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Frank - Olme Speck 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]).
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