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Вычислительные технологии
Том 6, № 1, 2001
COLLOCATION METHODS FOR SYSTEMS OF CAUCHY SINGULAR INTEGRAL EQUATIONS
Technische Universität Chemnitz Fakultät für Mathematik, Germany e-mail: peter.junghanns@mathematik.tu-chemnitz.de
Dedicated to Erhard Meister on the occasion of his seventieth birthday
Получены необходимые и достаточные условия устойчивости методов коллокации по узлам Чебышева первого и второго рода для скалярных сингулярных интегральных уравнений Коши с кусочно-непрерывными коэффициентами на отрезке и для систем таких уравнений. Рассматривается также поведение сингулярных значений матриц дискретных уравнений.
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
where a and b are given piecewise continuous functions and the equation is considered in an appropriate weighted L -space L2 . 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 {An} belonging to a C*-algebra A, which is generated by the
© P. Junghanns, S. Roch, B. Silbermann, 2001.
INTERVAL
P. Junghanns, S. Roch, B. Silbermann
1 < x < 1,
(1.1)
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(L2) , W : A —► L(L^) and n± : A —► L(€2) such that, in case of collocation w.r.t. Chebyshev nodes of second kind, a sequence {An} G A is stable if and only if the operators W {An} , W {An} , and n± {An} are invertible. In case of collocation w.r.t. Chebyshev nodes of first kind the invertibility of W {An} and W {An} is necessary and sufficient for the stability of {An} G 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 a(x) = (1 — x2)-1/2 and <^(x) = (1 — x2)1/2 denote the Chebyshev weights of first and second kind on the interval (—1, 1), respectively, and let L2 refer to the Hilbert space of all w.r.t. a on (—1,1) square integrable functions, equipped with inner product and norm
(u,v)2 = u(x)v(x)a(x) dx and ||u||2 := \J(u, u)2.
For u G (a, and n > 0, let p^ stand for the w.r.t. u orthonormal polynomial of degree n and abbreviate p^ and p^ to Tn and respectively. It is well known that
T0(x) = —= , Tn(cos s) = \/2/n cos ns , n > 1, s G (0, n) ,
and
. . /——sin(n + 1)s . .
Un (cos s) = V2/n---— , n > 0, s G (0, n).
sin s
Further define weighted polynomials wn := Both {Tn}n=0 and {uln}n°=0 form an orthonormal basis in L2 . The zeros of p^ are known to be
x2„ = cos —-n and x^„ = cos —— where j = 1, ... , n.
Jn 2n Jn n +1
Further, the Lagrange interpolation operator L^ acts on a function f : (—1,1) —► C by
ry __Ifi^ ( If I
x x kn __PnX^)
w (
f = \ ' f (x) = II ~ kn = _tinl
j=1 fc=1,fc=j jn
w _ (x _ )(pw)/(xw )
jn ^fcn V^ ^jnJKfnJ V"SW
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^, consider the Cauchy singular integral equation
a(x)u(x) + b(x) i1 dy = f (x), —1 <x< 1. (2.1)
ni J-i y — x
Both the Cauchy singular integral operator
S : L — L^ , u ~ -1 I'' UM dy
a n 7-1 y
and the multiplication operators al : L^ —► L^, u ^ au, belong to the algebra L(L^) of all linear and bounded operators in L^ which justifies to consider equation (2.1) on this space. For the approximate solution of (2.1), we look for a function u £ L^ of the form
a n-1
Un ^ ^ Ckn^k ; Cn [Cfcn]n=o ^ C k=0
which satisfies the collocation system
n)un(xn) + hj T dy = f(j), j = l,...,n. (2.2)
J=i - - j
If we introduce Fourier projections
n-1
Pn : La -* L a uk
k=0
and weighted interpolation operators := ^¿nV-1, then the collocation system (2.2) can be rewritten as an operator equation
Mn(al + bS)Pnun = Mnf , un £ im Pn . (2.3)
The reason for using instead of ¿n is that the range of coincides with that one of the Fourier projection Pn.
Our main concern is the stability of the sequence {An} with An = APn and A = al + bS. Recall that a sequence {An} is stable if there exists an n0 such that the operators An : im Pn —► im Pn are invertible for n > n0 and that their inverses A-1 are uniformly bounded:
sup llA-^PnlU_t 2 < ro .
n>n0 a a
If the sequence {An} is stable and if u* £ La and un £ im Pn are the solutions of (2.1) and (2.3), respectively, then the estimate
\PnU* - ux < IK^n^ ^ \\AnPnu* - Au* t + 11/ - M^J ll(J
shows that un converges to u* in the norm of L^ if the method (2.3) is consistent, i.e. if AnPn —► A (strong convergence) and if / —► / (convergence in L^). The stability result for the collocation method (2.3), which is a conclusion of Theorem 4.8, reads as follows.
(llAnPnU* - Aula + 11/ - M f IIa)
Theorem 2.1. Let a, b £ PC. Then the sequence (M2(aI + bS)Pn} is stable if and only if the operator aI + bS is invertible on L2, and the sequence (M^(a/ + bS)Pn} is stable if and only if the operators aI + bS and aI — bS are invertible on L^.
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 (An} of linear operators An : im Pn —► im Pn, provided with the supremum norm ||{Ara}||F := supra>! ||AJL2and with operations (An} + {£„} := (An + Bra} , {Ara}{£„} := (AnBn} , and (An}* := (An} . Further, let N be the two-sided closed ideal of F consisting of all sequences (Cn} £ F such that limn^^ ||CnPn||L2^L2 = 0. Then a simple Neumann series argument shows that the sequence (An} £ F is stable if and only if the coset (An} + 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
n— 1
Wn : L2 —► L2, u ^ in— 1—j>2Mj,
j=0
and consider the set FW of all sequences {An} £ F, for which the strong limits
W(An} := s-lim AnPn , (W(An})* = s-lim AnPn , (2.4)
and
W(An} := s-lim WnAnWn , (W(An})* = s-lim (WnAnWn)*Pn (2.5)
exist. Furthermore, let J refer to the collection of all sequences (An} of the form
An = PnKiPn + WnK2Wn + Cn with Kj £ K(L2), (Cn} £ N, where K(L^) C L(L^) stands for the ideal of all compact operators.
Lemma 2.2. (a) FW is a C*-subalgebra of F, and J is a closed two-sided ideal of FW.
(b) A sequence (An} £ FW is stable if and only if the operators W(An}, W(An} : L^ —► L^ and the coset (An} + J £ FW/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 (Wn} 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^ f —► f in L^ under suitable restrictions for f and that AnPn —► A strongly. The proof of the approximation properties of the interpolation operators M^ is based on the following auxiliary results.
Lemma 3.1 ([14], Theor. 9.25).. Let v be classical Jacobi weights with ^v G LL(-1,1) and let j G N be fixed. Then, for each polynomial q with deg q < jn,
n „ 1
X¡Aüq(xL)|v (xL) < consW |q(x)|^(x)v (x) dx, fc=i
where the constant does not depend on n and q and where x^n and A^n = f-L £^n(x)^(x) dx are the nodes and the Christoffel numbers of the Gaussian rule w.r.t. the weight respectively.
Let denote the Gaussian quadrature rule w.r.t. the weight
n
Qn/ = X/ Afcn/(xfcn), fc=i
and write R = R(-1,1) for the set of all functions / : (-1,1) —► C , which are bounded and Riemann integrable on each interval [a,^] C (-1,1).
Lemma 3.2 ([5], Satz III.1.6b and Satz III.2.1).. Let ^(x) = (1-x)7(1+x)á with y,¿>-1. If / G R satisfies
|/(x)| < const (1 - x)£-1-Y(1 + x)£-1- , -1 < x < 1 , for some £ > 0 , then lim Qn/ = /(x)^(x) dx . If even
n—J— 1
|/(x)| < const (1 - x)£-i++2(1 + x)£-^ , -1 < x < 1 , then limn—^ ||/ - L/= 0 .
Corollary 3.3. Let / G R and |/(x)| < const (1 - x2)£-4 , -1 < x < 1, for some £ > 0. Then M^/ —► / in L2 for u = ^ and u = a.
Proof. Since |/- M^/||CT = ||^-1/ - L^^-1/we can immediately apply the second assertion of Lemma 3.2 to get the assertion in case u = To consider the case u = a, introduce the quadrature rule
Qnf = (Lnf )(xMx) dx = X] aknf (xan) ,
k=1
where
= f1 Tn(x) y(x) = f1 Tn(x)(1 — x2)a(x) = n [1 — (xan)2]
°"kn x _ xa T' (xa ) dx (x _ xa )T/ (xa ) dx n
./-1 x xkn Tn(xkn) «/-1 (x xkn)Tn(xkn) n
for n > 2. Consequently,
n
Qnf = — X C1 — (xan)1 f (xan) • k=1
Since the nodes xan of the quadrature rule Qn are the zeros of 2Tn = Un — Un-2, the estimate
T |(Lnf )(x)|2 p(x) dx < 2Qn|f |2
holds true (see [5, Hilfssatz 2.4, § III.2]). As an immediate consequence we obtain
2 n n
||Mn2 f ||2 = ||Ln^—11 f < — £ If (x2n)|2 = 2 Qnif |2 • (3.1)
k=i
Now let 6 > 0 be arbitrary and p be a polynomial such that ||<^p — f ||2 < 6. For n > deg p we have ||M-f — f ||J < 2 (||M-(pp — f)||J + ||pp — f ||J) • Since, in view of Lemma 3.2, limn^ Qni^P — f I2 = ||^P — f ||2 , we get via (3.1) that limsup ||M*f — f ||J < 662 , which
n—><(
proves the assertion in second case, too.
The strong convergence of the sequence (M^ (a/ + bS)Pn} is part of the assertion of the following theorem. For the description of the occuring strong limits we need two further operators: the isometry
: L2 —► L2 , u —► 7n(u,in>2Tn , (3.2)
n=0
where y0 = \[2 and Yn = 1 for n > 1, and the shift operator
V : L2 —► L2 , u ^ in>2in+i (3.3)
n=0
with its adjoint V* : L2 —> L2, u ^™=0(u,M„+i)ffm
Theorem 3.4. For a,b £ PC, the sequence (A^} := (M^(a/ + bS)Pn} belongs to the algebra FW. In particular, W(A^} = a/ + bS and
a/ — bS, w = .
w(A-} ;
J-1(aJ2 + 1 bV *), w = a.
Proof. Step 1: Uniform boundedness. Let a,b £ PC and w £ (<^, a}. We have to verify the strong convergence of each of the sequences (M^(a/ + bS)Pn}, ((M^(a/ + bS)Pn)*}, (WnM^(a/ + bS)Wn} and ((WnM^(a/ + bS)Wn)*}. Let us start with showing the uniform boundedness of these sequences. Since || Wn|| = 1, it is sufficient to prove the uniform boundedness of the sequences (M^ (a/ + bS)Pn}.
We write M^bSPn as M^bPnMnSPn and consider first sequences of the form M^aPn where a is an arbitrary function with ||a||( := sup (|a(x)| : x £ [—1,1]} < to. Let un = £ imPn. Then, using the algebraic accuracy of a Gaussian rule, we get the estimate
||MnaujJ = ||L>vjJ = Q£|avn|2 < ||a||( ||vjJ = ||a||( ||ujJ . (3.4)
In case w = a we apply Relation (3.1) and Lemma 3.1 with ^ = a and v(x) = 1 — x2 to obtain
2 /( — 2 2 2
||mauJCT < 2 NL L1 - J |vn(x2„)| < const HalL ll^n"
k=i
and thus
hm2aunl^ < const llallœ "u„h2 , u„ G imP„ , (3.5)
where the constant does not depend on a , n , and un .
For the uniform boundedness of the sequences {M^SPn} we observe that
S^Un = iTn+1, n = 0,1, 2, ...,
(3.6)
which shows that, for G im Pn, the function := is a polynomial of degree not greater than n. Thus, Relation (3.1), Lemma 3.1, and the boundedness of S : L2 —► L2 yield
IIMnS«n||2 ^ 2Qniqn|2 < const ||qn||2 ^ const I|u«Il2 • In case u = ^ we also use Lemma 3.1 to obtain
IMSUn
Vl^ = XAfcn!qn(xL)|2t1 - (xL)2] 1 <const k
12 n N a
k=1
This verifies the uniform boundedness of all sequences under consideration. Their strong convergence on La follows once we have shown their convergence on all basis functions un of La.
Step 2: Convergence of {M^(aI + bS)Pn}. It is an immediate consequence of Corollary 3.3 and of (3.6) that
lim M^(aI + bS)Pnum = (aI + bS)um in La for all m = 0,1, 2, ...
n—
Step 3: Convergence of {(M^(aI + bS)Pn)*}. The determination of the adjoint sequence is based upon a formula for the Fourier coefficients of the interpolating function M^f. For this goal, we write
n-1
m / = x an/H-
j=0
and get in case of w = ^
a
Tn(f ) = (Mfj >a = f,Uj >„
n
n + 1
f (xfcn)u? (xfcn)
k=1
j = 0, ... , n — 1. In case of w = a we have for j = 0, ... , n — 2
(3.7)
a
jn
(f ) = (Ma f,Uj >a = (Ln^-1f,^2Uj >a = n £f (x*n)«j (*L) .
k=1
For j = n — 1 we use the three-term recurrence relation
Uk+1(x) = 2xUk(x) - Uk=1(x) , k =1, 2, ...,
as well as the relation
Tn+1(x) = 2 [Un+1(x) - Un=1(x)] , n = 0,1, 2, ..., U-1 = 0
to obtain
(1 - x2) Un=1 (x) = - [Tn=1(x) - Tn+1(x)] .
(3.8)
(3.9) (3.10)
2
a
a
Consequently, with ^(x) := x ,
<—i,n(/) = = 1 (L>-1№-i >2
2n ^(x(j ) Tn-i(xfcn) = 2n f (x2n)Mn-i(xfcn) •
Thus,
a2n(/)= j(xfc«)ui(x2n) , j = 0, •••,n - 1 , (3.11)
k=i
where j = 1 for j = 0, ... , n — 2 and j =1/2 for j = n — 1. As an immediate consequence of (3.7) we deduce for u,v £ L^
n— i n n— i
(MnfaPnu,v)2 = ^ n+T^ a(xfcn^ (u,M^>2 M^(xfcn K (xfcn)(v,u? >
n + 1
j=0 k=i ^=0
n— i n n— i
^ n+r S >2 (x№(xL)(u,M >2
n + 1
¿=0 k=i j=0 = (u,Mn^aPnV>2 •
Hence, the adjoint of M^aPn : L^ —► L^ is M^aPn : L^ —► L^ • In case u = a, (3.11) implies
n— i n n— i
j=0 k=i ¿=0
S n ^a(xfcn^ %n(v,Mj >2 (x2nMx2n)(u,^>2 ¿=0 k=i j=0
(u, (2Pn - Pn— i)M>2(Pn—i + Pn)v>2 •
Thus,
(M^aPn)* = M>Pn and (M2aPn)* = Pn — Pn—0 M-a(Pn—i + Pn), (3.12)
f^^D \ * _ Id _D lii /f 22
whence in both cases the strong convergence on L^ of (M^ aPn)* to a/. For the determination of the adjoint operator of M^SPn , we recall the Poincare-Bertrand commutation formula (see [13, Chapter II, Theorem 4.4]): If p(x) = (1 — x)a(1 + x)^ is a Jacobi weight with a, ft £ (—1,1)
2 r\ ■»-» /A r> 1 T 2
p-
(Su,v> = (u, Sv>, (3.13)
then, for u G L2 and v G L2_i ,
where (.,.> refers to the unweighted L2(—1,1) inner product. Thus, the adjoint operator of S : L2 —► L2 is 1/ : L2 —► L^ . Taking into account that SPnu is a polynomial of
degree at most n due to (3.6), we conclude that, for all u,v £ La ,
n
(M^SPnu,v)a = <L>-1SPnu^-1PnV}^ = X(SPnu)(xL)(PnV)(xL)
n + 1 k=1
= (SPnu, = (SPnu, ^Mn^-1PnV)a =
= <u,Pn^SMnV-1PnV)a •
Analogously we get, for j = 0, ... , n — 2 and u £ L
2
n
n
k=1
(Mn SPnU,Uj >2 = (Ln^ 1SPnU,^2Uj >CT = nX (SPnu)(xfcn)«j (Xfcn) = (SPnU,Lntij >CT
= (u, Pn^S^-1Ln«'j>CT and, again using Relation (3.10),
1 n n (MnSPnU,Mn-1 >2 = 2 (Ln^-1SPn«,^Un-2 — Un-3>2 = X(SPnU)(x2n)Mn-l(Xfcn)
k=1
= 2 (u,Pn ^S^ 1 LnUn-1 >2
Hence,
(M^SPn)* = Pn^SMnV-1Pn = M>SMnV-1Pn (3.14)
and
(MaSPn)* = 1 Pn^S^-1Ln(Pn-1 + Pn) = 2^S^-1Ln(Pn-1 + Pn), (3.15) where in (3.14) we took into account (3.6) and (3.9) and in (3.15) the relations
S^-1 Tn = —i Un-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^(aI + bS)Pn)*Pn} converges strongly on La to aI + in both cases u = a and u =
Step 4: Convergence of {WnM^(aI + bS)Wn}. We are going to verify the convergence of WnMnaWnum and WnMnSWnum for each fixed m > 0. Let n > m. With the help of (3.7), the identity
Un-1-m(x^n) = ^sin (n—+1fcn = (—1)k+1Um(x^n) , and Corollary 3.3 we get
n- 1
WnMn aWnttm = X an-1-j,n(aitn-1-m)« j=0
n- 1 n
X nn ^ a(xfcn)«n-1-m(xfcn)«n-1-j =
j=0 + k=1
2
Consequently,
WnM^aWn = M>Pn —► a/ in L2 . (3.17)
To describe the strong limit of WnMnaWn, we take into account that
„ . 2 , r— (n — m)(2k — 1)n r— 2k — 1 m(2k — 1)n in—1—m(xfcn) = V 2/n Sin-—- = v 2/n sin—2— n cos -—-
i. e.
in— 1—m(x2n) = (—1)fc+17mTm(x2n) (3.18)
and
n— 1 n
Lnf = X] a2n(f)Tj with j = - X f(x2n)Tj(x2n).
Then, from (3.11) and Lemma 3.2, we conclude
n— i
WnM2 aWnMm = = X an—i—i;n(atin—i—m)M
j=0
n—i n
^ ] £n— i—j,n~ ^ ] a(x2n)un— i—m(xfcn)Mn— i— j (xfcn)Mi j=0 k=i
n— i n
n \ ^ ^ rp I^a rp I^a
n— i— ?,n
n
j=0 k=i
^ ] en— i—j,n ~ ^ ] a(xfcn)7mTm(xfcn)7j(xfcn)Mj j=0 k=i
n—i n
^ ] ~ ^ ] a(x2n)(Jaum)(xfcn)Tj (xfcn) =
j=0 k=i
J<j iLnaJaum -* J(j iaJain L2 ,
where J2 is the isometry introduced in (3.2). Thus,
WnMnaWn = J—^naJ^Pn —^ J—1 aJ2 in L^ . (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} ,
(WnM^SWnim,ifc>2 = (M^Sin— 1—m,in— 1—k>2 = i ^n—m, Un—1—k=
2i v^ (n — m)jn . (n — k)jn x cos --— sin > -
E'
n + 1 ^^ n + 1 n +1
j=1
= —i 1Tm+1, Uk= —(M;fSPnim,ifc>2 .
Hence,
WnM^SWn = — M^SPn —> —S in L2 . (3.20)
Further, the identities
J2(in+2 — in) = Yn+2Tn+2 — YnTn = —2^in , n > 0 , (3.21)
(3.6), (3.9), and (3.19) imply for n > m > 1 , (in case n = m — 1 , note that LnTn = 0) WraMnSWra«m = 2 WnW„(Mm_i — Mm+i) = — 2 J-1 L>-1 J(Sm+1 — Um_l) =
= i J<r 1LnMm-1 -> iJ- 1um-1 in L2 .
Obviously, WnMnSWnMo = iWnM2Tn = 0. Hence, by means of the shift operator V introduced in (3.3) we can formulate the derived convergence result as follows:
WnM2SWn = i J-1LnV*Pn —^ iJ-V* in L2 . (3.22)
Step 5: Convergence of {(WnM^(a/ + bS)Wn)*}. In case u = 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 u = a we have, in view of (3.19),
n
(W Mn aWnU, = (LnaJPnU, J-* Pnv)CT = - J] a(4J( JPnuXj)(J"*Pnv)(xJn)
- j=i
= (u,J* LnaJ-* Pnv)CT
i. e. (WnMnaWn)* = J*LnaJ-*Pn —> J*aJ-* in L2 . Using (3.22), we get in the same manner
n
(WnMn SWnU,v)CT = i (LnV*PnU,J2Pnv)CT = n^X (V*Pnu)(xJn)(J-*Pnv)(x2n) =
j=i
= i (pV*PnU,Ln^_1 J-*Pn)2 = i (u, VMn J-*Pn)2 ,
whence the strong convergence of (WnM,2SWn)*. For further considerations we need Fredholm and invertibility conditions for the operator a/ + bS : L^ —► L^ , if a, b £ PC . For this goal, we define c := (a + b)/(a — b) on [—1, 1] and f (j) := exp(in(j — 1)/2) sin(n j/2) on [0, 1], and we associate to this operator the function
{c(x)(1 — j) + c(x + 0) j, j £ [0,1] , x £ (—1,1) , c(1) + [1 — c(1)]f (j), j £ [0,1] , x = 1, 1 + [c(—1) — 1]f(j), j £ [0,1] , x = —1.
Note that, for z1,z2 £ C , z1 + (z2 — z1)f (j) , j £ [0,1] , describes the half circle from z1 to z2 that lies to the right of the straight line from z1 to z2 . Thus, if c(x ± 0) is finite for all x £ [—1,1] , the image of c(x, j) is a closed curve in the complex plane which possesses a natural orientation, and by wind c(x, j) 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 = a/ + bS : L;2 —> L;2
is Fredholm if and only if a(x ± 0) — b(x ± 0) = 0 for all x £ [—1,1] and c(x, j) = 0 for all (x, j) £ [—1,1] x [0,1] . In this case, A is one-sided invertible and ind A = —wind c(x, j).
Lemma 3.6. Let a, b £ PC . Then the operator W{An} = JCT 1(aJ2 + ibV*) is invertible in L^ if and only if the operator W{An} = a/ + bS is invertible in L^ .
Proof. The invertibility of W{An} is equivalent to the invertibility of B = aJo- + ibV*. Since J2 = p/ — i—S and V* = —/ + ipS with — (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 = —/ — ipS and SpS = p/ + K0 , where K0u = — 1/v^2ff('U,uZ0)(J (see (4.3) below), we get
C = a(p/ — i—S)(—/ — ipS) + ib/ = —iap2S — i—2S + ib/ + K = i(b/ — aS) + K,
with a compact operator K : L^ —► L^. Now the assertion follows from b — a/b + a = — (a + b/a — b)-1 and Lemma 3.5.
4. Local theory of stability
Let A denote the smallest C*-subalgebra of which contains all sequences of the form {M^(a/ + bS)Pn} 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^(a/ + bS)Pn} and, more general, for arbitrary sequences {An} £ A. 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 {An}° := {An} + J in the quotient algebra A/J. In what follows we agree upon omitting the superscript u in all notations (such as in M^, which will be abbreviated to Mn) whenever the validity of the assertion where this notation is used does not depend on u = p or u = a.
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/J.
Lemma 4.1. If f £ C[—1,1], then the coset {MnfPn}° commutes with every coset {An}° £
A /J.
Proof. It is enough to verify that {Mnf Pn}° commutes with all cosets {MnaPn}° where a £ PC and with the coset {MnSPn}°. The first assertion is obvious; one even has {MnaPn}{MnfPn} = {MnfPn}{MnaPn} for arbitrary a, f £ PC. For the second assertion, note that the equalities Mnf PnMnSPn = Mnf SPn and MnSPnMnfPn = MnSMnf Pn hold. So, what remains to prove is
{Mnf SPn — MnSMnf Pn} £ 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 MnpPn-m = pPn-m for n > m. Consequently,
MnpSPn — MnSMnpPn = Mn(pS — Sp)Pn + MnS(/ — Mn)p(Pn — Pn-m) .
Obviously, the sequence {Mn(pS — Sp)Pn} belongs to J. Moreover, Pn — Pn-m = WnPmWn which implies that
MnS (/ — Mn)p(Pn — Pn-m) = MnS (/ — Mn)pPn WnPmWn = = (Mn(Sp — pS)Pn + MnpSPn — MnSPnMnpPn) WnPmWn and, hence, {MnS(/ — Mn)p(Pn — Pn-m)} £ J.
Together with the identities Mnf1Mnf2Pn = Mnf1f2Pn and (MnfPn)* = MnfPn , the preceding lemma shows that the set C- := ((MnfPn}° : f G C[— 1,1]} forms a C*-subalgebra of the center of A-/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 Bc be a closed subalgebra of the center of B containing the identity. For every maximal ideal x G M(Bc) , let J denote the smallest closed ideal of B which contains x, i.e.
( m ^
Jx = closg ^ ajCj : aj G B, Cj G x, m =1, 2, ... > .
Then an element a G B is invertible in B if and only if a + Jx is invertible in B/Jx for all x G M(Bc). (In case Jx = B we define that a + Jx is invertible.) Moreover, the mapping
M(Bc) —► [0, to) , x ^ ||b + Jx||
is upper semi-continuous for each b G B. In case B is a C*-algebra and Bc a is central C*-subalgebra of B, then all ideals Jx are proper ideals of B, and ||b|| = max(||b + JX|| : x G M(Bc)} for all b G B.
We will apply this local principle with A-/J and C- in place of B and Bc, respectively. The algebra C- is *-isomorphic to C[—1,1] via the isomorphism (MnfPn}° ^ f . This can be seen as follows: If f G C[— 1, 1] is invertible, then the coset (MnfPn}° is invertible. Conversely, assume this coset is invertible and choose one of its representatives (Mnf Pn + PnKPn + WnLWn + Gn} which then is invertible modulo J. An application of the homomorphism W yields the invertibility of f/ + K modulo compact operators, i.e. the Fredholmness of the multiplication operator f/. But Fredholm multiplication operators are invertible.
Consequently, the maximal ideal space of C is equal to (I^ : t G [—1,1]} with
Ir := {(MifPn}° : f G C[—1, 1], f(t) = 0} .
Let J- denote the smallest closed ideal of /J which contains the maximal ideal I^ of Cw, i. e.
J- = clos^(AnM^fjPn}0 : (An} G A, fj G C[—1,1], fj(t) = 0, m =1, 2,.. J .
Then the local principle of Allan/Douglas says that all ideals J- are proper in A- /J, and that a coset (An}0 is invertible in A-/J if and only if (An}0 + JT is invertible in (A-/J)/JTw for every t G [—1,1] .
Our next goal is the description of the local algebras (A-/J)/J^. First let —1 < t < 1. Let hT be the function which is 0 on [—1, t] and 1 on (t, 1]. Then, for every a G PC,
(MnaPn}0 + Jr = a(T + 0)(MnhrPn}° + a(T)(Mn(1 — hr)Pn}0 + Jr .
Consequently, the algebra (A-/J)/JTw is generated by its cosets e := (Pn}0 + JT ,
P := 1 ((Pn}° + (MnSPn}0) + Jr , and q := (MnhrPn}° + Jr
(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 1
SpS = p/ + K0 , where K0u = — (u, w0)o-T0 , (4.3)
which is a consequence of (3.6), (3.9), (3.21) and of the continuity of the operator SpS : L2 —► L2.Indeed,
i
SpSun = iSpTn+1 = 2 Sp(Un+1 — Un-1) = 1
2(Tn — Tn+2), n > 1, i p«n, n > 1 —1 T2, n = 0 , ( pu0 — 72 T0, n = 0 .
n- 1 n- 1
Further we recall that SPnu = i (u, )2Tk+1 = - (u, )2(Uk+1 — Uk-1) , which implies
fc=0 fc=0
M^pSPn = pSPn — 2 VWnP1Wn . (4.4)
Consequently, we have the identities M^SPnM^pSPn = M^SpSPn — 2M^SVPnWnP1Wn and M^pSPn = pSPn — ipJ(jVWnP1Wn. Thus, in both cases,
{MnSPn}°{MnSPn}° + Jr = -L- {MnSPn}o{MnpSPn}° + Jr =
p(r)
(4.5)
1 (MrapPra}° + Jt = (Pra}° + Jt , -1 < T < 1.
p(r)
The identities (3.6), (3.8), and (3.9) imply that, with —(x) = x,
V = —/ — i pS, V * = —/ + i pS. (4.6)
From this we can conclude that {MnSPn}° + JT is selfadjoint. Indeed,
ii {MnSPn}0 + Jt =----{MnipSPn}0 + Jt =---- ({V*Pn}° — {Mn—Pn}0) + Jt
p(t) p(t)
and, consequently,
({MnSPn}° + Jt)* = ({PnVPn}° — {Mn—Pn}°)+ Jt =
p(r)
" {Mn( —ip)SPn}° + Jt = {MnSPn}° + Jt .
p(T )
So we have seen that the local algebra (A/J)/J^ is generated by its identity element and by two projections in case —1 < t < 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 letp, q G B be projections (i. e. self-adjoint idempotent elements) such that (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 x 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
1 0 " 1 0
ß ^ 0 1 , ß ^ 0 0
and ^
ß vV(i - ß) \/ß(1 - ß) 1 - ß
(4.7)
respectively.
To apply this theorem, we have to check whether (pqp) = [0,1] for p and q defined
by (4.2). For this, let G be the smallest C*-subalgebra of L(L2) which contains all operators a/ + bS with a, b G PC[-1,1] and the ideal K = K(L2) of all compact operators on L2 . By , t G [-1,1] , we denote the smallest closed ideal of G/K , which contains all cosets // + K with / G C[-1,1] and /(t) = 0.
Lemma 4.3. If {An}° + JT is invertible in (A/J)/JT, then (W{An} + K) + JTG is invertible in (G/K)/JG .
Proof. Let {An} G A, and assume that there is a sequence {Bn} G A such that {Bn}o{Ara}o+ JT = {Pn}° + JT • Then BnAn = Pn + Jn + PnKPn + WnTWn + Cn with some operators K,T G K, some coset {Jn}° G JT and some sequence {Cn} G N. Further, given e > 0, there exist sequences } G A and functions / G C[-1,1] with / (t) = 0 such that
mE
||{Jn}° - {Bn}o||A/J < e for = X A^Mf Pn . Hence, there are operators K£,T£ G K(L2)
j=i
and a sequence (C;} G N such that
mE
J; P; - ^ A^M; / P; - P; K£ P; - W„T£W„ - Q P;
j=1
< e for all n > 1.
L(L2 )
Hence,
W(J„}-Em=i W(Aij)}/jI - k
L(L| )
< e, which implies W{Jn} + K G JTG . Thus,
because of W{Bn}W{An} = I + W{Jn} + K , the coset (W{An} + K) + JTG is invertible from the left in (G/K) / JG . Its invertibility from the right can be shown analogously.
Now we can complete the description of the local algebras in case —1 < t < 1. The product pqp is a non-negative element of (A/J)/JT , which implies that its spectrum 0"(a/j)/Jt (pqp) is contained in [0,1] . We prove that the spectrum of pqp coincides with this interval. Assume there is a À G (0,1) such that pqp — Ae is invertible in (A/J)/JT . The invertibility of pqp — Ae is equivalent to the invertibility of
(q — A)p — A(e — p) =
= 2{Mn(hr — A)P„}° ({P„}° + {M;SP„}°) — À ({Pn}° — {M„SP„}°) + Jr .
Lemma 4.3 implies that (A + K) + JTG := ((hr — A)(I + S) — A(I — S) + k) + Jf is invertible in (G/K)/JTG . If —1 < x < t, we have (A + K) + JxG = (—2AI + K) + JxG , and —2AI + K
is invertible in G/K. If t < x < 1 then (A + K) + = ((1 - 2A)i + S + k) + JxG , which
is also invertible in (G/K)/Jf . From the local principle of Allan and Douglas we conclude the Fredholmness of (hT — A)(i + S) — A(1 — S) in L2 . But this is in contradiction to (see
Lemma 3.5) 0 £ ^ ^ ^ — (t + 0) A — hT(t —
1 - 1 ■ 1
A A
Thus we can apply Halmos' two projections theorem to get that the local algebra (A/ J)/JT is *-isomorphic to the C*-algebra of the continuous 2 x 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 {Pn}° + JT ,
-((Pn}° + {MnSPn}o) + JT , and {MnhTPn}° + JT into the functions given in (4.7), respectively.
Now we turn our attention to the local algebras at t = ±1. Since {Mn(a/ + bS)Pn}o + JT = {Mn[a(T)1 + 6(t)S]Pn}° + JT , these local algebras are generated (as C*-algebras) by their cosets {Pn}° + JT (the identity element) and {MnSPn}° + JT . 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 u. For u = a we have the following result, the proof of which will be given in the Appendix.
Lemma 4.4. Let t = ±1. The coset {M2SPn}° + JT is a unitary element of the algebra (A2/J)/JT2 , and its spectrum is equal to TT , where T±1 = T if {t £ C : t > 0} .
Consequently, the algebra (A2/J)/JT2 is *-isomorphic to the algebra C(TT) of all complex valued continuous functions on TT, and the isomorphism can be chosen such that it sends {MnSPn}° + JT2 into the function t ^ t.
The treatment of the case u = p starts with the following lemma.
Lemma 4.5. The sequences {M^p 1 Pn} and {(M^p 1Pn)*} converge strongly to the multiplication operators p-1i : L^ —► L2 and pi : L2 —► L^ , respectively.
Proof. Convergence of M^p 1Pn: Since L2 is continuously embedded into L^ we have, due to Corollary 3.3,
lim Mn'p-1PnMTO = lim M^Um = Um = p-1ttm for all m > 0 in L2 .
Thus, it remains to show that the operators Mnp-1Pn : L^ —► L^ are uniformly bounded. Consider the quadrature rule
/1 n
(Ln/) (x)P(x) dx = X Pfcn/(xD , 1 fc=1
where p(x) = (1 — x2)3/2, and abbreviate x2n to xk. The quadrature weights pkn are equal to f1 Un(x) (1 — x2)p(x) ^
Pfcn =--TTK \- dx =
J-1 x — xfc Un(xfc)
/-. 2\ / 1 Un(x)p(x) dx 1 (', ,, . , . ,
(1 - Xk ) L (x - Xk (Xk )- Uncxk^y_iUn(x)<x+xk Mx) dx
(1 - xk)AL, n >1 •
Define ej := (Cj, jra)P for i, j = 1, • • • , n. We remark that U^(xk) = \j2/n---^
and compute
jn/P "iJ ■ ■ ■ > n ■ "" ^ v^w v " i 2
1 - xfc
f1 [Un (x)]2
nfc := -p(x) dx
j-i x — xfc
^ [Un(x)]2 -1
(1 — xk) I L <^(x) dx — (x + xk) [Un(x)]2 <^(x) dx = —xk ,
'-i x — xfc J-i
where we take into account the orthogonality and symmetry properties of Un. For i = j, it follows
£ij =
n(—1)i+j (1 — x2)(1 — x2) i1 [Un(x)]2 p(x) dx = 2(n + 1)2 J-1 (x — x»)(x — xj)
n(—1)*j(1 — x2)(1 — x2) ni — nj = n(—1)i+j+1(1 — x|)(1 — x2) 2(n +1)2 xi — xj 2(n +1)2
i. e.
sgn £ij = (—1)i+j+1 for i = j . (4.8)
For i = j, we get
[Un (xj )]2 j = J-1 (x — xj )2
[Un(x)]2 p(x) dx
j
= (1 — x22) C JUn(x)]22 p(x) dx — T [Un(x)]2 (x + xj)p(x) dx =
2
i —1 (x — xj) t/ — 1 x — xj
(1 — x2)Ajn [un(xj)]2 + 1 — 2 I ^^ x^(x) dx = (1 — x2)Ajn [un(xj)]2 + 1 —
[ [Un+1(x) + Un-1(x)] Un_(xr) ^(x) dx
-1 x — xj
t' m2
= (1 - Xj)Ajra [un(xj)] + 1 - Ajnu„-1 (xj(xj).
Since U„-i(Xj) = = 727n(-1)j+1, we have Aj„U„-i(Xj(xj) = 2 ,
which yields
ejj < (1 - xj) j = Pjn , j = 1 ..., n. (4.9)
2
Let now f : (—1,1) —► C be given. Then, due to (4.8) and (4.9) /:
|(Lf)(x)|2 p(x) dx = X Ef (xi)f (Xj <££jf(xi)||f (Xj )| j^ij-|<
i=1 j=1 i=1 j=1
< jf (Xi)|2eM - £ X(-1)i+j If (Xi)l If (Xj< i=1 i=1 j=1
1n
2
< 2^ Pinjf (Xi)|2 - X[(-1)f(Xi)Kn(x)] 2 p(x) dx <
i=1 *7-1 i=1 < 2Q
iraif |2
Using this estimate in combination with the explicit form of the quadrature weights pkn derived above we obtain, for = pvn £ im Pn,
IMp-1^^ = ||pL£p-1vra||J = ||L£p-1vra||;; <
< 2Qra 1v„| = 2^ ALMXL)|2 = 2 ||vjJ = 2 ||u
fc=1
which proves the desired uniform boundedness.
Convergence of (M^p 1Pn)*: The strong convergence of M^p implies the uniform boundedness of (M^p-1Pra)* : L2 —► L2 . Since {p-1Tm}m=0 forms an orthonormal basis in
n) ■ ^^ ^a ■
L2, it remains to prove that (M^p-1 Pn)*p-1Tm —► Tm in La . In view of
nnmj := ((Mra2p-1P„)*p-1Tm,ïïj)a = (p-1Tm, M^p-^) we have nnmj = 0 for j > n and, for n > m and j < n,
n
nnmj = (Tm,L2p-1Uj )2 = (xfc(xfc) = aJ„(Tm)
^ +1 k=1
taking into account Relation (3.7). Hence, due to Corollary 3.3, (Mn,p-1Pra)*p-1Tm = M^T^ —> Tm in L2 , and the lemma is completely proved.
We still need a consequence of the lifting principle Lemma 2.2.
Lemma 4.6. If {Ara}° £ A/ J and W{An} : L2 —► L2 as well as W{An} : L2 —► L2 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{An}, W{An} and {Ara}° are invertible from the same side, say from the right hand side, then the sequence {An} is stable from that side in the sense that there is a sequence {Bn} such that = Pn + Gn with a sequence {Gn} £ N. Since the are
matrices, this clearly implies the common stability of the sequence {An}. But then, due to Lemma 2.2, W{An}, W{An} and {Ara}° are two-sided invertible.
Corollary 4.7. One has {Mn,SPn}o{(Mn,SPn)*}° = {Pn}° and, hence,
{MSP„}o{(Mn'SPra)*}° + J2 = {P„}° + J2 for all t G [-1,1],
whereas
{(Mn^ n; / n
Proof. By (3.14), (4.4), and (4.3)
{(Mn'SPrar}o{Mn'SP4o + J/ = (Pn}° + Jr for T = ±1. (4.10)
m^SP„(M^SP„)* = --W„PiWJ M^-1P„
i
= P„ + - 2
It remains to show that the sequences
{Mn'KoMnV-^} and |W„PiW„MnV-1Pn} = |W„PiMnV-1PnW„} belong to the ideal J. This is a consequence of Lemma 4.5 and the relations
||P1«L = K«>S0>„| = >/2/*
and
u(x) dx
'-i
< const ||w||^
l|Kow||CT = -—= |(u,Mo)CT| < const ||u||2 ,
which imply the compactness of the operators Pi : L2 —► L2 and K0 : L2 —► L2 .
Now assume that (4.10) is not true for t = 1, for example. Then it is also not true for t = —1 which can be seen as follows. Set (Wf)(x) := f (—x). Then
WSW = — S, WPn = PnW , WMn2 = M^W , and {PnW}oJ^WPn}0 = J-1
(observe that Wiik = (—1)k). Hence, applying W to
{(Mn,SPn)T{Mn,SPn}° + J2 = {Pn}° + J
yields
{PnW(Mn2SPn)*WPn}o{Mn'(—S)Pn}° + J2i = {Pn}° + J2i.
Together with (4.5) and the local principle by Allan/Douglas, this leads to the invertibility of the coset {Mn2SPn}° in contradiction to Lemma 4.6.
Thus, the coset {(M^SPJ*}0 + J2 is an isometry in the local algebra (A2/J)/J2. 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 i2 of all square summable sequences of complex numbers. In particular, Coburn's theorem implies that the local algebra (A2/J)/JT2 is
* i^w^^V,^ n* ^f rtfj2
somorphic to the C*-subalgebra of L(€2) generated by the shift operator
£ : i ^ i , {x0, x 1, ...} ^ {0, x0, x 1, ...}
where the isomorphism sends {Mn2SPn}° + J2 into
£* : i2 ^ i2, {x0, x1, ...} ^ {x1, x2, ...} .
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 n^ from A/J onto a C*-algebra of bounded functions living on ((-1,1) x [0,1])U({±1} x TT) in case u = a and on ((-1,1) x [0,1])U{±1} in case u = This isomorphism sends the coset {M^aP„}° into
(x, ß)
a(x + 0)ß + a(x)(1 — ß) (a(x + 0) — a(x))^/ß(1 — ß) (a(x + 0) — a(x))^/ß(1 — ß) a(x + 0)(1 — ß) + a(x)ß
and the coset {M^SPn}° into
(x, ß)
10 01
for (x,ß) G (—1,1) x [0,1]. Moreover, for (x,t) G {±1} x TT
and, for x = ±1,
{Mn (a/ + bS)Pra}0(x, t) = a(x) + b(x)t
^{M^a/ + bS )Pra }o(x) = a(x)/ + b(x)E*
(b) The sequence {An} G A is stable if and only if the operators W{An}, W{An} : L
L2 are invertible and if, in case u = ^ , the operator n^{Ara}o(x) is invertible on for x = ±1.
We remark that the invertibility of W{An} already implies that det n^{Ara}o(x,^) = 0 for all (x,^) G (-1,1) x [0,1] and that n2{An}o(±1,t) = 0 for all t G T±1 (see Lemma 3.5).
For the stability of the sequence {A^} = {M2(al + bS)Pn}, this theorem yields the invertibility of W{A^} = al + bS and of W{A^} = J—1(aJCT + ibV*) as necessary and sufficient conditions. By Lemma 3.6, the invertibility of W{A^} is a consequence of the invertibility of W{An}. Similarly, the sequence {A^} = {M^(a/ + bS)Pn} proves to be stable if and only if the operators W{A^} = al + bS and W{A^} = al - bS on L2 and the operators a(±1)1 + b(±1)E* on 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 G C with |z| < 1 which, on its hand, is already a consequence of the invertibility of al ± 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
E
fc=i
ajfc (x)ufc (x) +
j(x) ni
i1 ifc (y) l-i y — x
fj (x), —1 <x< 1 = 1,
m,
(4.11)
with piecewise continuous coefficients ajk and bjk . Indeed, denote by {Anfc} the operator sequence of the collocation method for (1.1) with ajk and bjk instead of a and b, respectively. Then the collocation method for (4.11) is stable in (L2)m if and only if [W {A£fc}] m=1 and
W {Anfc }
x = ±1.
j,k=1
are invertible and if, in case u = , [n^ {Anfc}0 (x)] .™=1 is invertible for
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 An G Craxra will be denoted by 0 < < ... < aira), counted with respect to their multiplicity.
(a) (b)
а(х) = (1-х)1/2, b(x) = -ix а(ж) = (1.01—ж2)1/2, b(x) = -ix
20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200
The smallest three singular values of An = МП (ai + bS)Pr
If {An} £ F is a stable sequence of matrices, then there is a positive constant C such that the smallest singular value a(n) of An (hence, every singular value of An) is greater than C for all n, and conversely. Thus, if {An} is non-stable, then there is a subsequence of the sequence (a(n)) which tends to zero. Figures (a) and (b) illustrate this behaviour for the non-stable sequences {An} with An = Mnf(a/ + bS)Pn , where a(x) = л/1 — x , b(x) = — ix and a(x) = л/1-01 — x2 , b(x) = — ix, respectively. In both cases we observe that not only a subsequence of (a(n)) but the sequence itself tends to zero. Moreover, in Figure (b), also the sequence (a(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 An which tend to zero. Here we restrict ourselves to the case и = ^, although analogous considerations are possible for и = a.
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].
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 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 {An} £ A^ tends to zero and not only one of its proper subsequences.
Given a strongly monotonically increasing sequence n : N ^ N, let Fn refer to the C*-algebra of all bounded sequences {An} with An £ Cn(n)xn(n), and write for the ideal of all sequences {An} £ Fn which tend to zero in the norm. Further, let stand for the restriction mapping : F ^ Fn, {An} ^ {An(n)}. This mapping is a *-homomorphism from F onto Fn which moreover maps N onto . Given a C*-subalgebra A of F, let An denote the image of A under 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 n, there is a *-homomorphism Wn : An ^ B such that W = Wn Rn •
(b) The algebra A is fractal if the canonical homomorphism n : A ^ A/ (AflN) is fractal.
Thus, given a subsequence {An(n)} of a sequence {An} which belongs to a fractal algebra A,
it is possible to reconstruct the original sequence {An} from this subsequence modulo sequences in AnN. 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^.
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 {Wt}teT of unital and fractal *-homomorphisms Wt from A into unital C*-algebras such that the following equivalence holds for every sequence {An} G A: The coset {An} + AnN is invertible in A/(A H N) if and only if Wt{An} is invertible in Bt for every t G T.
To make the proof of the fractality of the algebra A^ more transparent, we introduce a few new notations and rewrite Theorem 4.8 as follows. Set T := {1, 2, 3, 4} and define *-homomorphisms W1, W2 : FW ^ L(L2) and W3, W4 : FW ^ L(/j) by W1 := W, W2 := W and
W3{Ara} := n^{A„}°(1), W4{Ara} := n^{Ara}o(-1).
Theorem 4.8'. (a) A sequence {An} G A^ is stable if and only if the operators Wt{An} are invertible for all t G T.
(b) The mapping
smb : A^ ^ L(L2) x L(L2) x L(/j) x L(/j), {Ara} ^ (W1{Ara}, Wj{Ara}, W3{Ara}, W4{Ara})
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^ is fractal.
Proof. By Theorem 5.2, we have to prove that all homomorphisms Wt are fractal. For W1 and W2, the fractality is evident: these homomorphisms act as strong limits, and the strong limit of a subsequence of {An} coincides with the strong limit of {An} itself. Concerning W3 and W4, 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^SF^ in place of the sequence {M^SPn} itself. Thus, Coburn's theorem again applies, yielding the fractality of W3 and W4.
Fredholm sequences. Let J(F) stand for the smallest closed subset of F which contains all sequences {Kn} for which sup dim Im 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 {An} G F is called a Fredholm sequence if it is invertible modulo the ideal J(F). If {An} is a Fredholm sequence then there is a number k such that liminfra^^ > 0 (see [18, Theorem 2]). The smallest number k with this property is called the a-number of the sequence {An} and will be denoted by a{An}. 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^.
Theorem 5.4. (a) A sequence {An} G A^ is Fredholm if and only if the operators Wt{An} are Fredholm operators for every t G T.
(b) If {An} G A^ is a Fredholm sequence, then
a{An} = dim Ker W1{An} + dim Ker W2{A„} + dim Ker W3{An} + dim Ker W4{An}.
(c) If {An} G A^ is Fredholm and k = a{An} > 0, then = 0.
Fredholm inverse closed subalgebras. Let A be a unital and fractal C*-subalgebra of F which contains the ideal N. A sequence {Kn} in A is said to be of central rank one if, for every sequence {An} G A, there is a sequence } G c (= the set of all convergent sequences of complex numbers) such that
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) = An 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 {Kn}, let J{Kn} refer to the smallest closed ideal of A which contains the sequence {Kn} and the ideal N. In [18] it is shown that, if {Kn} and {Ln} are sequences of essential rank one in A, then either J{Kn} = J{Ln} or J{Kn} H J{Ln} = N. Calling {Kn} and {Ln} 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 G S, there is associated a unique irreducible representation Ws of A into the algebra L(Hs) for some Hilbert space such that the ideal J{Kn} is mapped onto the ideal K(Hs) of the compact operators on and that the kernel of the mapping Ws : J{Kn} ^ K(Hs) 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 {An} 6 A is a Fredholm sequence, then the operators are Fredholm operators for every s 6 S, and a{An} = ^seSdim Ker Ws{An}.
(b) If {An} 6 A is Fredholm and k = a{An} > 0, then lim^^, = 0.
(c) If the family (Ws)seS is sufficient for the stability of sequences in A (in the sense that the invertibility of all operators Ws{An} implies the stability of {An}) and if all operators {An} are Fredholm for a sequence {An} 6 A, then this sequence is Fredholm.
We know already that 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 Ws with the corresponding homomorphisms Wt 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 {Kn} G A with dim Im < 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 {Kn} is a central rank one sequence in A then, since Nc A, every matrix Kn has rank one. Thus, {Kn} 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 {anPn} where {an} is in c, the set of all convergent sequences. Now let {Kn} G A be a sequence with dim ImKn < 1 for all n. The fractality of A further implies the existence of the limit a := lim ||Kn|| (see [17, Theorem 4]). If a = 0, then {Kn} is a zero sequence, hence in Nc K(A).
In case a = 0 we are going to show that {Kn} is a central rank one sequence. Assume {Kn} is not of central rank one. Then there are a sequence {An} G A and a non-convergent sequence (an) G such that
KnAnKn = anKn for all n.
Choose two partial limits ft = 7 of the sequence (an) as well as two subsequences ^ and n of the the positive integers such that
aM(n) ^ ft and an(n) ^ 7 as n ^ to.
Then both sequences {aM(n)KM(n) — ftK^(n)} and {an(n)Kn(n) — 7Kn(n)} tend to zero. Hence, again due to the fractality of A (see [17, Theorem 1]), both sequences {anKn — ftKn} and {anKn — YKn} are zero sequences. But then, also their difference (ft — 7){Kn} goes to zero. Since ||Kn|| ^ a = 0, this implies ft = 7 in contradiction to the choice of ft and 7.
Identification of the ideals J (A2) = K(A2). Our next objective is to characterize the image of the ideal J (A2) under the mapping smb introduced in Theorem 4.8'.
Theorem 5.7. The homomorphism smb maps J (A2) onto K (L^) x K (L^) x K (/2) x K (/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 G A, there is an element ^ in the center of A such that kak = ^k). In our setting, the homomorphisms Wt, 1 < t < 4 are irreducible since, in any case, the ideal of the compact operators belongs to the image of A2 under Wt. Thus,
smb(J(A2)) C K(L2) x K(L2) x K(/2) x K(/2).
For the reverse inclusion first recall that, by definition, the set J of all sequences {PnKPn + WnLWn + Cn} with K, L compact and {Cn} G N is contained in A2. Since every compact operator can approximated as closely as desired by an operator with finite dimensional range, we have J C K(A2) and thus, by Proposition 5.6, J C J (A2). Moreover, it is easy to check for the sequence {Kn} := {PnKPn + WnLWn + Cn} that
W1{Kn} = K and W2{Kn} = L,
and it is immediate from the definition of W3 and W4 that W3{Kn} = W4{Kn} = 0. Hence, K(L2) x K(L2) x {0} x {0} lies in smb(J(A2)). So it remains to show that smb(J(A2)) contains all quadrupels of the form (0, 0, K, 0) and (0, 0, 0, K) with K a compact operator on /2.
It is well known and easy to check that the smallest closed C*-subalgebra of L(/2) which contains the shift operator £ also contains all compact operators and that, in particular, K(/2)
is the smallest closed ideal of that algebra which contains the projection n := I — ££* acting by
n : /2 ^ /2, {x0, x1, ...} ^ {x0, 0, 0, ...}.
Because of Wt{Mn'SPn} = £* for t = 3 and t = 4, it is consequently sufficient to prove that the quadrupels (0, 0, n, 0) and (0, 0, 0, n) belong to smb(J(A2)). For this goal, we consider the sequences {An} and {Bn} with
An := Pn — (Mn,SPn)*Mn,SPn and Bn := M^VPn — (M2SPn)*M2SVPn
where, as above, V = ^I — ipS and ^(x) = x. For the sequence {An} we have W1{An} = I — pSp-1S (Section 2), and this operator is 0 as we have already mentioned several times. Similarly, W2{An} = I — (—pSp-1)(—S) = 0 due to Theorem 3.4. It is further immediate from the definitions that W3{An} = W4{An} = n; thus, smb {An} = (0, 0, n, n).
Concerning the sequence {Bn}, we will first show that it is indeed an element of the algebra A2. To see this, write SV = S(-0I — ipS) as
SV = ^S — ipl + (S^I — ^S) — iKo (5.12)
where the rank one operator K0 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 M2 Hence, {M?2'SVPn} G A2 and {Bn} G A2. To compute the operators Wt{Bn}, we recall from Section 2 that
W1{M2VPn} = w^m^i — ipS)Pn} = -0I — ipS = v, W2{Mn2VPn} = ^i + ipS,
W3{Mn2VPn} = W3{Mn2(^(1)I — ip(1)S )Pn} = I and W^M^VPn} = —I
and that
W1{Mra2SPn }* = —W2{Mra2SPn }* = pSp-1I and W3{M2SPn}* = W4{M£ SPn}* = £.
For the operators Wt{Mn2SVPn} we make use of identity (5.12) which together with the results of Section 2 yields
W^SVPn} = sv, W2{Mra2SVPn} = —ipi — ^s,
W3{M;2SVPn} = W3{Mn2(^(1)S — ip(1)I )Pn} = £* and W4{M2SVPn} = —£*. Puzzling these pieces together we find
W1{Bn} = V — pSp-1 SV = (I — pSp-1 S)V = 0,
and
W2{Bn} = (^I + ipS) + pSp-1(—^S — ipI) = pSp-1(S^I — ^S)
which is also 0 since the range of the commutator S^I — -0S consists of constant functions only and since the operator Sp-1 annihilates every constant function. Finally,
W3{Bn} = I — ££* = n and W4{Bn} = —I — £(—£*) = —n,
whence smb {Bn} = (0, 0, n, —n).
Thus, the homomorphism smb maps the sequences ^+ and ^— onto
the quadrupels (0, 0, n, 0) and (0, 0, 0, n), respectively. We show that these sequences are essential rank one sequences in An. For this goal, we determine the matrix representation of = — (MnSPn)*M—SPn with respect to the basis functions Uk, k = 0, ..., n — 1. For 0 < k, m < n — 1 we have
Tkm := ((M-Uk>. =
= msPnUm, M;nsp„ufc>CT = (M;nsUm, Mransufc>CT
and, since Sttm = = iTm+i, we obtain
Tfcm = (M-^Tm+i, M^Tk+i)^ = (Ln^-1Tm+i,Ln^-1Tfc+i>n =
n
n + 1
)Tfc+1 (Xn )
1=1
2 ^A (m +1)1n (k + 1)1n
cos-—;— cos
1=1
n
n + 1 1
n + 1
n + 1
n + 1
E
1=1
(m — k)1n (m + k + 2)1n cos-—;--+ cos ■
n + 1
n + 1
Short calculations using the well known identity
cos
sin ( n + 2 I X 1
1=1
o • x
2sin 2
yield in case k = m
whereas in case k = m
Tfcm
n + 1
n — 1 n + 1
if m + k is odd if m + k is even.
Summarizing we find that An =
2
n + 1
^fcm =
^fc.JLo with
0 if m + k is odd
1 if m + k is even,
and analogously one gets that the matrix representation of with respect to the same basis is
2
[£fc,m+i]fc 1=0- It is evident now that the sequences -+ Bn} and -— Bn}
1
1
Bn • n + - 1°fc,m=0' — " -------- —- .. ------ ---- - - --------- ^ L- -<* ■ — '«J ------ 2
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(L2) x K(L2) x K(/2) x K(/2) Ç smb(J(A^))
and of Theorem 5.7.
1
2
0
2
Fredholm inverse closedness of A^. To finish also the proof of Theorem 5.4, we have finally to show that the ideals J(A^) = K(A^) and J(F) Pi A^ of A^ 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 {Jn} be a sequence in J (F) PA. Then, for every irreducible representation W : A ^ L(K) of A, the operator W{Jn} 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 n : F ^ L(H) of F with a certain Hilbert space H, a subspace Hi of H and an isometry U from Hi onto K such that Hi is an invariant subspace for n{Jn} and
W {Jra} = Un{ J„}|Hx U *.
From [18, Theorem 3] we know that n{ Jn} is a compact operator on H. Since Hi is invariant for n{Jn}, this moreover implies that n{Jn}|Hl is a compact operator on Hi. Thus, W{Jn} is compact on K.
The Figures (a) and (b) revisited. Having Theorem 5.4 at our disposal, it is easy to explain the behaviour of the smallest singular values in Figures (a) and (b). In case A = a/+bS, a(x) = V1 — x , b(x) = — ix , we have for An = M^APn
dim Ker Wi{An} + dim Ker W2{An} + dim Ker W3{An} + dim Ker W4{An} =
=0+0+1+0=1
whereas the same quantity is 0 + 0 + 1 + 1 = 2 in case a(x) = Vl.01 — x2 , b(x) = — ix. Thus, in Figure (a) the lowest singular value tends to zero and the a2n remain bounded away from zero by a positive constant for all n, whereas in Figure (b) both lim= 0 and lima2n = 0.
66. Appendix: proof of lemma 4.4
Lemma 6.1. The coset {M2 SPn}° is a unitary element of A2 /J. Proof. We use (3.15) and (4.3) and get
M2 SP„(M-SPn)* = 1M2 SpSp-iLn(Pn-i + Pn) =
= 1M^ Ln (Pn-1 + Pn) + 1K (Pn -1 + Pn)
2(Pn-1 + Pn) + 2M:KoP-1 Ln(Pn-1 + Pn).
Now, from K0p 1u =---= (u, U0)CT
v2n
|LnPnf = nE t1 - |(p-1Pnf )(Xfcn)|2 <
k=1
< const J p(x) |(p iPn/)(x)|2 dx = const ||Pn
2
(see Lemma 3.1), and Pn-i = Pn — WnPiWn, it follows {M-SPn}o{(M-SPn)*}° = {Pn}°. On the other hand we have
(m« sp«)* m« sp« = ^-^«m« sp« — 2 ^s^1lnw„p1w„mn sp«
and SP« = ^S^"1SPn = P« due to (3.6) and (3.16).
For n > 1, let Q« : ¿2 —► ¿2 denote the projection Q«£ = (£0,... , 0,0,...}, and define
n
E± : im P„ —► im+ Q« by
and
Then, for £ G im Q«
= {yn^in)' • • • ' yn^n^' 0,
E« u H A/nu(xnn),...^nu(Xln)' 0'
n I--« I-
(E+rt = £ C =: E+«C and (EJ^C = £ C =: E_«£ :
fc=1 ' fc=1 '
where
(X) = <^(x)T«(x)
^(xfcn)(x Xfcn)Tra (xfcn)
Remark that, for n > 1
^ n(—1)k+1
T« (x) = nUn_1 (x) and T« (x^«)
n ^(xfc«)
In view of Lemma 3.1 and estimate (3.1), the sequences {E-±} and {E±n} are uniformly bounded, i. e. there are constants ci and c2 such that
n
7T
X^|u(Xfc«)|2 < c1 ||u||2 for all u G imP« (6.1)
and
n fc=1
«
n
7r
fc=1
< c^ |Ck_112 for all C G ¿2 . (6.2)
fc=1
Lemma 6.2. The sequences {E^M-SPnE±nQn} and {(E^M-SPnE±nQn)*} are strongly convergent on £2 .
2
2
Proof. The uniform boundedness of these sequences is obvious. So it remains to prove their convergence on the elements em = {imkof the standard basis of £2. For n > m > 1, one has
/mra : En Mn SPraE-raQraem-1 S^mra(xjra^ . i
(here we identify {£0,
S^fcn(x)
, £n-1, 0,...} £ with {£0,... , £n-1}). We compute, for x = x'
0
kn :
R (-i)fc+1 1
n
__L P(y)Tn(y)
7-1 (y - x0n)(y - x)
dy
n (-1)k+1 1
2 n(x0n- x) ni ./-A y - x0n y - x
P(y)Tn(y) dy
and, taking into account (3.16),
1 i 1
1 f1 1 - y2
n J-1 y - x
n J-1 y - x
Tn (y)^(y) dy
1
1
n-1
(1 - x )Un-1(y) - - I (y + x)Tn(y)a(y) dy
(1 - x2)Un-1(x).
Thus, for j = k,
With the help of
we get
It follows
(SC)(x0J
1 P(x0n) - (-1)j+kP(x0n)
ni
vfcn xjn
jn)__s(n)
d
— [(1 - x2)Un-1 (x)] = (1 - x2)Un_1(x) - 2xUn-1(x)
~ 1 x0
(S^fcn)(xfcn) - ~
(n)
ni P(x0n)
0 \ =• skk
(n) sjk
cos k+j_1 n _2n_
. . k+7_ 1
i n sin —77— n
2n
k—7
cos n
2n
and, consequently,
jl <
k—7
i n sin n
2n
1
even,
odd,
(6.3)
7k
1
even,
odd.
|k — j|
Thus, for fixed m, the sequences {/mn} = {sim,... , s^m, 0,...} are uniformly dominated by a square summable sequence, which implies
, . N (
2
fmn -A lim sjA =• {Sjm} in f
ln—<^ J 7=1
where sjk = lim sj^ , i. e.
n—j
s7k =
ni(j + k - 1) 2
ni(j - k)
even
odd
1 - (-1)j-k 1 - (-1)j+k
-1
ni(j - k) ni(j + k - 1)
(6.4)
1
1
1
2
Thus,
E-M-SP„E-raQra —^ S :=[Sjkj^i on €2 . Now it is easy to see that
(E-M-SP„E-raQra)* —^ S* = [sj=i on €2 ,
_ 1 — (—1)j-k 1 — (—1)j-k—i
where skj = -;-—--1--;-;-- . Hence, denoting by T(a) = [a7—kLT=0 and H(a) =
ni (j — k) ni (j + k — 1) j<
oo
[aj--k-i]j<fc>=0 the Toeplitz and Hankel operator w.r.t. the symbol a(t) = ^^ aktk , t G T,
k=—^o
respectively, we have
S = T(0) — H(0) and S* = T(0) + H(0) with 0(t) = sgn(St). Finally, from
(«)
Q
Dn+1_j,n+1_k
cos n ,
2n , j + k even,
k+j_1 i n sin —^— n
2n
cos V-n
2n j + k odd,
i n sin V-n
2n
we get
E_M« SP«EI«Q« —> —S and (E_M« SP«E_«Qn)* —^ — S*
in £2.
We remark that the assertion of the previous lemma is not directly used in the following but it essentially suggests the further considerations.
k k + 1
For k, n G Z and n > 1, let denote the characteristic function of the interval '
multiplied by -^/n. Then the operators
nn
E« : 4 —^ L2(R) , {Ck}£_« ~ X Ck
k=_œ
and
\ _ 1 • „_ p , f/2 \ ^ A , , r A 1 œ
E_« = (E«)_ : imE« — 4 , X CkP« ^ {Ck}r=_c
k=_œ
act as isometries. If we further denote the orthogonal projection from L2(R) onto im En by Ln then we get as a consequence of [7], Prop. 2.10 and Exerc. E2.11, the following lemma.
Lemma 6.3. The sequence EnSE—: L2(R) —► L 2(R) is strongly convergent.
Lemma 6.4. The sequences {E±nQnSQnE±Pn} belong to the algebra .
Proof. Obviously, the sequences under consideration are uniformly bounded. For k = 1,... , n , define functions
Îln kn k — 1 \ — , cos — < x < cos-n,
Vn n n
0, otherwise,
and let Rn, Sn : L2 —► L2 refer to the operators
— n n
nE(/,^V0n, Snf = £(f,pn)0pn. n k=1 k=1
Then, in view of (6.2),
llRnf ||0 < l(f,Pn)012 = C2 ||SnZ ||0 < C2
k=1
i. e. the sequence {Rn} C L(L^) is uniformly bounded. Moreover, for the characteristic function / = X[x,y] of an interval [x,y] C [—1,1] , we have
(/,pn>0 -\l(x0n)
kn n
n
n I k-1
/ (cos s) - / ( cos 2k2n 1 n
ds
(f,pn>0 -/(x0n)
which implies, again by (6.2),
» kn
n
kn k - 1
x, y £ I cos —, cos-n
nn
/ (cos s) - / ( cos 2k2n 1 n
ds
< W — , otherwise, n
||Rn/ - M
n0 / | 20
n
k=1
(/,Pn>0 -/(x0n)
/0
kn
2n
< C2 -
n
Consequently, Rn/ —► / in L^ for all / G L^ . In particular, we get the following equivalences (£ G C):
C / in L0 ^ lim
k=1
^ ] Cfc Ifcn ^n/"
k=1
n
lim
n-' ^
k=1 n
en - (/, pn)^
^ lim
n—TO
£\№n - Sn/ k=1v
E\/nenpn — / in l
k=1
Since Rn —► / in L^, the convergence E—„QraSQraE+P/ —► g in L^ for an / G L equivalent to
is
E+nQnSQnE+Rn/ = E+nQnSQn{(/,pn)0}k=1 g in L
2 0
0
n
n
2
0
0
0
2
0
0
2
0
and, due to the previous considerations, equivalent to
X^SjkV« —^ g in L2 . j = 1 k=1
(6.5)
The mapping T : L2 —► L2(0,1) defined by (T/)(s) = V^/(cosns) is an isometry, whereby T(- = . Consequently, (6.5) is equivalent to
X[0,1] X X Sjk(T/, ^«)L2(R)^n -> X[0,1]Tg in L
jez kez
(6.6)
The left-hand side of (6.6) can be written as EnSE—nL—X[0<i]T/ , and Lemma 6.3 guarantees the convergence of this sequence. Hence, we have proved that W{An} exists for = E+raQraSQraE—Pn .
To prove the existence of W{An}, we proceed as follows. By definitions and by taking into account (3.11) and (3.18) we find, for u G L^ and £ G £2 ,
E+W«u =
I— «_ 1
yYl (u,uj P«_1_j (xfcn )
. V n j=o
I- n— 1
y^X (u,Uj )i (—1)k+1Yj Tj (Xin)
k=1
n-1
¿fcn = Mn ¿fcn = „ gj«Uj (xfcn)Mj
j=0
and
W«E+«Q«C
re « — 1
n ^ y Ck _ 1 ^ y £«_ 1 _j,«U« _ 1 _j (xfc«)Mj n k=1 j=0
n 1
!>_ 1£ Y (—1)k+1Tj (x/n) k=1 j=0
n 1
J_ V ^Ck_ 1(—1)k+^ Tj (x/«)Tj
k=1
j=0
V _1(—1)k+1i;
"fcn .
k=1
Thus, if we define P« : La
► L/ , E« : im P« —► im Q« , and W : ¿2 —► ¿2 by
P«/ = £(u,Tk)/Tk , E«/ = {/(xk«)}k=1 , WC = {(—1)kCk}k^0 ,
: T 2 n:
« _ 1
k=0
(6.7)
respectively, then EnWn = WE- J2Pn = WE-P- J2 and WnE+n = J—iE2nW , where E2n := (E-)—i. Consequently,
WnE+nQnSQnE+W« = _ 1Ei«Q«W SWQ«E« P« J.
(6.8)
n
The strong convergence of this sequence can be proved as the strong convergence of the operators E+nQraSQraE+Pra.
Defining J„u = ^ ejn(u,Uj)
aUj (£jra :— 1 , j > n — 1) and taking into account (6.7), we
j=0
can write, for u £ L2 and f £ ,
I- n n — 1
(QnE—Pn ^ f ^ = Uj U(xfcn )ffc—1 =
' fc=1 j=0
I- n n — 1
= (u^/_ — 1_ S £jnUj(4n)U)2 =
V n z—' n z—'
v fc=1 j=0
= (u, E+nQnf )2 ,
which leads to (QnE+PnJn)* = E+nQn, i.e. (QnE+Pn)* = Jr—1E+nQn. Furthermore, again due to (6.7),
' - k=1
I— n n — 1
= V n S ^ — ^ (u'
V 7„_1 — H
j) 2 jn j
k=1 j=0 = (f,E+ PnJnu),2 ,
such that (E+nQn)* = E+PnJn . Hence,
(E—nQnSQnEn Pn) Jn E—nQnS QnEn Pn Jn .
Analogously, with the help of (6.8) one can show that
(WnE—nQnSQnE—Wn)* = J*E2nQnWS*WQnE:P2J— *.
Since Jn —► I in L2 we conclude also the strong convergence of AP and (WnAnWn)*Pn
+
QnEn P n I
and the proof of {E+nQnSQnE+Pnj £ FW is done.
For the second sequence we can use the same arguments taking into account the following two facts:
a) E- = and E -n = E—nVn , where Vnf = {fn — 1,... , f0,0,...} , f £ £2 . Consequently,
E—nQnSQnEn pn = E—nVraSVraEra pn ;
b) V^T(a)Vn = QnT(U)Qn , where U = a(t—1) , and H(0) belongs to the smallest closed subalgebra T of L(€2) containing all Toeplitz operators T(a) with a £ PC(T). Thus, VnSVn = QnS0Qn, S0 £ T, and Lemma 6.3 remains true for S0 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 McB is called a localizing class if 0 £ M and if, for all a^ a2 £ M, there exists an element a £ M such that
aaj = aj a = a for j = 1, 2 .
Let M be a localizing class. Two elements x, y £ B are called M-equivalent (in symbols:
M N x ~ y), if
inf ||a(x —y)||B = inf ||(x —y)a||B = 0 .
Further, x £ B is called M-invertible if there exist a1, a2 £ M and z1, z2 £ B such that
z1 xa1 = a1 and a2xz2 = a2 .
A system {MT}Ten of localizing classes (Q is an arbitrary index set) is said to be covering if, for each system {aT}Ten with aT £ MT , there exists a finite subsystem aTl,... , aTn such that aT1 + • • • + aTn is invertible in the algebra B.
Local principle of Gohberg and Krupnik ([6], Theorem XII.1.1). Let B be a unital
Banach algebra, {MT}Ten a covering system of localizing classes in B, x £ B and x " xT for all t £ Q. Then x is MT-invertible if and only if xT is MT-invertible. If x commutes with all elements from (JTen MT , then x is invertible in B if and only if xT is MT-invertible for all t £ Q.
For t £ [-1,1], let
mT := {/ £ C[— 1, 1] : 0 < / (x) < 1, / (x) = 1 in some neighborhood of t}
and define MT := {{M^/Pn}° : / £ mT} . Then {MT}Te[-1)1] forms a covering system of localizing classes in /J , which, due to Lemma 4.1, has the property that all elements of this system commute with all elements of the form {Mi(a/ + bS)Pn}°, a,b £ PC . The Relation (3.5) shows that, for a,a1,6,b1 £ PC , the cosets {M£(a/ + bS)Pn}° and {M£(a1/ + b1S)Pra}° are MT-equivalent if a(T ± 0) = a1(T ± 0) and b(T ± 0) = b1(T ± 0).
Lemma 6.5. The cosets {M^SPn}o and {±E±n
QnSQ„E±P„} are M±1 -equivalent. Proof. Let a £ m1 and B„ = E+M^aPraE+ra(E+M^SPraE+raQ„ — Q„SQ„). Then
= diag[a(xi„), . . . , a(xnJ][sjfc) — sjfc]j,fc=1
and, due to the uniform boundedness of the sequences {E+} and {E+n} ,
||MiaP„(MiSP„ — E+raQraSQ„E+Pra)||£(L|) < const ||BH^ .
Assume that supp a C [cos ne, 1] , 0< e < 1/4 . S ince the function g(z) = z cot z =1 — 3 z2 + • • • is analytic for |z | < n, we have |cot z — z-11 < const |z| for |z| < 3n/4, which leads to
(n)
< const —. It follows -2
' / 1 |Pn ||L(£2) < ^ 1 j - s jkl2 < const f e + -
1<j<ne+1 fc=1 ^
and, consequently,
||{M2aPra(M2SPra - £+raQraSQra£+Pra)}°|| < const v/i.
Analogously one can show that
||{(M2SPra - E+raQraSQra£+Pra)M2aPra}°|| < const Vi,
and the M ^equivalence of (M2SPra}o and SQraE+}o is proved. The proof of the M-i-
equivalence of (M2SPn}o and {—SQnE—}o is similar.
Lemma 6.6. The sequence {E±raSQraE±Pra — APn} is stable in L2 if and only if A £ D+ := (z £ C : |z| < 1, Sz > 0}.
Proof. Due to [2, Prop. 4.1], the sequence S*— AOn} is stable in £2 if and only if A £ D- := (z £ C : |z| < 1, Sz < 0}. This fact implies the assertion immediately (recall the uniform boundedness of and = (E±)-1).
Proof of Lemma 4.4. Let t = ±1. Lemma 3.5 and the local principle of Allan and Douglas imply that (S) = TT. Further, by Lemmas 4.3 and 6.1,
Tr c /j)/J ((Mra2SPra}° + J) C T.
Let A £ T\Tr . Due to Lemmas 6.6 and 6.4, the coset {±£-rnQnSQnE±Pn — APra}° is invertible in /J. By Lemma 6.5 and the local principle of Gohberg and Krupnik we get the MT-invertibility of (M2SPn — APra}° . 1 + x
Let x(x) = —^— and A £ T \ T1. Then (M2xSPn — APn}° is M1-equivalent to
(M2SPn — APn}°, and M-^equivalent to A (Pn}o. So, M1- and M-1-invertible. For t £ (—1,1) we use the fact that (A2/J)/J2 is *-isomorphic to a C*-algebra of continuous 2 x 2 matrix functions on [0,1] , which was shown in Section 4. This isomorphism sends
(M2xSPn - APra}° + J
1 + T
-2- M2 SPra - APra ¡> + J
into the function
1 + T
0
A
0
1 + T 2
A
which is invertible. Consequently, for each t £ (—1,1), there exist (BT} £ A2 and {T,Tfc}° £ J k = 1, 2 , such that
and
(B r (M^xSPn - APn r = (Pn r + {TM}
(M- xSPn - APn }o (bt }o = (Pn }o + {t;2}c
Since {T^k}° is MT-equivalent to the zero element of A2/J, we get the MT-invertibility of (M2xSPn — APn}° also for t £ (—1,1). The local principle of Gohberg and Krupnik gives the invertibility of (M2xSPn — APn}° in /J. Because of the inverse closedness of C*-subalgebras, the inverse of (M2xSPn — APn}° belongs to A2/J, which implies, due to the local principle of Allan and Douglas, the invertibility of (M2xSPn — APn}° + J2 .
The invertibility of (M2xSPra — APra}° + J -21 for A £ T \ T-1 can be shown analogously.
2
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Received for publication August 31, 2000
