DEMONSTRATIO MATHEMATICA
Vol. XVIII No 3 1985
Frank - Olme Speck
ON GENERAL WIENER-HOPF OPERATOR THEORY AND RECENT APPLICATIONS
Dedicated to Professor Donald Cecil Pack on the occasion of his 65th birthday
1. Introduction We oonaider 3 &aneraljffiecar~Hopf operator (WHO) defined by
(1.1) W t* PAjpi where X le assumed to be a Banach space, P,AeL(X) linear p bounded operators on X, P = P a projector,, and A invertible, I.e. A"1 e l(X). This notion introduced by M. Shinbrot [14..2] comprehends several Glasses of singular operators in the sen- se of S. ProBSdorf [13], of. section 2„ A crucial point in WHO theory consists in the equivalence of the "regularity" of W (invertible, IPredholm, ...) to a par- ticular factorization of A « A.,A,,... with certain "factor pro- perties" of A< {e.g. having invariant subspaces PX,...), which «J _ -i Ay yieldmeanins g a to"pseudoinverse present the "complet of W e(i nsolutio termsn otf o Pth, e WHA ^equatio ) In nth e
The results of this paper were presented during the 3rd Symposium on Integral Equations and Their Applications held at the Teohnioal University of Warsaw, December 3-6, 1984.
- 725 2 F.-0. Speck
(1.2) WU = V 6 PX.
Various applications admit the same operator theoretic facto- rization type due to similar operator algebra structures (sym- bol calculus), cf. the monographs [5, 13» 7, 11]. We always shall assume a cross factorization of A (with respect to X and P) to be given, i.e.
(1.3) A = A_CA+ holds with invertible factors and
A+PX ® PX, A_QX = QX, P + Q = I (1.4) 1 1 p1 C" PCP P? = PP.,. P3 := C" QCQ = P^ = QP3. The author proved [15, 16] that this assumption is equivalent to the generalized invertibility of W [12],
(1.5) WW~W = W with a suitable W~E L(PX) obtained from (1.3)» for instanoe, by
(1.6) W A;1PC"1PA:1 PX A croes faotor C splits off the space X twioe into complemen- tary subspaces
x1 + X0 = PX, Xg + x3 QX (1.7) Y + Y QX 1 2 K. *o + Y3 such that
(1.8) x4 ¡ - 0,1,2,3 Xi are bijective, and the projectors on X^, Yj along the corres- ponding complements are given by the so-called Ids of C
- 726 - Wiener-Hopf operator theory
1 1 1 1 p1 - C" PCP, p0 = C~ QCP, p2 - C~ PCQ, p3 = C~ QC<3 (1.9) 1 1 1 -1 q1 = CPC~ P, q2 « CQC~ P, qQ = CPC~ Q, = CQC Q la this paper we will discuss the Hilbert space case in order to obtain the least squares approximate solution of the incon- sistent equation (1.2) and the least norm solution of the con- sistent one. These are given by ff+v where W+ denotes the Moore-Penrose inverse (MPI) of a WHO with closed range defin- ed by [12] Wff+W = ff, W+ffW+ = w+ (1.10) (WW+)* = WW+, (W+W)* = ff+W.
There is not a simple formula like (1.6) for the MPI, since the reverse order law
+ (T2T1) * T|T+ does not hold true in general and even in the case where one factor is invertible. So we need some new formulas for the MPI of a product of operators (§ 3). The representation of W+ de- pends on whether P, pQ, q^ are orthogonal or not (§ 4-6). In the second oase, the orthogonal projectors P', p^, q^ on the ranges of P, pQ, q1 or a corresponding refactorization of (1.3) have to be found, which involves existence and unique- ness questions for such factorizations (§ 5). Finally we give some comments about extensions of our re- sults (§ 6). Applications will be discussed in § 2, 4 and 6, which show the practical use of the formulas (4.3-4), (6.1-2) for W+.
2. Examples of cross factorizations Example 1. The classical WH equation of second kind [19,8]
- 727 - 4 g.-O. Speck
po u(x) + J k(x-y)u(y)dy • v(x), x >0
(2.1) Vu(x)
P+(I+k*) u(x) Lp(H+) can be solved by factorisation of the Fourier symbol i * 1+ Fk in the Wiener algebra,
(2.2) tUi - $ (-|fî)Uï+(U, if where the middle factor û is the symbol of a orosa factor C « For p » 2, C is unitary, which implies p^, q^ to be orthogonal. W is always Fredholm and one-sided invertible in t|ils case. Example 2* For the half-spaoe analogue [16] we find
(2.3) UK) •*.(** ^ U'.^6®
teC(Rn"1, [0,°°) ), M « IRn"1 — Z measurable with a unitary cross factor C again, but generaliz- ed invertible V. The defect numbers are either zero or infi- nite. B x a m p 1 e 3. The quarter plane WHO [17, 10] with L1 kernel and nonvanishing symbol is Fredholm-Eiesz (IndW=0), In general, there does not exist a cross factorization of convolutional type. Bxample 4. Toeplitz operators in one dimension with continuous symbols behave similar as example 1.
(2.4) iJSi^i+U). t*r0 » I4I- 1 p involves a unitary C in the oase of X - 1 (Z).
- 728 - Wiener-Hopf operator theory 1
Example 5. For finite dimensional WHOs [6], a cross factorisation can be obtained by use of the Gauss algorithm
(2.5) where C is the linear transformation ooncerned with a change of lines matrix BA, whloh is unitary. Example 6. Singular integral operators of Cauahy type on a olosed Ljapounov curve system r
(2.6) Tu(t) - a(t)u(t) b(t) dr, t«r are usually written in the form [5, 13]
T - al + bS_ « (a+b)Pr + (a-b)Q- - (2.7) r r r « A^P + AgQ - AgiPA^A^ + Q)(I + QAP)
1 with A : = Ag A1f if the coefficients are invertible. The equi- valence to a (general) WHO was pointed out independently by I. Gohberg [5] and U. Shinbrot [14] in 1964. Factorization of a rational function A * A^A.. with respeot to r yieldB a cross •* * —1 factorization as in example 4. But here P = P, C = C hold only for X « I»2( r_) (or r=IR and the Hilbert transformation S^ os L(K)). w is Fredholm and one-sided invertible. Example 7. Systems of singular integral opera- tors lead us to a matrix factorization [11]
1 (2.8) (a-b)~ (a+b) - «= A » A_ " • . u ) A+ (OeD+).
1 Again, P* = P, C* » C" hold only for r = rQ (orr=IR). ff is Fredholm, but not one-sided invertible in general.
729 - 6 F.-O. Speok
3. Soma results about Moore-Penrose inverses In what follows, X denotes a Hilbert space, L(X) are assumed to be normall? solvable, i.e. R(T) « R(T) etc., and Pz denotes the orthogonal projector on a olosed subspaoe ZCX, i.e. P^ • Pz holds. Looking for the MPI of a produot of ope- rators we make use of normal faotorlaation
(3.1) « T*(TT*)+ = (T*T)+T* and of full rank factorization
1 1 (3.2) (TjjT,)* - TJ(T1T![)- (T2T2)- T2 for T^T" = I = TgTj, i.e. if T1 is right invertible and Tg is left invertible, of. [l]. Proposition 1. T+ may be represented in terms of any generalized inverse T" and the orthogonal projectors on the range R(T) and the carrier C(T) • H(T)1 of T by
+ (3.3) T = Pc(!r) ®"PR(Tj*
Proof.
+ + + + (3.4) Pc(Tj T"PR(Tj - T TT~TT » T TT » T. Proposition 2. T =» Tj^i aßd T^^ = I = = Tyield
+ (3.5) T .Tl(Pc(T2)TlTl r1 T F T T C(I2) 2( R(T2) 3 3| • R(T2) (where restricted operators have to be continued by zero ex- tension on the orthogonal oomplement). Proof. Proposition 1 implies
+ p T T T P (3.6) T = PC(T) T"PR(T) - c(T) 1 2 3 R(T) with
(3.7) P = P a P C(T) C(TgT^) C(T2PC(T2>T1
- 730 - Wlener-Hopf operator theory 7
Since PC(t2)T1 X —C(T2) is right invertible by T^ c(t2) and T, is left invertible by Tg, we obtain c(t2)
(3.8) PC(T) = PC(PC(T jl,) = (PC(T2)T1,+T1-
Normal factorisation and positivity of T1T1 yield the left factor in (3*5). The right one follows by analogy. Proposition 3. If are even invertible, then
(3.9) I+ - [i - ^„(T^VV 1 T*-lj T- . n(T2) -1 *?] [l - T«-1 T-1T*-1 (PrD(Tg ) 3 a3 d(T2, holds true where D(T2) ° R(T2)1 denotes the defect spaoe of T2 that is orthogonal to the range. Proof* This formula follows also from Proposition 1, if we rewrite
1 (3.10) I-P, ,(t: P.
m-1/p m*-1T-1 £-1
due to normal factorization and I - PR(T) = Pq(>x) by analogy. Remark. In general, the inverses in equation (3.5) etc. are given by a Neumann series according to the positivity of T^ and ||PC(l2) I - 1, of. [10].
4. The representation of W"*" in the case of P* «= P and C* Now let
(4.1) A o A_CA+6 L(I)
- 731 - 8 F.-O. Speofc be a arose factorization with respect to X and an orthogonal projector P, and let C be unitary or just
(4.2) Pi = P-) • be fulfilled, of. equations (1.4), (1.9). Theorem 1 (generalizes some results in [10]). The Mil of W - FA? reads
. -1 * (4.3) W+ = PA+(PlA+PA* J-1 C-Mq^PA. ) 1 A_P p^X and (4.4) TV"1 P x ]
W-[p-At-1(q2A:lA:"1 )-1 a;1]. q2X where V" is given by equation (1.6) for instance. Proof. This is a direot consequence of
(4.5) W - PAP « PA_CA+P = PA_PCPA+P « PA_q1Cp1A+P Tj^l and Propositions 2 and 3, respectively. Remark. The second formula (4.4) is more interest- ing in view of the applications mentioned in $ 2, sinae the inverses of finite dimensional operators are simply given by matrix inversion. Note also that the middle factor P in (4.3) does not appear In (4.4). Example 1. Consider the first example of § 2 with «>0. In this caise the cross factor
(4.6) fulfils CP = PCP( since the symbol is holomorphie in the upper half plane of (4.7) p, - C,- 1 PCP - P, pe - 732 - Wiener-Hopf operator theory holds true and the left factor la (4*4) equals F (W is left invertible). The orthogonal defect spaoe of ?C|pX reads 1 1 (4.8) q2X « CQC" PI « F" span , « span I f^ t... ,fw| t where the functions f1 may be orthonormalized, of. [10, 16]. This yields u (4.9) q2v - 2 ^ 1=1 A lA 1q v A 1 1 »-11- + 1 1 1 1 1 1 (4.12) W v - A; PC" PA: (T - A*" ^ •{* " (*lt...,xn) - (x',xfl), xfl>o} and A a translation in- variant operator with symbol $, whioh admits a factorization (2.3). So we may write (4.13) A - A_CA+ - where A+ are assumed to fulfil (1.4a). Furthermore let - 733 - 10 F.-0. Speck t(V) = V$'2 + 1. K'e 1 1 (4.14) Bk -{V e JRf " » mes (Bk - B^.) = 0, kez. It has been shown [16] that (4.15) P0«W->jix2 2 U')^). k=1 1=1 holds where $ denotes the n-1 dimensional Fourier transform of $ and l+1/2 n 1 4>kle H" (R " ), supp$klc B_k, 1= 1,...,keN (4.16) J 1 T+U) = * it(V)]" are fulfilled. The projector q2 looks the same with being supported in Bk, keN. If we orthonormalize {r+} with respect to the last variab> le by the S. Schmidt procedure, we obtain 1 Pu (4.17) p0u - F- ^ < »ri> rfxjc k,l as an orthogonal projection where %k denotes the characteris- tic funotion of Ek and <.,.> means the scalar product with respect to the last variable* From this follows oo k k 1 1 1 (4.18) p0A*- A; p0u = F" ^ 2 2 «j'liXk k=1 1=1 j=1 2 with := < | $+| ~ -r^»r^>.Let (aj^) be the Inverse matrix, which exists uniformly with respect to t,' due to strict 2 positiveness of |$+|~ , of. [l6]. - 734 - Wieoer-Hopf operator theory 11 So we get (4.19) (P0A^-1A*-1 r1 W = P"1 2 (4.20) then act on a matrix M in the way that some matrix elements have to be dropped, e.g. '0 0 *\ * * (4.21) Hp0 M P<|M = 0 P2M ,0 o *y and so on. Example 7. for systems of singular integral equations, the cross factor in (2.8) may be splitted off into I u, o\ 0 \ (4.22) C = m with = 2 10 j = 2 This imPlie£ (4.23) p = C QCP = C-1_ QC_, P - 735 12 g.-O. Speok 'zw1\ M 0 0 Po1 epan< 0 com \0 0 and a representation of q2 by analogy. The factors of equa- tion (4.4) are obtained as before. 5. The case of P P and Pi * Pi or q* j q1 Instead of assumption (4*2), we introduce the orthogonal projectors s P + 'l " C(PCP) * (PCP) PCP, p'0 P - p!, (5.1) J P + := P - q <1 " R(PCP) " *CP(PCP) , q'2 1 and obtain by the same idea as before. Theorem 2. The formulas (4.3)» (4.4) in Theo- rem 1 hold true, if the Ids of C are replaced by p!|,...,q'2. Remark. p^ does not mean the orthogonal projector on p^X (whioh is not useful here) but that projector on the orthogonal complement of p X with respect to PX. It oan be shown easily that p^X p^X holds. Furthermore, C is a cross factor with Ids q*,...,P3 instead of P0,...,q.j, of. eqs. (1.4), (1.9). Now we take into aooount the orthogonal projectors p^, qlj, qp and p^ on four components pQX, q^X, qQX and p^X, respecti- vely, as well as the complementary projectors, with respect to PX or QX denoted by p^, q'2, q^ and p2, respectively. The additional ones are concerned with the operator QC~1Q asso- ciated with PCP, cf. [16]. Proposition 4. The following representation formulas hold true, - 736 - Tiener-Hopf operator theory 12 1 p, + - c*(pcc*|pi)~ C q!, • q; - C(PC*C|pj)"^ C* (5.2) 1 1 1 p; • pi - c- (Qc*-V|QIr (f- 1 1 1 1 qg + - C*" (QC'V" )|qx)- C- and the single projectors are given by (5.3) P-J - PÎP^+Pg) - (PÎ+P^)P - Pip'-j+PgJP eto. Proof. Obviously, all these operators are self- adjoint (thus orthogonal) projectors). According to the ixip- vertibility of the factors on X, PX or QX, respectively, the ranges read 1 (5.4) R(C*(PCC*|pxr C) - R(C*P) - (p*+p|)X - 1 i - C(PC) » N(PC) = (p0X+p3X) « (p^+pgjx etc. The combinations in (5.2) always contain one projector on a subspace of PX and another one on a subspace of QX, vhioh yields (5.3), etc. Proposition 5. The MPI of the projection of a cross faotor (with nonorthogonal Ids, but P* = P) reads + 1 1 (5.5) (PCP) - C*(PCC*|px)~ C(jC*c|Kr C* = 1 1 1 1 1 1 1 1 1 1 - [p-c" (Qc*-V |QIr c^- ]pc- p[p-c^- (Qc- c^- |QXr (r ]. Proof. This is a consequence of the last result, Proposition 1, equations (5.1) and (1.6). Bow we discuss existence and uniqueness questions for oross factorizations with unitary C provided P is orthogonal. Proposition 6. If W = PA|px has a closed range (A be invertible, P2 = P = P* on a Hilbert space X), - 737 - 14 P.-O. Speck there is exactly one oroae factorization of left (right) stan- dard type A *> (|I + QB_P) C(I + PB+Q) (5.6) qoB-Q2 " 0 POB+p2 " rs8P*> with orthogonal Ids P0,...,q3 of C. Proof. As a result in ring theory all factoriza- tions of this type are in one-to-one correspondence with the pairs (W^1'2', wy*2') of reflexive generalized inverses of W and the associate operator W^ « |QX» C1^» chapter 6] where 1 (1,2, 1 1 2 (5.7) C~ QCP = p0 » P - W W,...,CQC~ Q = q-J = WJ » ^ are fulfilled. Therefore the only oross faotorization of type (5.6) with orthogonal Ids IB concerned with (W*,^). Proposition 7. Under the same assumptions, there exists a cross faotorization with unitary cross faotor. Proof. We start with the factorization (5.6) and factorize C by polar decomposition (5.8) C = CS, S = (C*C)1/2 suoh that C = CS~1 = C*~1S is unitary. Sinoe the Ids of C are orthogonal, the restrictions of (5.9) C*CJ PJX—C*Q3X= p.jX, j = 0,1,2,3 on the single complements are bisections. Thus the same holds true for S, of. [18, § 7.3]» which implies (5.10) S = PSP + QSQ to have invariant subspaces PX and QX. On the other hand, C represents a oross factor due to equations (1.4) and 1 1 1 1 P1 C~ PCP = SC" PCS" P = SPlS- = Pp., (5.11) 1 1 P3 := C" QCQ Sp3S" = Qp3. 738 - Wiener-Hopf operator theory 15 So we end up vith (5.12) A * A_CA+ (I+QBJ>)C S(I+PB+Q) where A+PX = PX is obviously fulfilled. Remark 8. 1. Cross factorizations of standard type (5.6a) are characterized by W • PC| 2. In general, there is not a cross factorization of stan- dard type with C* «= C~1 (if W is not a partial isometry). 3. All aross factorizations of A can be obtained from the standard ones by splitting off invertible factors of type PBP + QBQ and 6. The case of P* 4 P Finally we assume X to be a Hilbert space, P2 = P, A in- vertible on X, and a cross factorization A = A_CA+ to be given. Theorem 3. The MPI of W is represented by (6.1) W+ « P* A* ( pi A PP*A* I )~1 C"1«(q'A*P*PA J )~1A*P* + 1 + +Ip^x 1 " "1 q i x and 1 1 1 1 1 (6.2) WR* = [P'-A; (P;A*- A; #X)" A*" JW"- i i i 1 1 1 • [p'-Ar (q2A: (q'2A: Ar |q/x)- A; ], where P' denotes the orthogonal projeotor on PX and PQ,.*. are defined by (5.1). Proof. Again, we apply Ptropositions 2 and 3 to (4.5). Example . For (systems of) Cauchy type singular integral operators on closed Ljapounov systems, the adjoint Cauchy operator Sp e L(L2(r)) is represented by S* = -HpSpHp, Hpi(t) - hp(t)$(t), hp(t) - a16^, * e L2( D, it r where e(t) - 739 - 16 L.-O. Speck. denotes the angle between the tangent sector in t and the positive x-axis [11, II, § 5]. Furthermore, P « Ip • j (I+Sr) and Szego projectors concerned with P' are involved, and the finite dimensional orthogonal projectors p^» q'2 can be eva- luated as usual, Pinal remarksi The method of this paper can be extended in order to treat more complicated problems« 1. If an approximation factor with positiva real part occurs in the factorisation (continuous or locally sectorial functions instead of rational symbol»), and if W is one-Elded invartible, the formulas can be modified easily» Sometimos it makes sanse to consider factorisations A » A —U "AJOLA 0 + +. with remainder A„0,' ReA 0 >0, cross factors U+L with QU.+P • 0 « Pü Q and some compatibility conditions [16]7 In this case, the corresponding fórmalas for W* ars quits long, but not -acre complicated In prinaipl®. 2. Asymmetric WHOs W «= P^Al with different Hilbert £' IP.,2 spaces Xj, A e l{21,2?) and projectors F^ s L.(X,J can be traatsd by analogy, cf. !j6, chapter 2]* 3« It would be interesting to consider also ¡¿abounded operators, cf. papers by A. Devinatz, M» Shinbrot, ¥. Pelle- grini, J, Reedsr cited In [9] or M.S. Bashed [12]. 4, Bisinguls» aparators [~4] and two-dimensional Toeplits: operator® [3] lead us on representations of the MPI, ?fhich contain infinite products of projectors as in the casa of the quarter plane probliem [10] • 5, Patead singular operators [13] are equivalent to WHOe, if the coefficients are invartible» cf. eq* (2.7). But, as a significient difference, the carrier of T * A^P + A?Q pends on both factors A+ and Am (and C, indeed). Results about will be given elsewhere. - 740 - Wiener-Hopf operator theory 17 REFERENCES [1] R.H. B o u 1 d i a : Generalized inverses and faoto- rizations (im Recent Applications of Generalized Inver- ses). London 1982, 233-249. [2] A. Devinatz, M. S.hinhrot: General Wiener-Hopf operators, Trans. Amer. Math. Soc. 145 (1969) 467-494. [3] R. Douglas: Banach algebra techniques in ope- rator theory. New York 1972. [4] R. Duduchava: On bisingular integral opera- tors on convolution operators on a quadrant, Soviet. Math. Dokl. 16 (1975) 330-334. [5] I.Z. Gochberg, I.A. F e 1 d m a n : Fal- tungsgleichungen und Projektionsverfahren zu ihrer Lö- sung. Basel 1974 (Russ. 1971). [6] I. Gohberg, S. Goldberg: Finite-di- mensional Wiener-Hopf equations and factorization of matrices, Linear Algebra and Appl. 48 (1982) 219-236. [7] I. Gohberg, N. Krupniks Einführung in die Theorie der eindimensionalen singulären Integral- operatoren. Basel 1979 (Süss. 1973). [8j M.G. K r e i n : Integral equations on a half-line with kernel depending upon the differenoe of the argu- ments. Amer. Math. Soc. Trans1. 22 (1962) 163-188 (Russ. 1958). [9] S. Meister, F.-O. Speck: Some multi-di- mensional Wiener-Hopf equations with applications, Trends Appl. Pure Math. Mech. 2 (1977) 217-262. [10] S. Meister, F.-O. Speck: The Mooire- -Penrose inverse of Wiener-Hopf operators on the half- -axis and the quarter-plane. To appear in Int. Sqs. [11] S.G. Miohlin, S. Prössdorf: Singu- lare Integraloperatoren. Berlin 1980. [12] M.Z. Bashed (ed.): Generalized inverse and appli- cations. New York 1976. - 741 18 f.-O. Speok [13] S. Prössdorf: Einige Klassen singulärer Gleichungen. Berlin 1974, Basel 1974* [14] M. Shinbrot: On singular integral operators, J. Math. Meoh. 13 (1964) 395-406. [153 F«-0. Speck: On the generalized invertibility of Wiener-Hopf operators in Banach spaces, Int. Eqs. Op. Th. 6 (1983) 458-465. [l6j F.-O. Speck: General Wiener-Hopf factorization methods. London 1985. [17JG. Strang: Toeplitz operators in a quarter-pla- ne, Bull. Amer. Math. Soo. 76 (1970) 1303-1307. [18] J. Weidmann: Lineare Operatoren in Hilbert- raumen. Stuttgart 1976. [19] N. Wiener, S. Hopf: Über eine Klasse singulärer Integralgleichungen. S.-B. Preuse. Akad. Wies. Berlin. Phys.-Math. Kl. 30/32 (1931) 696-706. ABTEILUNG MATHEMATIK, TECHNISCHE UNIVERSITÄT DARMSTADT, 6100 DARMSTADT, BRD Received December 6, 1984. - 742 -