI + bS : L_{σ}^{2} ⟶ L_{σ}^{2} is Fredholm if and only if a(x ± 0) − b(x ± 0) ≠ 0 for all x ∈ [−1, 1] and c(x, μ) ≠ 0 for all (x, μ) ∈ [−1, 1] × [0, 1] . In this case, A is one-sided invertible and ind A = −wind c(x, μ) .

Lemma 3.6 Let a, b ∈ PC . Then the operator W̃{A_{n}^{σ}} = J_{σ}^{−1}(aJ_{σ} + ibV^{∗}) is invertible in L_{σ}^{2} if and only if the operator W {A_{n}^{σ}} = aI + bS is invertible in L_{σ}^{2} .

Proof. The invertibility of W̃{A_{n}^{σ}} is equivalent to the invertibility of B = aJ_{σ} + ibV^{∗} . Since J_{σ} = φI − iψS and V^{∗} = ψI + iφS with ψ(x) = x (this follows from (3.6), (3.9), and (3.8)), the operator B is again a singular integral operator. Thus, the invertibility of B is equivalent to the Fredholmness of B with index 0 , or to the Fredholmness of C = BV with index −1. With the help of V = ψI − iφS and SφS = φI + K_{0} , where K_{0}u = −\frac{1}{\sqrt{2π}}⟨u, ũ_{0}⟩_{σ} (see (4.3) below), we get

C = a(φI − iψS)(ψI − iφS) + ibI = −iaφ^{2}S − iψ^{2}S + ibI + K = i(bI − aS) + K ,

with a compact operator K : L_{σ}^{2} ⟶ L_{σ}^{2}. Now the assertion follows from \frac{b−a}{b+a} = − (\frac{a+b}{a−b})^{−1} and Lemma 3.5.

4 Local theory of stability

Let A^{ω} denote the smallest C^{∗}-subalgebra of F^{W} which contains all sequences of the form {M_{n}^{ω}(aI + bS)P_{n}} with a, b ∈ PC and the ideal J . The aim of this section is to derive necessary and sufficient conditions for the stability of the sequences {M_{n}^{ω}(aI + bS)P_{n}} and, more general, for arbitrary sequences {A_{n}} ∈ A^{ω}. Our approach to these results is essentially based on the application of the local principles by Allan/Douglas and Gohberg/Krupnik to study the invertibility of a coset {A_{n}}^{o} := {A_{n}} + J in the quotient algebra A^{ω}/J. In what follows we agree upon omitting the superscript ω in all notations (such as in M_{n}^{ω}, which will be abbreviated to M_{n}) whenever the validity of the assertion where this notation is used does not depend on ω = φ or ω = σ.

The applicability of a local principle in a Banach algebra depends on the existence of sufficiently many elements which commute with every other element of the algebra, i.e. which belong to the center of the algebra. The following lemma establishes the existence of such elements for the algebra A^{ω}/J.

Lemma 4.1 If f ∈ C[−1, 1], then the coset {M_{n}fP_{n}}^{o} commutes with every coset {A_{n}}^{o} ∈ A^{ω}/J.

Proof. It is enough to verify that {M_{n}fP_{n}}^{o} commutes with all cosets {M_{n}aP_{n}}^{o} where a ∈ PC and with the coset {M_{n}SP_{n}}^{o}. The first assertion is obvious; one even has {M_{n}aP_{n}}{M_{n}fP_{n}} = {M_{n}fP_{n}}{M_{n}aP_{n}} for arbitrary a, f ∈ PC. For the second assertion, note that the equalities M_{n}fP_{n}M_{n}SP_{n} = M_{n}fSP_{n} and M_{n}SP_{n}M_{n}fP_{n} = M_{n}SM_{n}fP_{n} hold. So, what remains to prove is

{M_{n}fSP_{n} − M_{n}SM_{n}fP_{n}} ∈ J for all f ∈ C[−1, 1] . (4.1)

We show that (4.1) holds true for all algebraic polynomials p in place of f. Then the assertion follows from the closedness of J and from (3.4) and (3.5). So let p be a polynomial of degree not greater than m . Then M_{n}pP_{n−m} = pP_{n−m} for n > m. Consequently,

M_{n}pSP_{n} − M_{n}SM_{n}pP_{n} = M_{n}(pS − Sp)P_{n} + M_{n}S(I − M_{n})p(P_{n} − P_{n−m}) .

Obviously, the sequence {M_{n}(pS − Sp)P_{n}} belongs to J . Moreover, P_{n} − P_{n−m} = W_{n}P_{m}W_{n} , which implies that

M_{n}S(I − M_{n})p(P_{n} − P_{n−m}) = M_{n}S(I − M_{n})pP_{n}W_{n}P_{m}W_{n}
= (M_{n}(Sp − pS)P_{n} + M_{n}pSP_{n} − M_{n}SP_{n}M_{n}pP_{n})W_{n}P_{m}W_{n}

and, hence, {M_{n}S(I − M_{n})p(P_{n} − P_{n−m})} ∈ J.

Together with the identities M_{n}f_{1}M_{n}f_{2}P_{n} = M_{n}f_{1}f_{2}P_{n} and (M_{n}fP_{n})^{∗} = M_{n}\overline{f}P_{n} , the preceding lemma shows that the set C^{ω} := {{M_{n}fP_{n}}^{o} : f ∈ C[−1, 1]} forms a C^{∗}-subalgebra of the center of A^{ω}/J . This offers the applicability of the local principle by Allan/Douglas which we recall here for the reader’s convenience.

Local principle of Allan and Douglas. (comp. [1, 4]) Let B be a unital Banach algebra and let B_{c} be a closed subalgebra of the center of B containing the identity. For every maximal ideal x ∈ M(B_{c}) , let J_{x} denote the smallest closed ideal of B which contains x , i.e.

J_{x} = clos_{B} {∑_{j=1}^{m} a_{j}c_{j} : a_{j} ∈ B, c_{j} ∈ x, m = 1, 2, …} .

Then an element a ∈ B is invertible in B if and only if a + J_{x} is invertible in B/J_{x} for all x ∈ M(B_{c}) . (In case J_{x} = B we define that a + J_{x} is invertible.) Moreover, the mapping

M(B_{c}) ⟶ [0, ∞) , x ↦ ∥b + J_{x}∥

is upper semi-continuous for each b ∈ B . In case B is a C^{∗}-algebra and B_{c} a is central C^{∗}- subalgebra of B, then all ideals J_{x} are proper ideals of B, and ∥b∥ = max{∥b + J_{x}∥ : x ∈ M(B_{c})} for all b ∈ B.

We will apply this local principle with A^{ω}/J and C^{ω} in place of B and B_{c}, respectively. The algebra C^{ω} is^{∗}-isomorphic to C[−1, 1] via the isomorphism {M_{n}fP_{n}}^{o} ↦ f . This can be seen as follows: If f ∈ C[−1, 1] is invertible, then the coset {M_{n}fP_{n}}^{o} is invertible. Conversely, assume this coset is invertible and choose one of its representatives {M_{n}fP_{n} + P_{n}KP_{n} + W_{n}LW_{n} + G_{n}} which then is invertible modulo J. An application of the homomorphism W yields the invertibility of fI + K modulo compact operators, i.e. the Fredholmness of the multiplication operator fI. But Fredholm multiplication operators are invertible. Consequently, the maximal ideal space of C is equal to {I_{τ}^{ω} : τ ∈ [−1, 1]} with

I_{τ}^{ω} := {{M_{n}^{ω}fP_{n}}^{o} : f ∈ C[−1, 1], f(τ) = 0} .

Let J_{τ}^{ω} denote the smallest closed ideal of A^{ω}/J which contains the maximal ideal I_{τ}^{ω} of C^{ω}, i.e.

J_{τ}^{ω} = clos_{A^{ω}/J} {∑_{j=1}^{m}{A_{n}^{j}M_{n}^{ω}f_{j}P_{n}}^{o} : {A_{n}^{j}} ∈ A^{ω}, f_{j} ∈ C[−1, 1], f_{j}(τ) = 0, m = 1, 2, …} .

Then the local principle of Allan/Douglas says that all ideals J_{τ}^{ω} are proper in A^{ω}/J, and that a coset {A_{n}}^{o} is invertible in A^{ω}/J if and only if {A_{n}}^{o} + J_{τ} is invertible in (A^{ω}/J)/J_{τ}^{ω} for every τ ∈ [−1, 1] .

Our next goal is the description of the local algebras (A^{ω}/J)/J_{τ}^{ω}. First let −1 < τ < 1. Let h_{τ} be the function which is 0 on [−1, τ] and 1 on (τ, 0]. Then, for every a ∈ PC,

{M_{n}aP_{n}}^{o} + J_{τ} = a(τ + 0){M_{n}h_{τ}P_{n}}^{o} + a(τ){M_{n}(1 − h_{τ})P_{n}}^{o} + J_{τ} .

Consequently, the algebra (A^{ω}/J)/J_{τ}^{ω} is generated by its cosets e := {P_{n}}^{o} + J_{τ} ,

p := \frac{1}{2} (\{P_{n}\}^{o} + \{M_{n}SP_{n}\}^{o}) + J_{τ} , and q := \{M_{n}h_{τ}P_{n}\}^{o} + J_{τ} . (4.2)

Obviously, q is a selfadjoint projection. In order to see that the same is true for p, we make use of the relation

SφS = φI + K_{0} , where K_{0}u = −\frac{1}{\sqrt{2}} ⟨u, ũ_{0}⟩_{σ}T_{0} , (4.3)

which is a consequence of (3.6), (3.9), (3.21) and of the continuity of the operator SφS : L_{σ}^{2} ⟶ L_{σ}^{2} . Indeed,

SφSũ_{n} = iSφT_{n+1} = \frac{i}{2} Sφ(U_{n+1} − U_{n−1})

= \{ \begin{matrix}\frac{1}{2}(T_{n} − T_{n+2}) & , & n ≥ 1, \\ −\frac{1}{2} T_{2} & , & n = 0 , \end{matrix} = \{ \begin{matrix}φũ_{n} & , & n ≥ 1, \\ φũ_{0} − \frac{1}{\sqrt{2}} T_{0} & , & n = 0 . \end{matrix}

Further we recall that SP_{n}u = i ∑_{k=0}^{n−1}⟨u, ũ_{k}⟩_{σ}T_{k+1} = \frac{i}{2} ∑_{k=0}^{n−1}⟨u, ũ_{k}⟩_{σ}(U_{k+1} − U_{k−1}) , which implies

M_{n}^{φ}φSP_{n} = φSP_{n} − \frac{i}{2}V W_{n}P_{1}W_{n} . (4.4)

Consequently, we have the identities M_{n}^{φ}SP_{n}M_{n}^{φ}φSP_{n} = M_{n}^{φ}SφSP_{n} − \frac{i}{2}M_{n}^{φ}SV P_{n}W_{n}P_{1}W_{n} and M_{n}^{σ}φSP_{n} = φSP_{n} − iφJ_{σ}V W_{n}P_{1}W_{n} . Thus, in both cases,

\{M_{n}SP_{n}\}^{o}\{M_{n}SP_{n}\}^{o} + J_{τ} = \frac{1}{φ(τ)} \{M_{n}SP_{n}\}^{o}\{M_{n}φSP_{n}\}^{o} + J_{τ}

(4.5)

= \frac{1}{φ(τ)} \{M_{n}φP_{n}\}^{o} + J_{τ} = \{P_{n}\}^{o} + J_{τ} , −1 < τ < 1 .

The identities (3.6), (3.8), and (3.9) imply that, with ψ(x) = x ,

V = ψI − iφS , V^{∗} = ψI + iφS . (4.6)

From this we can conclude that {M_{n}SP_{n}}^{o} + J_{τ} is selfadjoint. Indeed,

\{M_{n}SP_{n}\}^{o} + J_{τ} = −\frac{i}{φ(τ)}\{M_{n}iφSP_{n}\}^{o} + J_{τ} = −\frac{i}{φ(τ)} (\{V^{∗}P_{n}\}^{o} − \{M_{n}ψP_{n}\}^{o}) + J_{τ}

and, consequently,

(\{M_{n}SP_{n}\}^{o} + J_{τ})^{∗} = \frac{i}{φ(τ)} (\{P_{n}V P_{n}\}^{o} − \{M_{n}ψP_{n}\}^{o}) + J_{τ}

= \frac{i}{φ(τ)}\{M_{n}(−iφ)SP_{n}\}^{o} + J_{τ} = \{M_{n}SP_{n}\}^{o} + J_{τ} .

So we have seen that the local algebra (A^{ω}/J)/J_{τ}^{ω} is generated by its identity element and by two projections in case −1 < τ < 1. Algebras of this kind are described by the following result.

Theorem 4.2 (Halmos’ two-projections theorem, [8]) Let B be a unital C^{∗}-algebra, and let p, q ∈ B be projections (i.e. self-adjoint idempotent elements) such that σ_{B}(pqp) = [0, 1] . Then the smallest closed subalgebra of B , which contains p , q , and the identity element e , is ∗-isomorphic to the C^{∗}-algebra of all continuous 2 × 2 matrix functions on [0, 1] , which are diagonal at 0 and 1 . The isomorphism can be chosen in such a way that it sends e , p , and q into the functions

μ ↦ [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] , μ ↦ [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] , and μ ↦ [ \begin{matrix}μ & \sqrt{μ(1 − μ)} \\ \sqrt{μ(1 − μ)} & 1 − μ \end{matrix} ] , (4.7)

respectively.

To apply this theorem, we have to check whether σ_{(A/J)/J_{τ}} (pqp) = [0, 1] for p and q defined by (4.2). For this, let G be the smallest C^{∗}-subalgebra of L(L_{σ}^{2}) which contains all operators aI + bS with a, b ∈ PC[−1, 1] and the ideal K = K(L_{σ}^{2}) of all compact operators on L_{σ}^{2} . By J_{τ}^{G} , τ ∈ [−1, 1] , we denote the smallest closed ideal of G/K , which contains all cosets fI + K with f ∈ C[−1, 1] and f(τ) = 0.

Lemma 4.3 If {A_{n}}^{o} + J_{τ} is invertible in (A/J)/J_{τ}, then (W {A_{n}} + K) + J_{τ}^{G} is invertible in (G/K)/J_{τ}^{G} .

Proof. Let {A_{n}} ∈ A, and assume that there is a sequence {B_{n}} ∈ A such that {B_{n}}^{o}{A_{n}}^{o} + J_{τ} = {P_{n}}^{o} + J_{τ} . Then B_{n}A_{n} = P_{n} + J_{n} + P_{n}KP_{n} + W_{n}T W_{n} + C_{n} with some operators K, T ∈ K , some coset {J_{n}}^{o} ∈ J_{τ} and some sequence {C_{n}} ∈ N. Further, given ε > 0, there exist sequences {A_{n}^{(j)}} ∈ A and functions f_{j} ∈ C[−1, 1] with f_{j}(τ) = 0 such that

∥{J_{n}}^{o} − {B′_{n}}^{o}∥_{A/J} < ε for B′_{n} = ∑_{j=1}^{m_{ε}} A_{n}^{(j)}M_{n}f_{j}P_{n} . Hence, there are operators K_{ε}, T_{ε} ∈ K(L_{σ}^{2})

and a sequence {C_{n}^{ε}} ∈ N such that

∥J_{n}P_{n} − ∑_{j=1}^{m_{ε}} A_{n}^{(j)}M_{n}f_{j}P_{n} − P_{n}K_{ε}P_{n} − W_{n}T_{ε}W_{n} − C_{n}^{ε}P_{n}∥_{L(L_{σ}^{2})} < ε for all n ≥ 1.

Hence, ∥W {J_{n}} − ∑_{j=1}^{m_{ε}} W {A_{n}^{(j)}}f_{j}I − K_{ε}∥_{L(L_{σ}^{2})} ≤ ε , which implies W {J_{n}} + K ∈ J_{τ}^{G} . Thus,

because of W {B_{n}}W {A_{n}} = I + W {J_{n}} + K , the coset (W {A_{n}} + K) + J_{τ}^{G} is invertible from the left in (G/K)/J_{τ}^{G} . Its invertibility from the right can be shown analogously.

Now we can complete the description of the local algebras in case −1 < τ < 1. The product pqp is a non-negative element of (A/J)/J_{τ} , which implies that its spectrum σ_{(A/J)/J_{τ}} (pqp) is contained in [0, 1] . We prove that the spectrum of pqp coincides with this interval. Assume there is a λ ∈ (0, 1) such that pqp − λe is invertible in (A/J)/J_{τ} . The invertibility of pqp − λe is equivalent to the invertibility of

(q − λ)p − λ(e − p)

= \frac{1}{2}\{M_{n}(h_{τ} − λ)P_{n}\}^{o} (\{P_{n}\}^{o} + \{M_{n}SP_{n}\}^{o}) − \frac{λ}{2} (\{P_{n}\}^{o} − \{M_{n}SP_{n}\}^{o}) + J_{τ} .

Lemma 4.3 implies that (A + K) + J_{τ}^{G} := ((h_{τ} − λ)(I + S) − λ(I − S) + K) + J_{τ}^{G} is invertible in (G/K)/J_{τ}^{G} . If −1 ≤ x < τ, we have (A + K) + J_{x}^{G} = (−2λI + K) + J_{x}^{G} , and −2λI + K is invertible in G/K . If τ < x ≤ 1 then (A + K) + J_{x}^{G} = ((1 − 2λ)I + S + K) + J_{x}^{G} , which is also invertible in (G/K)/J_{x}^{G} . From the local principle of Allan and Douglas we conclude the Fredholmness of (h_{τ} − λ)(I + S) − λ(I − S) in L_{σ}^{2} . But this is in contradiction to (see Lemma 3.5) 0 ∈ [1 − \frac{1}{λ}, 1] = [\frac{λ−h_{τ} (τ+0)}{λ}, \frac{λ−h_{τ} (τ−0)}{λ}] . Thus we can apply Halmos’ two projections theorem to get that the local algebra (A/J)/J_{τ} is ^{∗}-isomorphic to the C^{∗}-algebra of the continuous 2 × 2 matrix functions on [0, 1] which are diagonal at 0 and 1 . The isomorphism can be chosen in such a way that it sends {P_{n}}^{o} + J_{τ} , \frac{1}{2}(\{P_{n}\}^{o} +\{M_{n}SP_{n}\}^{o})+J_{τ} , and \{M_{n}h_{τ}P_{n}\}^{o} +J_{τ} into the functions given in (4.7), respectively.

Now we turn our attention to the local algebras at τ = ±1. Since {M_{n}(aI + bS)P_{n}}^{o} + J_{τ} = {M_{n}[a(τ)I +b(τ)S]P_{n}}^{o}+J_{τ} , these local algebras are generated (as C^{∗}-algebras) by their cosets {P_{n}}^{o} + J_{τ} (the identity element) and {M_{n}SP_{n}}^{o} + J_{τ} . It turns out that the properties of the latter coset (and, thus, the behaviour of the algebras generated by it) depends heavily on the weight function ω. For ω = σ we have the following result, the proof of which will be given in the Appendix.

Lemma 4.4 Let τ = ±1 . The coset {M_{n}^{σ}SP_{n}}^{o} + J_{τ} is a unitary element of the algebra (A^{σ}/J)/J_{τ}^{σ} , and its spectrum is equal to 𝕋_{τ} , where 𝕋_{±1} = 𝕋 ∩ {t ∈ ℂ : ±ℑt ≥ 0} .

Consequently, the algebra (A^{σ}/J)/J_{τ}^{σ} is ^{∗}-isomorphic to the algebra C(𝕋_{τ}) of all complex valued continuous functions on 𝕋_{τ}, and the isomorphism can be chosen such that it sends {M_{n}^{σ}SP_{n}}^{o} + J_{τ}^{σ} into the function t ↦ t .

The treatment of the case ω = φ starts with the following lemma.

Lemma 4.5 The sequences {M_{n}^{φ}φ^{−1}P_{n}} and {(M_{n}^{φ}φ^{−1}P_{n})^{∗}} converge strongly to the multiplication operators φ^{−1}I : L_{σ}^{2} ⟶ L_{φ}^{2} and φI : L_{φ}^{2} ⟶ L_{σ}^{2} , respectively.

Proof. Convergence of M_{n}^{φ}φ^{−1}P_{n}: Since L_{σ}^{2} is continuously embedded into L_{φ}^{2} we have, due to Corollary 3.3,

lim_{n→∞} M_{n}^{φ}φ^{−1}P_{n}ũ_{m} = lim_{n→∞} M_{n}^{φ}U_{m} = U_{m} = φ^{−1}ũ_{m} for all m ≥ 0 in L_{φ}^{2} .

Thus, it remains to show that the operators M_{n}^{φ}φ^{−1}P_{n} : L_{σ}^{2} ⟶ L_{φ}^{2} are uniformly bounded. Consider the quadrature rule

Q̃_{n}f = ∫_{−1}^{1} (L_{n}^{φ}f) (x)ρ(x) dx = ∑_{k=1}^{n} ρ_{kn}f(x_{kn}^{φ}) ,

where ρ(x) = (1 − x^{2})^{\frac{3}{2}} , and abbreviate x_{kn}^{φ} to x_{k}. The quadrature weights ρ_{kn} are equal to

ρ_{kn} = ∫_{−1}^{1} \frac{U_{n}(x)}{x − x_{k}} \frac{(1 − x^{2})φ(x)}{U′_{n}(x_{k})} dx

= (1 − x_{k}^{2}) ∫_{−1}^{1} \frac{U_{n}(x)φ(x) dx}{(x − x_{k})U′_{n}(x_{k})} − \frac{1}{U′_{n}(x_{k})} ∫_{−1}^{1} U_{n}(x)(x + x_{k})φ(x) dx
= (1 − x_{k}^{2})λ_{kn}^{φ} , n > 1 .

COLLOCATION METHODS FOR SYSTEMS OF SIE 17

Define ε_{ij} := ⟨ℓ_{in}^{φ}, ℓ_{jn}^{φ}⟩_{ρ} for i, j = 1, …, n . We remark that U′_{n}(x_{k}) = \sqrt{\frac{2}{π}} \frac{(−1)^{k+1}(n + 1)}{1 − x_{k}^{2}} and

compute

η_{k} := ∫_{−1}^{1} \frac{[U_{n}(x)]^{2}}{x − x_{k}} ρ(x) dx
= (1 − x_{k}^{2}) ∫_{−1}^{1} \frac{[U_{n}(x)]^{2}}{x − x_{k}} φ(x) dx − ∫_{−1}^{1}(x + x_{k}) [U_{n}(x)]^{2} φ(x) dx = −x_{k} ,

where we take into account the orthogonality and symmetry properties of U_{n}. For i ≠ j, it follows

ε_{ij} = \frac{π(−1)^{i+j}(1 − x_{i}^{2})(1 − x_{j}^{2})}{2(n + 1)^{2}} ∫_{−1}^{1} \frac{[U_{n}(x)]^{2} ρ(x) dx}{(x − x_{i})(x − x_{j})}
= \frac{π(−1)^{i+j}(1 − x_{i}^{2})(1 − x_{j}^{2})}{2(n + 1)^{2}} \frac{η_{i} − η_{j}}{x_{i} − x_{j}} = \frac{π(−1)^{i+j+1}(1 − x_{i}^{2})(1 − x_{j}^{2})}{2(n + 1)^{2}} ,

i.e.

sgn ε_{ij} = (−1)^{i+j+1} for i ≠ j . (4.8) For i = j, we get

[U′_{n}(x_{j})]^{2} ε_{jj} = ∫_{−1}^{1} \frac{[U_{n}(x)]^{2} ρ(x) dx}{(x − x_{j})^{2}}
= (1 − x_{j}^{2}) ∫_{−1}^{1} \frac{[U_{n}(x)]^{2}}{(x − x_{j})^{2}} φ(x) dx − ∫_{−1}^{1} \frac{[U_{n}(x)]^{2}}{x − x_{j}} (x + x_{j})φ(x) dx
= (1 − x_{j}^{2})λ_{jn}^{φ} [U′_{n}(x_{j})]^{2} + 1 − 2 ∫_{−1}^{1} \frac{[U_{n}(x)]^{2}}{x − x_{j}} xφ(x) dx
= (1 − x_{j}^{2})λ_{jn}^{φ} [U′_{n}(x_{j})]^{2} + 1

− ∫_{−1}^{1} [U_{n+1}(x) + U_{n−1}(x)] \frac{U_{n}(x)}{x − x_{j}} φ(x) dx

= (1 − x_{j}^{2})λ_{jn}^{φ} [U′_{n}(x_{j})]^{2} + 1 − λ_{jn}^{φ}U_{n−1}(x_{j})U′_{n}(x_{j}) .

Since U_{n−1}(x_{j}) = \sqrt{\frac{2}{π}} \begin{matrix}sin & \frac{njπ}{n+1} \\ sin & \frac{jπ}{n+1} \end{matrix} = \sqrt{\frac{2}{π}} (−1)^{j+1}, we have λ_{jn}^{φ}U_{n−1}(x_{j})U′_{n}(x_{j}) = 2 , which yields

ε_{jj} < (1 − x_{j}^{2})λ_{jn}^{φ} = ρ_{jn} , j = 1, …, n . (4.9) Let now f : (−1, 1) ⟶ ℂ be given. Then, due to (4.8) and (4.9),

∫_{−1}^{1} |(L_{n}^{φ}f)(x)|^{2} ρ(x) dx = ∑_{i=1}^{n} ∑_{j=1}^{n} f(x_{i})\overline{f(x_{j})}ε_{ij} ≤ ∑_{i=1}^{n} ∑_{j=1}^{n} |f(x_{i})| |f(x_{j})| |ε_{ij}|

≤ 2 ∑_{i=1}^{n} |f(x_{i})|^{2}ε_{ii} − ∑_{i=1}^{n} ∑_{j=1}^{n}(−1)^{i+j}|f(x_{i})| |f(x_{j})|ε_{ij}

≤ 2 ∑_{i=1}^{n} ρ_{in}|f(x_{i})|^{2} − ∫_{−1}^{1} ∑_{i=1}^{n} [(−1)^{i}|f(x_{i})|ℓ_{in}^{φ}(x)]^{2} ρ(x) dx

≤ 2Q̃_{n}|f|^{2} .

Using this estimate in combination with the explicit form of the quadrature weights ρ_{kn} derived above we obtain, for u_{n} = φv_{n} ∈ im P_{n} ,

∥M_{n}^{φ}φ^{−1}u_{n}∥_{φ}^{2} = ∥φL_{n}^{φ}φ^{−1}v_{n}∥_{φ}^{2} = ∥L_{n}^{φ}φ^{−1}v_{n}∥_{ρ}^{2}

≤ 2Q̃_{n} |φ^{−1}v_{n}|^{2} = 2 ∑_{k=1}^{n} λ_{kn}^{φ}|v_{n}(x_{kn}^{φ})|^{2} = 2 ∥v_{n}∥_{φ}^{2} = 2 ∥u_{n}∥_{σ}^{2} ,

which proves the desired uniform boundedness.

Convergence of (M_{n}^{φ}φ^{−1}P_{n})^{∗}: The strong convergence of M_{n}^{φ}φ^{−1}P_{n} implies the uniform boundedness of (M_{n}^{φ}φ^{−1}P_{n})^{∗} : L_{φ}^{2} ⟶ L_{σ}^{2} . Since {φ^{−1}T_{m}}_{m=0}^{∞} forms an orthonormal basis in L_{φ}^{2}, it remains to prove that (M_{n}^{φ}φ^{−1}P_{n})^{∗}φ^{−1}T_{m} ⟶ T_{m} in L_{σ}^{2} . In view of

η_{nmj} := ⟨(M_{n}^{φ}φ^{−1}P_{n})^{∗}φ^{−1}T_{m}, ũ_{j}⟩_{σ} = ⟨φ^{−1}T_{m}, M_{n}^{φ}φ^{−1}P_{n}ũ_{j}⟩_{φ},

we have η_{nmj} = 0 for j ≥ n and, for n > m and j < n ,

η_{nmj} = ⟨T_{m}, L_{n}^{φ}φ^{−1}U_{j}⟩_{φ} = \frac{π}{n + 1} ∑_{k=1}^{n} T_{m}(x_{k})ũ_{j}(x_{k}) = α_{jn}^{φ}(T_{m})

taking into account Relation (3.7). Hence, due to Corollary 3.3, (M_{n}^{φ}φ^{−1}P_{n})^{∗}T_{m} = M_{n}^{φ}T_{m} ⟶ T_{m} in L_{σ}^{2} , and the lemma is completely proved.

We still need a consequence of the lifting principle Lemma 2.2.

Lemma 4.6 If {A_{n}}^{o} ∈ A/J and W {A_{n}} : L_{σ}^{2} ⟶ L_{σ}^{2} as well as W̃{A_{n}} : L_{σ}^{2} ⟶ L_{σ}^{2} are invertible from the same side, then they are invertible from both sides.

Proof. A closer look at the proof of Lemma 2.2 shows that also a one-sided version of that lemma holds: if W {A_{n}}, W̃{A_{n}} and {A_{n}}^{o} are invertible from the same side, say from the right hand side, then the sequence {A_{n}} is stable from that side in the sense that there is a sequence {B_{n}} such that A_{n}B_{n} = P_{n} + G_{n} with a sequence {G_{n}} ∈ N. Since the A_{n} are matrices, this clearly implies the common stability of the sequence {A_{n}}. But then, due to Lemma 2.2, W {A_{n}}, W̃{A_{n}} and {A_{n}}^{o} are two-sided invertible.

Corollary 4.7 One has {M_{n}^{φ}SP_{n}}^{o}{(M_{n}^{φ}SP_{n})^{∗}}^{o} = {P_{n}}^{o} and, hence,

{M_{n}^{φ}SP_{n}}^{o}{(M_{n}^{φ}SP_{n})^{∗}}^{o} + J_{τ}^{φ} = {P_{n}}^{o} + J_{τ}^{φ} for all τ ∈ [−1, 1],

whereas

{(M_{n}^{φ}SP_{n})^{∗}}^{o}{M_{n}^{φ}SP_{n}}^{o} + J_{τ}^{φ} ≠ {P_{n}}^{o} + J_{τ}^{φ} for τ = ±1 . (4.10)

Proof. By (3.14), (4.4), and (4.3),

M_{n}^{φ}SP_{n}(M_{n}^{φ}SP_{n})^{∗} = (M_{n}^{φ}SφSP_{n} − \frac{i}{2}M_{n}^{φ}SV P_{n}W_{n}P_{1}W_{n}) M_{n}^{φ}φ^{−1}P_{n}

= P_{n} + M_{n}^{φ}K_{0}M_{n}^{φ}φ^{−1}P_{n} − \frac{i}{2}M_{n}^{φ}SV P_{n}W_{n}P_{1}W_{n}M_{n}^{φ}φ^{−1}P_{n}.

It remains to show that the sequences

{M_{n}^{φ}K_{0}M_{n}^{φ}φ^{−1}P_{n}} and {W_{n}P_{1}W_{n}M_{n}^{φ}φ^{−1}P_{n}} = {W_{n}P_{1}M_{n}^{φ}φ^{−1}P_{n}W_{n}}

belong to the ideal J. This is a consequence of Lemma 4.5 and the relations

∥P_{1}u∥_{σ} = |⟨u, ũ_{0}⟩_{σ}| = \sqrt{\frac{2}{π}} |∫_{−1}^{1} u(x) dx| ≤ const ∥u∥_{φ}

and

∥K_{0}u∥_{σ} = \frac{1}{\sqrt{2}} |⟨u, ũ_{0}⟩_{σ}| ≤ const ∥u∥_{φ} ,

which imply the compactness of the operators P_{1} : L_{φ}^{2} ⟶ L_{σ}^{2} and K_{0} : L_{φ}^{2} ⟶ L_{σ}^{2} .

Now assume that (4.10) is not true for τ = 1 , for example. Then it is also not true for τ = −1 which can be seen as follows. Set (W f)(x) := f(−x). Then

W SW = −S , W P_{n} = P_{n}W , W M_{n}^{φ} = M_{n}^{φ}W , and {P_{n}W }^{o}J_{1}^{φ}{W P_{n}}^{o} = J_{−1}^{φ}

(observe that W ũ_{k} = (−1)^{k}ũ_{k}). Hence, applying W to

{(M_{n}^{φ}SP_{n})^{∗}}^{o}{M_{n}^{φ}SP_{n}}^{o} + J_{1}^{φ} = {P_{n}}^{o} + J_{1}^{φ}

yields

{P_{n}W (M_{n}^{φ}SP_{n})^{∗}W P_{n}}^{o}{M_{n}^{φ}(−S)P_{n}}^{o} + J_{−1}^{φ} = {P_{n}}^{o} + J_{−1}^{φ} .

Together with (4.5) and the local principle by Allan/Douglas, this leads to the invertibility of the coset {M_{n}^{φ}SP_{n}}^{o} in contradiction to Lemma 4.6.

Thus, the coset {(M_{n}^{φ}SP_{n})^{∗}}^{o} + J_{τ}^{φ} is an isometry in the local algebra (A^{φ}/J)/J_{τ}^{φ}. Thanks to a result by Coburn [3], C^{∗}-algebras generated by an isometry possess a nice description in terms of shift operators on the Hilbert space ℓ^{2} of all square summable sequences of complex numbers. In particular, Coburn’s theorem implies that the local algebra (A^{φ}/J)/J_{τ}^{φ} is ^{∗}-isomorphic to the C^{∗}-subalgebra of L(ℓ^{2}) generated by the shift operator

Σ : ℓ^{2} → ℓ^{2}, {x_{0}, x_{1}, …} ↦ {0, x_{0}, x_{1}, …}

where the isomorphism sends {M_{n}^{φ}SP_{n}}^{o} + J_{τ}^{φ} into

Σ^{∗} : ℓ^{2} → ℓ^{2}, {x_{0}, x_{1}, …} ↦ {x_{1}, x_{2}, …} .

Applying the local principle of Allan and Douglas together with Lemma 2.2 and Lemma 3.6, we can summarize the considerations of this section.

Theorem 4.8 (a) There is a *-isomorphism η_{ω} from A^{ω}/J onto a C^{∗}-algebra of bounded functions living on ((−1, 1) × [0, 1])∪({±1} × 𝕋_{τ}) in case ω = σ and on ((−1, 1) × [0, 1])∪{±1} in case ω = φ. This isomorphism sends the coset {M_{n}^{ω}aP_{n}}^{o} into

(x, μ) ↦ [ \begin{matrix}a(x + 0)μ + a(x)(1 − μ) & (a(x + 0) − a(x))\sqrt{μ(1 − μ)} \\ (a(x + 0) − a(x))\sqrt{μ(1 − μ)} & a(x + 0)(1 − μ) + a(x)μ \end{matrix} ]

and the coset {M_{n}^{ω}SP_{n}}^{o} into

(x, μ) ↦ [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ]

for (x, μ) ∈ (−1, 1) × [0, 1]. Moreover, for (x, t) ∈ {±1} × 𝕋_{τ},

η_{σ}{M_{n}^{σ}(aI + bS)P_{n}}^{o}(x, t) = a(x) + b(x)t

and, for x = ±1,

η_{φ}{M_{n}^{φ}(aI + bS)P_{n}}^{o}(x) = a(x)I + b(x)Σ^{∗}.

(b) The sequence {A_{n}} ∈ A^{ω} is stable if and only if the operators W {A_{n}}, W̃{A_{n}} : L_{σ}^{2} ⟶ L_{σ}^{2} are invertible and if, in case ω = φ , the operator η_{φ}{A_{n}}^{o}(x) is invertible on ℓ^{2} for x = ±1 .

We remark that the invertibility of W {A_{n}} already implies that det η_{ω}{A_{n}}^{o}(x, μ) ≠ 0 for all (x, μ) ∈ (−1, 1) × [0, 1] and that η_{σ}{A_{n}}^{o}(±1, t) ≠ 0 for all t ∈ 𝕋_{±1} (see Lemma 3.5).

For the stability of the sequence {A_{n}^{σ}} = {M_{n}^{σ}(aI +bS)P_{n}}, this theorem yields the invertibility of W {A_{n}^{σ}} = aI + bS and of W̃{A_{n}^{σ}} = J_{σ}^{−1}(aJ_{σ} + ibV^{∗}) as necessary and sufficient conditions. By Lemma 3.6, the invertibility of W̃{A_{n}^{σ}} is a consequence of the invertibility of W {A_{n}^{σ}}. Similarly, the sequence {A_{n}^{φ}} = {M_{n}^{φ}(aI +bS)P_{n}} proves to be stable if and only if the operators W {A_{n}^{φ}} = aI + bS and W̃{A_{n}^{φ}} = aI − bS on L_{σ}^{2} and the operators a(±1)I + b(±1)Σ^{∗} on ℓ^{2} are invertible. It is easy to see that the invertibility of the latter operators is equivalent to the condition a(±1) + b(±1)z ≠ 0 for all z ∈ ℂ with |z| ≤ 1 which, on its hand, is already a consequence of the invertibility of aI ± bS. This proves Theorem 2.1.

The assertion (b) of Theorem 4.8 can be easily translated into the case of a system of CSIE’s

∑_{k=1}^{m} [a_{jk}(x)u_{k}(x) + \frac{b_{jk}(x)}{πi} ∫_{−1}^{1} \frac{u_{k}(y)}{y − x} dy] = f_{j}(x) , −1 < x < 1 , j = 1, …, m , (4.11)

with piecewise continuous coefficients a_{jk} and b_{jk} . Indeed, denote by {A_{n}^{j,k}} the operator sequence of the collocation method for (1.1) with a_{jk} and b_{jk} instead of a and b , respectively. Then the collocation method for (4.11) is stable in (L_{σ}^{2})^{m} if and only if [W {A_{n}^{j,k}}]_{j,k=1}^{m} and

[W̃ {A_{n}^{j,k}}]_{j,k=1}^{m} are invertible and if, in case ω = φ , [η_{φ} {A_{n}^{j,k}}^{o} (x)]_{j,k=1}^{m} is invertible for x = ±1 .

Figure 1: The smallest three singular values of A_{n} = M_{n}^{φ}(aI + bS)P_{n} .

5 Behaviour of the smallest singular values

The singular values of a matrix A are the non-negative square roots of the eigenvalues of A^{∗}A. The singular values of a matrix A_{n} ∈ ℂ^{n×n} will be denoted by 0 ≤ σ_{1}^{(n)} ≤ … ≤ σ_{n}^{(n)}, counted with respect to their multiplicity.

If {A_{n}} ∈ F is a stable sequence of matrices, then there is a positive constant C such that the smallest singular value σ_{1}^{(n)} of A_{n} (hence, every singular value of A_{n}) is greater than C for all n, and conversely. Thus, if {A_{n}} is non-stable, then there is a subsequence of the sequence (σ_{1}^{(n)}) which tends to zero. Figures 1 and 2 illustrate this behaviour for the non-stable sequences \{A_{n}\} with A_{n} = M_{n}^{φ}(aI + bS)P_{n} , where a(x) = \sqrt{1 − x} , b(x) = −ix and a(x) = \sqrt{1.01 − x^{2}} , b(x) = −ix , respectively. In both cases we observe that not only a subsequence of (σ_{1}^{(n)}) but the sequence itself tends to zero. Moreover, in Figure 2, also the sequence (σ_{2}^{(n)}) of the second singular values goes to zero, whereas all other singular values are uniformly bounded from below by a positive constant. It is the goal of the present section to explain this effect and to derive a formula for the number of the singular values of A_{n} which tend to zero. Here we restrict ourselves to the case ω = φ , although analogous considerations are possible for ω = σ .

The desired results are closely related with a Fredholm theory for approximation sequences which has been developed in [20] and [18]. For the reader’s convenience, we start with recalling some definitions and results from [20] and [18].

Figure 2: The smallest three singular values of A_{n} = M_{n}^{φ}(aI + bS)P_{n} .

Fractal algebras. This class of subalgebras of F has been introduced and studied in [19, 17]. We will see in a moment that the algebra A^{φ} is fractal, and that the property of fractality is responsible for the fact that the complete sequence of the smallest singular values of a non-stable sequence {A_{n}} ∈ A^{φ} tends to zero and not only one of its proper subsequences.

Given a strongly monotonically increasing sequence η : ℕ → ℕ, let F_{η} refer to the C^{∗}-algebra of all bounded sequences {A_{n}} with A_{n} ∈ ℂ^{η(n)×η(n)}, and write N_{η} for the ideal of all sequences {A_{n}} ∈ F_{η} which tend to zero in the norm. Further, let R_{η} stand for the restriction mapping R_{η} : F → F_{η}, {A_{n}} ↦ {A_{η(n)}}. This mapping is a ^{∗}-homomorphism from F onto F_{η} which moreover maps N onto N_{η}. Given a C^{∗}-subalgebra A of F, let A_{η} denote the image of A under R_{η} which is a C^{∗}-algebra again.

Definition 5.1 Let A be a C^{∗}-subalgebra of the algebra F.

(a) A ^{∗}-homomorphism W : A → B of A into a C^{∗}-algebra B is fractal if, for every strongly monotonically increasing sequence η, there is a ^{∗}-homomorphism W_{η} : A_{η} → B such that W = W_{η}R_{η}.

(b) The algebra A is fractal if the canonical homomorphism π : A → A/(A ∩ N) is fractal.

Thus, given a subsequence {A_{η(n)}} of a sequence {A_{n}} which belongs to a fractal algebra A, it is possible to reconstruct the original sequence {A_{n}} from this subsequence modulo sequences in A∩N. This assumption is very natural for sequences arising from discretization procedures. On the other hand, the algebra F of all bounded sequences fails to be fractal. The following theorem is shown in [17] and will easily imply the fractality of the algebra A^{φ}.

Theorem 5.2 Let A be a unital C^{∗}-subalgebra of F. The algebra A is fractal if and only if there exists a family {W_{t}}_{t∈T} of unital and fractal ^{∗}-homomorphisms W_{t} from A into unital C^{∗}-algebras B_{t} such that the following equivalence holds for every sequence {A_{n}} ∈ A: The coset {A_{n}} + A ∩ N is invertible in A/(A ∩ N) if and only if W_{t}{A_{n}} is invertible in B_{t} for every t ∈ T.

To make the proof of the fractality of the algebra A^{φ} more transparent, we introduce a few new notations and rewrite Theorem 4.8 as follows. Set T := {1, 2, 3, 4} and define ∗- homomorphisms W_{1}, W_{2} : F^{W} → L(L_{σ}^{2}) and W_{3}, W_{4} : F^{W} → L(l^{2}) by W_{1} := W, W_{2} := W̃ and

W_{3}{A_{n}} := η_{φ}{A_{n}}^{o}(1), W_{4}{A_{n}} := η_{φ}{A_{n}}^{o}(−1).

Theorem 4.8’ (a) A sequence {A_{n}} ∈ A^{φ} is stable if and only if the operators W_{t}{A_{n}} are invertible for all t ∈ T.

(b) The mapping

smb : A^{φ} → L(L_{σ}^{2}) × L(L_{σ}^{2}) × L(l^{2}) × L(l^{2}),

{A_{n}} ↦ (W_{1}{A_{n}}, W_{2}{A_{n}}, W_{3}{A_{n}}, W_{4}{A_{n}})

is a ^{∗}-homomorphism with kernel N.

The first assertion is just a reformulation of Theorem 4.8, and the second one is a simple consequence of the fact that every ^{∗}-homomorphism between C^{∗}-algebras which preserves spectra also preserves norms.

Corollary 5.3 The algebra A^{φ} is fractal.

Proof. By Theorem 5.2, we have to prove that all homomorphisms W_{t} are fractal. For W_{1} and W_{2}, the fractality is evident: these homomorphisms act as strong limits, and the strong limit of a subsequence of {A_{n}} coincides with the strong limit of {A_{n}} itself. Concerning W_{3} and W_{4}, a closer look at the proof of Corollary 4.7 shows that the assertion of that corollary remains valid for every infinite subsequence of {M_{n}^{φ}SP_{n}} in place of the sequence {M_{n}^{φ}SP_{n}} itself. Thus, Coburn’s theorem again applies, yielding the fractality of W_{3} and W_{4}.

Fredholm sequences. Let J (F) stand for the smallest closed subset of F which contains all sequences {K_{n}} for which sup dim Im K_{n} is finite. The set J (F) is a closed two-sided ideal of F which contains the ideal N of the zero sequences. A sequence {A_{n}} ∈ F is called a Fredholm sequence if it is invertible modulo the ideal J (F). If {A_{n}} is a Fredholm sequence then there is a number k such that lim inf_{n→∞} σ_{k+1}^{(n)} > 0 (see [18, Theorem 2]). The smallest number k with this property is called the α-number of the sequence {A_{n}} and will be denoted by α{A_{n}}. This number plays the same role in the Fredholm theory of approximation sequences as the number dim Ker A plays in the common Fredholm theory for operators A on a Hilbert space.

The remainder of this section is devoted to the proof of the following result which characterizes the Fredholm sequences in A^{φ}.

Theorem 5.4 (a) A sequence {A_{n}} ∈ A^{φ} is Fredholm if and only if the operators W_{t}{A_{n}} are Fredholm operators for every t ∈ T.

(b) If {A_{n}} ∈ A^{φ} is a Fredholm sequence, then

α{A_{n}} = dim Ker W_{1}{A_{n}} + dim Ker W_{2}{A_{n}} + dim Ker W_{3}{A_{n}} + dim Ker W_{4}{A_{n}}.

(c) If {A_{n}} ∈ A^{φ} is Fredholm and k = α{A_{n}} > 0, then lim_{n→∞} σ_{k}^{(n)} = 0.

Fredholm inverse closed subalgebras. Let A be a unital and fractal C^{∗}-subalgebra of F which contains the ideal N. A sequence {K_{n}} in A is said to be of central rank one if, for every sequence {A_{n}} ∈ A, there is a sequence {μ_{n}} ∈ c (= the set of all convergent sequences of complex numbers) such that

K_{n}A_{n}K_{n} = μ_{n}K_{n}.

The smallest closed two-sided ideal of A which contains all sequences of central rank one will be denoted by J (A). The algebra A is called Fredholm inverse closed in F if J (A) = A ∩ J (F).

Sequences of essential rank one. Let A be as before. A central rank one sequence of A is said to be of essential rank one if it does not belong to the ideal N. For every essential rank one sequence {K_{n}}, let J{K_{n}} refer to the smallest closed ideal of A which contains the sequence {K_{n}} and the ideal N. In [18] it is shown that, if {K_{n}} and {L_{n}} are sequences of essential rank one in A, then either J{K_{n}} = J{L_{n}} or J{K_{n}} ∩ J{L_{n}} = N. Calling {K_{n}} and {L_{n}} equivalent in the first case we get a splitting of the sequences of essential rank one into equivalence classes, which we denote by S. Further, with every s ∈ S, there is associated a unique irreducible representation W_{s} of A into the algebra L(H_{s}) for some Hilbert space H_{s} such that the ideal J{K_{n}} is mapped onto the ideal K(H_{s}) of the compact operators on H_{s} and that the kernel of the mapping W_{s} : J{K_{n}} → K(H_{s}) is N. The main result of [18] reads as follows:

Theorem 5.5 Let A be a unital, fractal and Fredholm inverse closed C^{∗}-subalgebra of F which contains the ideal N.

(a) If {A_{n}} ∈ A is a Fredholm sequence, then the operators W_{s}{A_{n}} are Fredholm operators for every s ∈ S, and α{A_{n}} = ∑_{s∈S} dim Ker W_{s}{A_{n}}.

(b) If {A_{n}} ∈ A is Fredholm and k = α{A_{n}} > 0, then lim_{n→∞} σ_{k}^{(n)} = 0.

(c) If the family (W_{s})_{s∈S} is sufficient for the stability of sequences in A (in the sense that the invertibility of all operators W_{s}{A_{n}} implies the stability of {A_{n}}) and if all operators W_{s}{A_{n}} are Fredholm for a sequence {A_{n}} ∈ A, then this sequence is Fredholm.

We know already that A^{φ} is a fractal algebra. Thus, once we have shown that this algebra is Fredholm inverse closed and once we have identified the set S with T = {1, 2, 3, 4} as well as the representations W_{s} with the corresponding homomorphisms W_{t} figuring in Theorem 5.2, then Theorem 5.4 is proved.

Another type of ‘compact’ sequences. Let again A refer to a unital C^{∗}-subalgebra of F. Besides the ideal J (A) we consider a further ideal, K(A), which is the smallest closed two-sided ideal of A containing all sequences {K_{n}} ∈ A with dim Im K_{n} ≤ 1 for all n.

Proposition 5.6 Let A be a unital and fractal C^{∗}-subalgebra of F which contains N. Then, J (A) = K(A).

Proof. If {K_{n}} is a central rank one sequence in A then, since N ⊆ A, every matrix K_{n} has rank one. Thus, {K_{n}} belongs to K(A).

For the reverse inclusion, first observe that, under the made assumptions, the center of A consists exactly of all sequences of the form {α_{n}P_{n}} where {α_{n}} is in c, the set of all convergent sequences. Now let {K_{n}} ∈ A be a sequence with dim Im K_{n} ≤ 1 for all n. The fractality of A further implies the existence of the limit α := lim ∥K_{n}∥ (see [17, Theorem 4]). If α = 0, then {K_{n}} is a zero sequence, hence in N ⊆ K(A).

In case α ≠ 0 we are going to show that {K_{n}} is a central rank one sequence. Assume {K_{n}} is not of central rank one. Then there are a sequence {A_{n}} ∈ A and a non-convergent sequence (α_{n}) ∈ l^{∞} such that

K_{n}A_{n}K_{n} = α_{n}K_{n} for all n.

Choose two partial limits β ≠ γ of the sequence (α_{n}) as well as two subsequences μ and η of the the positive integers such that

α_{μ(n)} → β and α_{η(n)} → γ as n → ∞.

Then both sequences {α_{μ(n)}K_{μ(n)} − βK_{μ(n)}} and {α_{η(n)}K_{η(n)} − γK_{η(n)}} tend to zero. Hence, again due to the fractality of A (see [17, Theorem 1]), both sequences {α_{n}K_{n} − βK_{n}} and {α_{n}K_{n} − γK_{n}} are zero sequences. But then, also their difference (β − γ){K_{n}} goes to zero. Since ∥K_{n}∥ → α ≠ 0, this implies β = γ in contradiction to the choice of β and γ.

Identification of the ideals J (A^{φ}) = K(A^{φ}). Our next objective is to characterize the image of the ideal J (A^{φ}) under the mapping smb introduced in Theorem 4.8’.

Theorem 5.7 The homomorphism smb maps J (A^{φ}) onto K(L_{σ}^{2}) × K(L_{σ}^{2}) × K(l^{2}) × K(l^{2}).

Proof. It is shown in [18, Theorem 3] that every irreducible representation of a C^{∗}-algebra A maps every central rank one element of A onto a compact operator (an element k of a C^{∗}- algebra A is of central rank one if, for every a ∈ A, there is an element μ in the center of A such that kak = μk). In our setting, the homomorphisms W_{t}, 1 ≤ t ≤ 4 are irreducible since, in any case, the ideal of the compact operators belongs to the image of A^{φ} under W_{t}. Thus,

smb(J (A^{φ})) ⊆ K(L_{σ}^{2}) × K(L_{σ}^{2}) × K(l^{2}) × K(l^{2}).

For the reverse inclusion first recall that, by definition, the set J of all sequences {P_{n}KP_{n} + W_{n}LW_{n} + C_{n}} with K, L compact and {C_{n}} ∈ N is contained in A^{φ}. Since every compact operator can approximated as closely as desired by an operator with finite dimensional range, we have J ⊆ K(A^{φ}) and thus, by Proposition 5.6, J ⊆ J (A^{φ}). Moreover, it is easy to check for the sequence {K_{n}} := {P_{n}KP_{n} + W_{n}LW_{n} + C_{n}} that

W_{1}{K_{n}} = K and W_{2}{K_{n}} = L,

and it is immediate from the definition of W_{3} and W_{4} that W_{3}{K_{n}} = W_{4}{K_{n}} = 0. Hence, K(L_{σ}^{2}) × K(L_{σ}^{2}) × {0} × {0} lies in smb(J (A^{φ})). So it remains to show that smb(J (A^{φ})) contains all quadrupels of the form (0, 0, K, 0) and (0, 0, 0, K) with K a compact operator on l^{2}.

It is well known and easy to check that the smallest closed C^{∗}-subalgebra of L(l^{2}) which contains the shift operator Σ also contains all compact operators and that, in particular, K(l^{2}) is the smallest closed ideal of that algebra which contains the projection Π := I − ΣΣ^{∗} acting by

Π : l^{2} → l^{2}, {x_{0}, x_{1}, …} ↦ {x_{0}, 0, 0, …}.

Because of W_{t}{M_{n}^{φ}SP_{n}} = Σ^{∗} for t = 3 and t = 4, it is consequently sufficient to prove that the quadrupels (0, 0, Π, 0) and (0, 0, 0, Π) belong to smb(J (A^{φ})). For this goal, we consider the sequences {A_{n}} and {B_{n}} with

A_{n} := P_{n} − (M_{n}^{φ}SP_{n})^{∗}M_{n}^{φ}SP_{n} and B_{n} := M_{n}^{φ}V P_{n} − (M_{n}^{φ}SP_{n})^{∗}M_{n}^{φ}SV P_{n}

where, as above, V = ψI − iφS and ψ(x) = x. For the sequence {A_{n}} we have W_{1}{A_{n}} = I − φSφ^{−1}S (Section 2), and this operator is 0 as we have already mentioned several times. Similarly, W_{2}{A_{n}} = I −(−φSφ^{−1})(−S) = 0 due to Theorem 3.4. It is further immediate from the definitions that W_{3}{A_{n}} = W_{4}{A_{n}} = Π; thus, smb {A_{n}} = (0, 0, Π, Π).

Concerning the sequence {B_{n}}, we will first show that it is indeed an element of the algebra A^{φ}. To see this, write SV = S(ψI − iφS) as

SV = ψS − iφI + (SψI − ψS) − iK_{0} (5.12)

where the rank one operator K_{0} is given by (4.3). Thus, SV is the sum of a singular integral operator with continuous coefficients and of a compact operator, which maps into the convergence manifold of the M_{n}^{φ}. Hence, {M_{n}^{φ}SV P_{n}} ∈ A^{φ} and {B_{n}} ∈ A^{φ}. To compute the operators W_{t}{B_{n}}, we recall from Section 2 that

W_{1}{M_{n}^{φ}V P_{n}} = W_{1}{M_{n}^{φ}(ψI − iφS)P_{n}} = ψI − iφS = V, W_{2}{M_{n}^{φ}V P_{n}} = ψI + iφS,

W_{3}{M_{n}^{φ}V P_{n}} = W_{3}{M_{n}^{φ}(ψ(1)I − iφ(1)S)P_{n}} = I and W_{4}{M_{n}^{φ}V P_{n}} = I

and that

W_{1}{M_{n}^{φ}SP_{n}}^{∗} = −W_{2}{M_{n}^{φ}SP_{n}}^{∗} = φSφ^{−1}I and W_{3}{M_{n}^{φ}SP_{n}}^{∗} = W_{4}{M_{n}^{φ}SP_{n}}^{∗} = Σ.

For the operators W_{t}{M_{n}^{φ}SV P_{n}} we make use of identity (5.12) which together with the results of Section 2 yields

W_{1}{M_{n}^{φ}SV P_{n}} = SV, W_{2}{M_{n}^{φ}SV P_{n}} = −iφI − ψS,
W_{3}{M_{n}^{φ}SV P_{n}} = W_{3}{M_{n}^{φ}(ψ(1)S − iφ(1)I)P_{n}} = Σ^{∗} and W_{4}{M_{n}^{φ}SV P_{n}} = −Σ^{∗}.

Puzzling these pieces together we find

W_{1}{B_{n}} = V − φSφ^{−1}SV = (I − φSφ^{−1}S)V = 0,

and

W_{2}{B_{n}} = (ψI + iφS) + φSφ^{−1}(−ψS − iφI) = φSφ^{−1}(SψI − ψS)

which is also 0 since the range of the commutator SψI − ψS consists of constant functions only and since the operator Sφ^{−1} annihilates every constant function. Finally,

W_{3}{B_{n}} = I − ΣΣ^{∗} = Π and W_{4}{B_{n}} = −I − Σ(−Σ^{∗}) = −Π,

whence smb {B_{n}} = (0, 0, Π, −Π).

Thus, the homomorphism smb maps the sequences \frac{1}{2}\{A_{n} + B_{n}\} and \frac{1}{2}\{A_{n} − B_{n}\} onto the quadrupels (0, 0, Π, 0) and (0, 0, 0, Π), respectively. We show that these sequences are essential rank one sequences in A^{φ}. For this goal, we determine the matrix representation of A_{n} = P_{n} − (M_{n}^{φ}SP_{n})^{∗}M_{n}^{φ}SP_{n} with respect to the basis functions ũ_{k}, k = 0, …, n − 1. For 0 ≤ k, m ≤ n − 1 we have

τ_{km} := ⟨(M_{n}^{φ}SP_{n})^{∗}M_{n}^{φ}SP_{n}ũ_{m}, ũ_{k}⟩_{σ}
= ⟨M_{n}^{φ}SP_{n}ũ_{m}, M_{n}^{φ}SP_{n}ũ_{k}⟩_{σ} = ⟨M_{n}^{φ}Sũ_{m}, M_{n}^{φ}Sũ_{k}⟩_{σ}

and, since Sũ_{m} = SφU_{m} = iT_{m+1}, we obtain

τ_{km} = ⟨M_{n}^{φ}T_{m+1}, M_{n}^{φ}T_{k+1}⟩_{σ} = ⟨L_{n}^{φ}φ^{−1}T_{m+1}, L_{n}^{φ}φ^{−1}T_{k+1}⟩_{φ}
= \frac{π}{n + 1} ∑_{l=1}^{n} T_{m+1}(x_{ln}^{φ})T_{k+1}(x_{ln}^{φ})

= \frac{2}{n + 1} ∑_{l=1}^{n} cos \frac{(m + 1)lπ}{n + 1} cos \frac{(k + 1)lπ}{n + 1}

= \frac{1}{n + 1} ∑_{l=1}^{n} (cos \frac{(m − k)lπ}{n + 1} + cos \frac{(m + k + 2)lπ}{n + 1} ) .

Short calculations using the well known identity

∑_{l=1}^{n} cos lx = \frac{sin(n + \frac{1}{2})x}{2 sin \frac{x}{2}} − \frac{1}{2}

yield in case k = m

τ_{mm} = \frac{n − 1}{n + 1},

whereas in case k ≠ m

τ_{km} = \{ \begin{matrix}0 & if m + k is odd \\ −\frac{2}{n+1} & if m + k is even. \end{matrix}

Summarizing we find that A_{n} =: \frac{2}{n+1} [ε_{km}]_{k,m=0}^{n−1} with

ε_{km} = \{ \begin{matrix}0 & if m + k is odd \\ 1 & if m + k is even, \end{matrix}

and analogously one gets that the matrix representation of B_{n} with respect to the same basis is B_{n} =: \frac{2}{n+1} [ε_{k,m+1}]_{k,m=0}^{n−1}. It is evident now that the sequences \frac{1}{2}\{A_{n} + B_{n}\} and \frac{1}{2}\{A_{n} − B_{n}\} consist of matrices with rank one only. Thus, by Proposition 5.6, these sequences are of essential rank one, and this observation finishes the proof of the inclusion

K(L_{σ}^{2}) × K(L_{σ}^{2}) × K(l^{2}) × K(l^{2}) ⊆ smb(J (A^{φ}))

and of Theorem 5.7.

Fredholm inverse closedness of A^{φ}. To finish also the proof of Theorem 5.4, we have finally to show that the ideals J (A^{φ}) = K(A^{φ}) and J (F) ∩ A^{φ} of A^{φ} coincide. This equality is a simple consequence of the following result which, on its hand, is a generalization of [18, Theorem 3].

Theorem 5.8 Let A be a C^{∗}-subalgebra of F and let {J_{n}} be a sequence in J (F) ∩ A. Then, for every irreducible representation W : A → L(K) of A, the operator W {J_{n}} is compact.

Proof. The proof is based on [15, Prop. 4.1.8] which states that, under the above assumptions, there exist an irreducible representation π : F → L(H) of F with a certain Hilbert space H, a subspace H_{1} of H and an isometry U from H_{1} onto K such that H_{1} is an invariant subspace for π{J_{n}} and

W {J_{n}} = Uπ{J_{n}}|_{H_{1}}U^{∗}.

From [18, Theorem 3] we know that π{J_{n}} is a compact operator on H. Since H_{1} is invariant for π{J_{n}}, this moreover implies that π{J_{n}}|_{H_{1}} is a compact operator on H_{1}. Thus, W {J_{n}} is compact on K.

The Figures 1 and 2 revisited. Having Theorem 5.4 at our disposal, it is easy to explain the behaviour of the smallest singular values in Figures 1 and 2. In case A = aI + bS , a(x) = \sqrt{1 − x} , b(x) = −ix , we have for A_{n} = M_{n}^{φ}AP_{n}

dim Ker W_{1}{A_{n}} + dim Ker W_{2}{A_{n}} + dim Ker W_{3}{A_{n}} + dim Ker W_{4}{A_{n}}

= 0 + 0 + 1 + 0 = 1

whereas the same quantity is 0 + 0 + 1 + 1 = 2 in case a(x) = \sqrt{1.01 − x^{2}} , b(x) = −ix . Thus, in Figure 1 the lowest singular value tends to zero and the σ_{2}^{(n)} remain bounded away from zero by a positive constant for all n , whereas in Figure 2 both lim σ_{1}^{(n)} = 0 and lim σ_{2}^{(n)} = 0.

6 Appendix: Proof of Lemma 4.4

Lemma 6.1 The coset {M_{n}^{σ}SP_{n}}^{o} is a unitary element of A^{σ}/J .

Proof. We use (3.15) and (4.3) and get

M_{n}^{σ}SP_{n}(M_{n}^{σ}SP_{n})^{∗} = \frac{1}{2}M_{n}^{σ}SφSφ^{−1}L_{n}^{σ}(P_{n−1} + P_{n})

= \frac{1}{2}M_{n}^{σ}L_{n}^{σ}(P_{n−1} + P_{n}) + \frac{1}{2}M_{n}^{σ}K_{0}φ^{−1}L_{n}^{σ}(P_{n−1} + P_{n})

= \frac{1}{2}(P_{n−1} + P_{n}) + \frac{1}{2}M_{n}^{σ}K_{0}φ^{−1}L_{n}^{σ}(P_{n−1} + P_{n}) .

Now, from K_{0}φ^{−1}u = −\frac{1}{\sqrt{2π}}⟨u, U_{0}⟩_{σ} ,

∥L_{n}^{σ}P_{n}f∥_{σ}^{2} = \frac{π}{n} ∑_{k=1}^{n} [1 − (x_{kn}^{σ})^{2}] |(φ^{−1}P_{n}f)(x_{kn}^{σ})|^{2}

≤ const ∫_{−1}^{1} φ(x) |(φ^{−1}P_{n}f)(x)|^{2} dx = const ∥P_{n}f∥_{σ}^{2}

(see Lemma 3.1), and P_{n−1} = P_{n} − W_{n}P_{1}W_{n}, it follows {M_{n}^{σ}SP_{n}}^{o}{(M_{n}^{σ}SP_{n})^{∗}}^{o} = {P_{n}}^{o} . On the other hand we have

(M_{n}^{σ}SP_{n})^{∗}M_{n}^{σ}SP_{n} = φSφ^{−1}L_{n}^{σ}M_{n}^{σ}SP_{n} − \frac{1}{2}φSφ^{−1}L_{n}^{σ}W_{n}P_{1}W_{n}M_{n}^{σ}SP_{n}

and φSφ^{−1}L_{n}^{σ}M_{n}^{σ}SP_{n} = φSφ^{−1}SP_{n} = P_{n} due to (3.6) and (3.16).

For n ≥ 1, let Q_{n} : ℓ^{2} ⟶ ℓ^{2} denote the projection Q_{n}ξ = {ξ_{0}, …, ξ_{n−1}, 0, 0, …}, and define E_{n}^{±} : im P_{n} ⟶ im Q_{n} by

E_{n}^{+}u = \{\sqrt{\frac{π}{n}}u(x_{1n}^{σ}), …, \sqrt{\frac{π}{n}}u(x_{nn}^{σ}), 0, …\}

and

E_{n}^{−}u = \{\sqrt{\frac{π}{n}}u(x_{nn}^{σ}), …, \sqrt{\frac{π}{n}}u(x_{1n}^{σ}), 0, …\} .

Then, for ξ ∈ im Q_{n} ,

(E_{n}^{+})^{−1}ξ = ∑_{k=1}^{n} \sqrt{\frac{n}{π}}ξ_{k−1}ℓ̃_{kn}^{σ} =: E_{−n}^{+}ξ and (E_{n}^{−})^{−1}ξ = ∑_{k=1}^{n} \sqrt{\frac{n}{π}}ξ_{n−k}ℓ̃_{kn}^{σ} =: E_{−n}^{−}ξ ,

where

ℓ̃_{kn}^{σ}(x) = \frac{φ(x)T_{n}(x)}{φ(x_{kn}^{σ})(x − x_{kn}^{σ})T′_{n}(x_{kn}^{σ})} .

Remark that, for n ≥ 1 ,

T′_{n}(x) = nU_{n−1}(x) and T′_{n}(x_{kn}^{σ}) = \sqrt{\frac{2}{π}} \frac{n(−1)^{k+1}}{φ(x_{kn}^{σ})} .

In view of Lemma 3.1 and estimate (3.1), the sequences {E_{n}^{±}} and {E_{−n}^{±}} are uniformly bounded, i.e. there are constants c_{1} and c_{2} such that

\frac{π}{n} ∑_{k=1}^{n} |u(x_{kn}^{σ})|^{2} ≤ c_{1} ∥u∥_{σ}^{2} for all u ∈ im P_{n} (6.1)

and

∥\sqrt{\frac{n}{π}} ∑_{k=1}^{n} ξ_{k−1}ℓ̃_{kn}^{σ}∥_{σ}^{2} ≤ c_{2} ∑_{k=1}^{n} |ξ_{k−1}|^{2} for all ξ ∈ ℓ^{2} . (6.2)

Lemma 6.2 The sequences {E_{n}^{±}M_{n}^{σ}SP_{n}E_{−n}^{±}Q_{n}} and {(E_{n}^{±}M_{n}^{σ}SP_{n}E_{−n}^{±}Q_{n})^{∗}} are strongly convergent on ℓ^{2} .

Proof. The uniform boundedness of these sequences is obvious. So it remains to prove their convergence on the elements e_{m} = {δ_{mk}}_{k=0}^{∞} of the standard basis of ℓ^{2}. For n > m ≥ 1, one has

f_{mn} := E_{n}^{+}M_{n}^{σ}SP_{n}E_{−n}^{+}Q_{n}e_{m−1} = {Sℓ̃_{mn}^{σ}(x_{jn}^{σ})}_{j=1}^{n}

(here we identify {ξ_{0}, …, ξ_{n−1}, 0, …} ∈ ℓ^{2} with {ξ_{0}, …, ξ_{n−1}}). We compute, for x ≠ x_{kn}^{σ} ,

Sℓ̃_{kn}^{σ}(x) = \sqrt{\frac{2}{π}} \frac{(−1)^{k+1}}{n} \frac{1}{πi} ∫_{−1}^{1} \frac{φ(y)T_{n}(y)}{(y − x_{kn}^{σ})(y − x)} dy

= \sqrt{\frac{π}{2}} \frac{(−1)^{k+1}}{n(x_{kn}^{σ} − x)} \frac{1}{πi} ∫_{−1}^{1} (\frac{1}{y − x_{kn}^{σ}} − \frac{1}{y − x}) φ(y)T_{n}(y) dy

and, taking into account (3.16),

\frac{1}{π} ∫_{−1}^{1} \frac{1}{y − x} φ(y)T_{n}(y) dy = \frac{1}{π} ∫_{−1}^{1} \frac{1 − y^{2}}{y − x} T_{n}(y)σ(y) dy

= (1 − x^{2})U_{n−1}(y) − \frac{1}{π} ∫_{−1}^{1}(y + x)T_{n}(y)σ(y) dy
= (1 − x^{2})U_{n−1}(x) .

Thus, for j ≠ k ,

(Sℓ̃_{kn}^{σ})(x_{jn}^{σ}) = \frac{1}{ni} \frac{φ(x_{kn}^{σ}) − (−1)^{j+k}φ(x_{jn}^{σ})}{x_{kn}^{σ} − x_{jn}^{σ}} =: s_{jk}^{(n)} .

With the help of

\frac{d}{dx} [(1 − x^{2})U_{n−1}(x)] = (1 − x^{2})U′_{n−1}(x) − 2xU_{n−1}(x)

we get

(Sℓ̃_{kn}^{σ})(x_{kn}^{σ}) = −\frac{1}{ni} \frac{x_{kn}^{σ}}{φ(x_{kn}^{σ})} =: s_{kk}^{(n)} .

COLLOCATION METHODS FOR SYSTEMS OF SIE 31 It follows

s_{jk}^{(n)} = \{ \begin{matrix}−\frac{cos \frac{k+j−1}{2n}π}{in sin \frac{k+j−1}{2n}π} & , & j + k even, \\ −\frac{cos \frac{k−j}{2n} π}{in sin \frac{k−j}{2n} π} & , & j + k odd, \end{matrix} (6.3)

and, consequently,

|s_{jk}^{(n)}| ≤ \{ \begin{matrix}\frac{1}{k + j − 1} & , & j + k even, \\ \frac{1}{|k − j|} & , & j + k odd. \end{matrix}

Thus, for fixed m , the sequences {f_{mn}} = {s_{1m}^{(n)}, …, s_{nm}^{(n)}, 0, …} are uniformly dominated by a square summable sequence, which implies

f_{mn} ⟶ { lim_{n→∞} s_{jm}^{(n)}}_{j=1}^{∞} =: {s_{jm}} in ℓ^{2} ,

where s_{jk} = lim_{n→∞} s_{jk}^{(n)} , i.e.

s_{jk} = \{ \begin{matrix}−\frac{2}{πi(j + k − 1)} & , & j + k even \\ \frac{2}{πi(j − k)} & , & j + k odd \end{matrix} \} = \frac{1 − (−1)^{j−k}}{πi(j − k)} − \frac{1 − (−1)^{j+k−1}}{πi(j + k − 1)} . (6.4)

Thus,

E_{n}^{+}M_{n}^{σ}SP_{n}E_{−n}^{+}Q_{n} ⟶ S := [s_{jk}]_{j,k=1}^{∞} on ℓ^{2} .

Now it is easy to see that

(E_{n}^{+}M_{n}^{σ}SP_{n}E_{−n}^{+}Q_{n})^{∗} ⟶ S^{∗} = [\overline{s_{kj}}]_{j,k=1}^{∞} on ℓ^{2} ,

where \overline{s_{kj}} = \frac{1 − (−1)^{j−k}}{πi(j − k)} + \frac{1 − (−1)^{j+k−1}}{πi(j + k − 1)} . Hence, denoting by T (a) = [a_{j−k}]_{j,k=0}^{∞} and H(a) =

[a_{j+k+1}]_{j,k=0}^{∞} the Toeplitz and Hankel operator w.r.t. the symbol a(t) = ∑_{k=−∞}^{∞} a_{k}t^{k} , t ∈ 𝕋 , respectively, we have

S = T (ϕ) − H(ϕ) and S^{∗} = T (ϕ) + H(ϕ) with ϕ(t) = sgn(ℑt) . Finally, from

s_{n+1−j,n+1−k}^{(n)} = \{ \begin{matrix}\frac{cos \frac{k+j−1}{2n}π}{in sin \frac{k+j−1}{2n}π} & , & j + k even , \\ −\frac{cos \frac{j−k}{2n}π}{in sin \frac{j−k}{2n}π} & , & j + k odd , \end{matrix}

we get

E_{n}^{−}M_{n}^{σ}SP_{n}E_{−n}^{−}Q_{n} ⟶ −S and (E_{n}^{−}M_{n}^{σ}SP_{n}E_{−n}^{−}Q_{n})^{∗} ⟶ −S^{∗}

in ℓ^{2} .

We remark that the assertion of the previous lemma is not directly used in the following, but it essentially suggests the further considerations.

For k, n ∈ ℤ and n ≥ 1, let φ̃_{k}^{n} denote the characteristic function of the interval [\frac{k}{n}, \frac{k+1}{n}) multiplied by \sqrt{n}. Then the operators

Ẽ_{n} : ℓ_{ℤ}^{2} ⟶ L^{2}(ℝ) , {ξ_{k}}_{k=−∞}^{∞} ↦ ∑_{k=−∞}^{∞} ξ_{k}φ̃_{k}^{n}

and

Ẽ_{−n} = (Ẽ_{n})^{−1} : im Ẽ_{n} ⟶ ℓ_{ℤ}^{2} , ∑_{k=−∞}^{∞} ξ_{k}φ̃_{k}^{n} ↦ {ξ_{k}}_{k=−∞}^{∞} .

act as isometries. If we further denote the orthogonal projection from L^{2}(ℝ) onto im Ẽ_{n} by L̃_{n}, then we get as a consequence of [7], Prop. 2.10 and Exerc. E2.11, the following lemma.

Lemma 6.3 The sequence Ẽ_{n}SẼ_{−n}L̃_{n} : L^{2}(ℝ) ⟶ L^{2}(ℝ) is strongly convergent.

Lemma 6.4 The sequences {E_{−n}^{±}Q_{n}SQ_{n}E_{n}^{±}P_{n}} belong to the algebra F^{W} .

Proof. Obviously, the sequences under consideration are uniformly bounded. For k = 1, …, n , define functions

φ_{k}^{n}(x) = \{ \begin{matrix}\sqrt{\frac{n}{π}} & , & cos \frac{kπ}{n} ≤ x < cos \frac{k − 1}{n}π, \\ 0 & , & otherwise , \end{matrix}

and let R_{n}, S_{n} : L_{σ}^{2} ⟶ L_{σ}^{2} refer to the operators

R_{n}f = \sqrt{\frac{π}{n}} ∑_{k=1}^{n}⟨f, φ_{k}^{n}⟩_{σ}ℓ̃_{kn}^{σ} , S_{n}f = ∑_{k=1}^{n}⟨f, φ_{k}^{n}⟩_{σ}φ_{k}^{n} .

Then, in view of (6.2),

∥R_{n}f∥_{σ}^{2} ≤ c_{2} ∑_{k=1}^{n} |⟨f, φ_{k}^{n}⟩_{σ}|^{2} = c_{2} ∥S_{n}f∥_{σ}^{2} ≤ c_{2} ∥f∥_{σ}^{2} ,

i.e. the sequence {R_{n}} ⊂ L(L_{σ}^{2}) is uniformly bounded. Moreover, for the characteristic function f = χ_{[x,y]} of an interval [x, y] ⊂ [−1, 1] , we have

|⟨f, φ_{k}^{n}⟩_{σ} − \sqrt{\frac{π}{n}}f(x_{kn}^{σ})| = |\sqrt{\frac{n}{π}} ∫\frac{k−1}{n} π^{\frac{kπ}{n}} [f(cos s) − f (cos \frac{2k − 1}{2n}π)] ds|

= 0 , x, y ∉ (cos \frac{kπ}{n} , cos \frac{k − 1}{n}π) ,

≤ \sqrt{\frac{π}{n}} ,  otherwise,

which implies, again by (6.2),

∥R_{n}f − M_{n}^{σ}f∥_{σ}^{2} = ∥\sqrt{\frac{n}{π}} ∑_{k=1}^{n} [⟨f, φ_{k}^{n}⟩_{σ} − \sqrt{\frac{π}{n}}f(x_{kn}^{σ})] ℓ̃_{kn}^{σ}∥_{σ}^{2} ≤ c_{2} \frac{2π}{n} .

Consequently, R_{n}f ⟶ f in L_{σ}^{2} for all f ∈ L_{σ}^{2} . In particular, we get the following equivalences (ξ_{k}^{n} ∈ ℂ)

∑_{k=1}^{n} ξ_{k}^{n}ℓ̃_{kn}^{σ} ⟶ f in L_{σ}^{2} ⇔ lim_{n→∞} ∥∑_{k=1}^{n} ξ_{k}^{n}ℓ̃_{kn}^{σ} − R_{n}f∥_{σ} = 0

⇔ lim_{n→∞} ∑_{k=1}^{n} |\sqrt{\frac{π}{n}}ξ_{k}^{n} − ⟨f, φ_{k}^{n}⟩_{σ}|^{2} = 0

⇔ lim_{n→∞} ∥∑_{k=1}^{n} ξ_{k}^{n}φ_{k}^{n} − S_{n}f∥_{σ} = 0

⇔ ∑_{k=1}^{n} ξ_{k}^{n}φ_{k}^{n} ⟶ f in L_{σ}^{2} .

Since R_{n} ⟶ I in L_{σ}^{2}, the convergence E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}P_{n}f ⟶ g in L_{σ}^{2} for an f ∈ L_{σ}^{2} is equivalent to

E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}R_{n}f = \sqrt{\frac{n}{π}} E_{−n}^{+}Q_{n}SQ_{n}\{⟨f, φ_{k}^{n}⟩_{σ}\}_{k=1}^{n} ⟶ g in L_{σ}^{2}

and, due to the previous considerations, equivalent to

∑_{j=1}^{n} ∑_{k=1}^{n} s_{jk}⟨f, φ_{k}^{n}⟩_{σ}φ_{j}^{n} ⟶ g in L_{σ}^{2} . (6.5)

The mapping T : L_{σ}^{2} ⟶ L^{2}(0, 1) defined by (T f)(s) = \sqrt{π} f(cos πs) is an isometry, whereby T φ_{k}^{n} = φ̃_{k}^{n} . Consequently, (6.5) is equivalent to

∑_{j∈ℤ} ∑_{k∈ℤ}s_{jk}⟨χ_{[0,1]}T f, φ̃_{k}^{n}⟩_{L^{2}(ℝ)}φ̃_{j}^{k} ⟶ χ_{[0,1]}T g in L^{2}(ℝ) . (6.6)

The left-hand side of (6.6) can be written as Ẽ_{n}SẼ_{−n}L̃_{n}χ_{[0,1]}T f , and Lemma 6.3 guarantees the convergence of this sequence. Hence, we have proved that W {A_{n}} exists for A_{n} = E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}L_{n} .

To prove the existence of W̃{A_{n}}, we proceed as follows. By definitions and by taking into account (3.11) and (3.18) we find, for u ∈ L_{σ}^{2} and ξ ∈ ℓ^{2} ,

E_{n}^{+}W_{n}u = \{\sqrt{\frac{π}{n}} ∑_{j=0}^{n−1}⟨u, ũ_{n−1−j}⟩_{σ}ũ_{n−1−j}(x_{kn}^{σ})\}_{k=1}^{n}

= \{\sqrt{\frac{π}{n}} ∑_{j=0}^{n−1}⟨u, ũ_{n−1−j}⟩_{σ}(−1)^{k+1}γ_{j}T_{j}(x_{kn}^{σ})\}_{k=1}^{n} ,

ℓ̃_{kn}^{σ} = M_{n}^{σ}ℓ̃_{kn}^{σ} = \frac{π}{n} ∑_{j=0}^{n−1} ε_{jn}ũ_{j}(x_{kn}^{σ})ũ_{j} , (6.7)

and

W_{n}E_{−n}^{+}Q_{n}ξ = \sqrt{\frac{π}{n}} ∑_{k=1}^{n} ξ_{k−1} ∑_{j=0}^{n−1} ε_{n−1−j,n}ũ_{n−1−j}(x_{kn}^{σ})ũ_{j}

= \sqrt{\frac{π}{n}} ∑_{k=1}^{n} ξ_{k−1} ∑_{j=0}^{n−1} γ_{j}(−1)^{k+1}T_{j}(x_{kn}^{σ})ũ_{j}

= J_{σ}^{−1}\sqrt{\frac{π}{n}} ∑_{k=1}^{n} ξ_{k−1}(−1)^{k+1} ∑_{j=0}^{n−1} T_{j}(x_{kn}^{σ})T_{j}

= J_{σ}^{−1}\sqrt{\frac{n}{π}} ∑_{k=1}^{n} ξ_{k−1}(−1)^{k+1}ℓ_{kn}^{σ} .

Thus, if we define P_{n}^{σ} : L_{σ}^{2} ⟶ L_{σ}^{2} , E_{n}^{σ} : im P_{n}σ ⟶ im Q_{n} , and W̃ : ℓ^{2} ⟶ ℓ^{2} by

P_{n}^{σ}f = ∑_{k=0}^{n−1}⟨u, T_{k}⟩_{σ}T_{k} , E_{n}^{σ}f = {f(x_{kn}^{σ})}_{k=1}^{n} , W̃ξ = {(−1)^{k}ξ_{k}}_{k=0}^{∞} ,

respectively, then E_{n}W_{n} = W̃E_{n}^{σ}J_{σ}P_{n} = W̃E_{n}^{σ}P_{n}^{σ}J_{σ} and W_{n}E_{−n}^{+} = J_{σ}^{−1}E_{−n}^{σ}W̃ , where E_{−n}^{σ} := (E_{n}^{σ})^{−1} . Consequently,

W_{n}E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}W_{n} = J_{σ}^{−1}E_{−n}^{σ}Q_{n}W̃SW̃Q_{n}E_{n}^{σ}P_{n}^{σ}J_{σ}. (6.8)

The strong convergence of this sequence can be proved as the strong convergence of the operators E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}P_{n} .

Defining J_{n}u = ∑_{j=0}^{∞} ε_{jn}⟨u, ũ_{j}⟩_{σ}ũ_{j} (ε_{jn} := 1 , j > n − 1) and taking into account (6.7), we can

write, for u ∈ L_{σ}^{2} and ξ ∈ ℓ^{2} ,

⟨Q_{n}E_{n}^{+}P_{n}J_{n}u, ξ⟩_{ℓ^{2}} = \sqrt{\frac{π}{n}} ∑_{k=1}^{n} ∑_{j=0}^{n−1} ε_{jn}⟨u, ũ_{j}⟩_{σ}ũ(x_{kn}^{σ})\overline{ξ}_{k−1}

= ⟨u, \sqrt{\frac{n}{π}} ∑_{k=1}^{n} ξ_{k−1} \frac{π}{n} ∑_{j=0}^{n−1} ε_{jn}ũ_{j}(x_{kn}^{σ})ũ_{j}⟩_{σ}

= ⟨u, E_{−n}^{+}Q_{n}ξ⟩_{σ} ,

which leads to (Q_{n}E_{n}^{+}P_{n}J_{n})^{∗} = E_{−n}^{+}Q_{n} , i.e. (Q_{n}E_{n}^{+}P_{n})^{∗} = J_{n}^{−1}E_{−n}^{+}Q_{n} . Furthermore, again due to (6.7),

⟨E_{−n}^{+}Q_{n}ξ, u⟩_{σ} = \sqrt{\frac{n}{π}} ∑_{k=1}^{n} ξ_{k−1}⟨ℓ̃_{kn}^{σ}, P_{n}u⟩_{σ}

= \sqrt{\frac{n}{π}} ∑_{k=1}^{n} ξ_{k−1} ∑_{j=0}^{n−1} \overline{⟨u, ũ_{j}⟩_{σ}}ε_{jn}ũ(x_{kn}^{σ})

= ⟨ξ, E_{n}^{+}P_{n}J_{n}u⟩_{ℓ^{2}} ,

such that (E_{−n}^{+}Q_{n})^{∗} = E_{n}^{+}P_{n}J_{n} . Hence,

(E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}P_{n})^{∗} = J_{n}^{−1}E_{−n}^{+}Q_{n}S^{∗}Q_{n}E_{n}^{+}P_{n}J_{n} .

Analogously, with the help of (6.8) one can show that

(W_{n}E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}W_{n})^{∗} = J_{σ}^{∗}E_{−n}^{σ}Q_{n}W̃S^{∗}W̃Q_{n}E_{n}^{σ}P_{n}^{σ}J_{n}^{−∗} .

Since J_{n} ⟶ I in L_{σ}^{2} we conclude also the strong convergence of A_{n}^{∗}P_{n} and (W_{n}A_{n}W_{n})^{∗}P_{n} , and the proof of {E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}P_{n}} ∈ F^{W} is done.

For the second sequence we can use the same arguments taking into account the following two facts:

a) E_{n}^{−} = V_{n}E_{n}^{+} and E_{−n}^{−} = E_{−n}^{+}V_{n} , where V_{n}ξ = {ξ_{n−1}, …, ξ_{0}, 0, …} , ξ ∈ ℓ^{2} . Consequently, E_{−n}^{−}Q_{n}SQ_{n}E_{n}^{−}P_{n} = E_{−n}^{+}V_{n}SV_{n}E_{n}^{+}P_{n} .

b) V_{n}T (a)V_{n} = Q_{n}T (ã)Q_{n} , where ã = a(t^{−1}) , and H(ϕ) belongs to the smallest closed subalgebra T of L(ℓ^{2}) containing all Toeplitz operators T (a) with a ∈ PC(𝕋) . Thus, V_{n}SV_{n} = Q_{n}S_{0}Q_{n} , S_{0} ∈ T , and Lemma 6.3 remains true for S_{0} instead of S .

This completes the proof of the lemma.

In what follows we will use the local principle of Gohberg and Krupnik (see below). Although it is possible to apply the local principle of Allan and Douglas (cf. Section 4) equivalently, we decided to go this other way with the aim of a little more clear presentation. Let B be a unital Banach algebra. A subset M ⊂ B is called a localizing class if 0 ∉ M and if, for all a_{1}, a_{2} ∈ M, there exists an element a ∈ M such that

aa_{j} = a_{j}a = a for j = 1, 2 .

Let M be a localizing class. Two elements x, y ∈ B are called M-equivalent (in symbols:

x ∼^{M} y), if

inf_{a∈M} ∥a(x − y)∥_{B} = inf_{a∈M} ∥(x − y)a∥_{B} = 0 .

Further, x ∈ B is called M-invertible if there exist a_{1}, a_{2} ∈ M and z_{1}, z_{2} ∈ B such that

z_{1}xa_{1} = a_{1} and a_{2}xz_{2} = a_{2} .

A system {M_{τ}}_{τ∈Ω} of localizing classes (Ω is an arbitrary index set) is said to be covering if, for each system {a_{τ}}_{τ∈Ω} with a_{τ} ∈ M_{τ} , there exists a finite subsystem a_{τ_{1}}, …, a_{τ_{n}} such that a_{τ_{1}} + ⋯ + a_{τ_{n}} is invertible in the algebra B .

Local principle of Gohberg and Krupnik ([6, Theorem XII.1.1]) Let B be a unital Banach algebra, {M_{τ}}_{τ∈Ω} a covering system of localizing classes in B , x ∈ B and x ∼^{M_{τ}} x_{τ} for all τ ∈ Ω.

Then x is M_{τ}-invertible if and only if x_{τ} is M_{τ}-invertible. If x commutes with all elements from ⋃_{τ∈Ω} M_{τ} , then x is invertible in B if and only if x_{τ} is M_{τ}-invertible for all τ ∈ Ω .

For τ ∈ [−1, 1], let

m_{τ} := {f ∈ C[−1, 1] : 0 ≤ f(x) ≤ 1 , f(x) ≡ 1 in some neighborhood of τ}

and define M_{τ} := {{M_{n}^{σ}fP_{n}}^{o} : f ∈ m_{τ}} . Then {M_{τ}}_{τ∈[−1,1]} forms a covering system of localizing classes in F^{W} /J , which, due to Lemma 4.1, has the property that all elements of this system commute with all elements of the form {M_{n}^{σ}(aI + bS)P_{n}}^{o} , a, b ∈ PC . The Relation (3.5) shows that, for a, a_{1}, b, b_{1} ∈ PC , the cosets {M_{n}^{σ}(aI + bS)P_{n}}^{o} and {M_{n}^{σ}(a_{1}I + b_{1}S)P_{n}}^{o} are M_{τ}-equivalent if a(τ ± 0) = a_{1}(τ ± 0) and b(τ ± 0) = b_{1}(τ ± 0) .

Lemma 6.5 The cosets {M_{n}^{σ}SP_{n}}^{o} and {±E_{−n}^{±}Q_{n}SQ_{n}E_{n}^{±}P_{n}}^{o} are M_{±1}-equivalent.

Proof. Let a ∈ m_{1} and B_{n} = E_{n}^{+}M_{n}^{σ}aP_{n}E_{−n}^{+}(E_{n}^{+}M_{n}^{σ}SP_{n}E_{−n}^{+}Q_{n} − Q_{n}SQ_{n}) . Then

B_{n} = diag[a(x_{1n}^{σ}), …, a(x_{nn}^{σ})][s_{jk}^{(n)} − s_{jk}]_{j,k=1}^{n}

and, due to the uniform boundedness of the sequences {E_{n}^{+}} and {E_{−n}^{+}} ,

∥M_{n}^{σ}aP_{n}(M_{n}^{σ}SP_{n} − E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}P_{n})∥_{L(L_{σ}^{2})} ≤ const ∥B_{n}∥_{L(ℓ^{2})} .

Assume that supp a ⊂ [cos πε, 1] , 0 < ε < 1/4 . Since the function g(z) = z cot z = 1− \frac{1}{3}z^{2} +⋯ is analytic for |z| < π , we have |cot z − z^{−1}| ≤ const |z| for |z| ≤ 3π/4 , which leads to

|s_{jk}^{(n)} − s_{jk}| ≤ const \frac{k + j}{n^{2}} . It follows

∥B_{n}∥_{L(ℓ^{2})}^{2} ≤ ∑_{1≤j≤nε+ \frac{1}{2}} ∑_{k=1}^{n} |s_{jk}^{(n)} − s_{jk}|^{2} ≤ const (ε + \frac{1}{n})

and, consequently,

∥\{M_{n}^{σ}aP_{n}(M_{n}^{σ}SP_{n} − E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}P_{n})\}^{o}∥ ≤ const \sqrt{ε} .

Analogously one can show that

∥\{(M_{n}^{σ}SP_{n} − E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}P_{n})M_{n}^{σ}aP_{n}\}^{o}∥ ≤ const \sqrt{ε} ,

and the M_{1}-equivalence of {M_{n}^{σ}SP_{n}}^{o} and {E_{−n}^{+}Q_{n}SQ_{n}E_{n}^{+}}^{o} is proved. The proof of the M_{−1}-equivalence of {M_{n}^{σ}SP_{n}}^{o} and {−E_{−n}^{−}Q_{n}SQ_{n}E_{n}^{−}}^{o} is similar.

Lemma 6.6 The sequence {E_{−n}^{±}Q_{n}SQ_{n}E_{n}^{±}P_{n} − λP_{n}} is stable in L_{σ}^{2} if and only if λ ∉ 𝔻_{+} := {z ∈ ℂ : |z| ≤ 1, ℑz ≥ 0}.

Proof. Due to [2, Prop. 4.1], the sequence {Q_{n}S^{∗}Q_{n} − λO_{n}} is stable in ℓ^{2} if and only if λ ∉ 𝔻_{−} := {z ∈ ℂ : |z| ≤ 1, ℑz ≤ 0}. This fact implies the assertion immediately (recall the uniform boundedness of E_{n}^{±} and E_{−n}^{±} = (E_{n}^{±})^{−1}).

Proof of Lemma 4.4. Let τ = ±1 . Lemma 3.5 and the local principle of Allan and Douglas imply that σ_{(G/K)/J_{τ}^{G}}(S) = 𝕋_{τ}. Further, by Lemmas 4.3 and 6.1,

𝕋_{τ} ⊂ σ_{(A^{σ}/J)/J_{τ}^{σ}} ({M_{n}^{σ}SP_{n}}^{o} + J_{τ}^{σ}) ⊂ 𝕋 .

Let λ ∈ 𝕋\𝕋_{τ} . Due to Lemmas 6.6 and 6.4, the coset {±E_{−n}^{±}Q_{n}SQ_{n}E_{n}^{±}P_{n} − λP_{n}}^{o} is invertible in F^{W} /J . By Lemma 6.5 and the local principle of Gohberg and Krupnik we get the M_{τ}- invertibility of {M_{n}^{σ}SP_{n} − λP_{n}}^{o} .

Let χ(x) = \frac{1+x}{2} and λ ∈ 𝕋\𝕋_{1} . Then \{M_{n}^{σ}χSP_{n} − λP_{n}\}^{o} is M_{1}-equivalent to \{M_{n}^{σ}SP_{n} − λP_{n}\}^{o}, and M_{−1}-equivalent to λ {P_{n}}^{o}. So, M_{1}- and M_{−1}-invertible. For τ ∈ (−1, 1) we use the fact that (A^{σ}/J)/J_{τ}^{σ} is *-isomorphic to a C^{∗}-algebra of continuous 2 × 2 matrix functions on [0, 1] , which was shown in Section 4. This isomorphism sends

\{M_{n}^{σ}χSP_{n} − λP_{n}\}^{o} + J_{τ}^{σ} = \{\frac{1 + τ}{2} M_{n}^{σ}SP_{n} − λP_{n}\}^{o} + J_{τ}^{σ}

into the function

μ ↦ [ \begin{matrix}\frac{1+τ}{2} − λ & 0 \\ 0 & −\frac{1+τ}{2} − λ \end{matrix} ] ,

which is invertible. Consequently, for each τ ∈ (−1, 1) , there exist {B_{n}^{τ}} ∈ A^{σ} and {T_{n,k}^{τ}}^{o} ∈ J_{τ}^{σ} , k = 1, 2 , such that

{B_{n}^{τ}}^{o} {M_{n}^{σ}χSP_{n} − λP_{n}}^{o} = {P_{n}}^{o} + {T_{n,1}^{τ}}^{o}

and

{M_{n}^{σ}χSP_{n} − λP_{n}}^{o} {B_{n}^{τ}}^{o} = {P_{n}}^{o} + {T_{n,2}^{τ}}^{o} .

Since {T_{n,k}^{τ}}^{o} is M_{τ}-equivalent to the zero element of A^{σ}/J , we get the M_{τ}-invertibility of {M_{n}^{σ}χSP_{n} − λP_{n}}^{o} also for τ ∈ (−1, 1) . The local principle of Gohberg and Krupnik gives the invertibility of {M_{n}^{σ}χSP_{n} − λP_{n}}^{o} in F^{W} /J . Because of the inverse closedness of C^{∗}- subalgebras, the inverse of {M_{n}^{σ}χSP_{n} − λP_{n}}^{o} belongs to A^{σ}/J , which implies, due to the local principle of Allan and Douglas, the invertibility of {M_{n}^{σ}χSP_{n} − λP_{n}}^{o} + J_{1}^{σ} .

The invertibility of {M_{n}^{σ}χSP_{n} − λP_{n}}^{o} + J_{−1}^{σ} for λ ∈ 𝕋 \ 𝕋_{−1} can be shown analogously.

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Peter Junghanns, Bernd Silbermann Technische Universiät Chemnitz Fakultät für Mathematik D-09107 Chemnitz Germany peter.junghanns@mathematik.tu-chemnitz.de bernd.silbermann@mathematik.tu-chemnitz.de

Steffen Roch Technische Universität Darmstadt Fachbereich Mathematik Schloßgartenstraße 7 D-64289 Darmstadt Germany roch@mathematik.tu-darmstadt.de