Convergence of the Neumann series in higher norms

Transcription

1 Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann series for (Id +A) 1 converges in higher norm opologies. 1 The general case In applicaions one ofen considers equaions of he form (Id +A)u = v (1) where A is a bounded linear operaor. In his noe we consider he convergence of he Neumann series soluion for his ype of equaion. We firs prove a fairly absrac resul and hen consider a variey of special cases. Suppose ha {X k } is a nesed collecion of Banach spaces, X 0 X 1... X k X k+1..., wih norms { k }. Le A denoe he operaor norm of A : X 0 X 0, Au 0 A = sup. () X 0 u 0 u 0 If A < α 0 < 1 hen i is well known ha he Neumann series for (Id +A) 1 converges, see [1]. Indeed, if we define he sequence u 0 = v u j = v Au j 1 (3) hen u j is he jh parial sum of he Neumann series. A simple inducion argumen shows ha u j+1 u j 0 α j 0 u 1 u 0 0. (4) Research parially suppored by NSF gran DMS and he Francis J. Carey chair. Address: Deparmen of Mahemaics, Universiy of Pennsylvania, Philadelphia, PA. cle@mah.upenn.edu Keywords: Neumann series, higher norms, Marchenko equaion 1

2 1 The general case If we furher suppose ha A : X k X k is a bounded operaor for k K, and v X k, for such a k, hen he ieraes {u j }, defined in equaion (3), belong o X k. Our main heorem gives condiions under which he soluion u o (1) also belongs o X k, and he ieraes defined in (3) converge o u in he X k -opology. Theorem 1. Suppose ha A : X 0 X 0 is a bounded linear operaor and K is a posiive ineger or. Assume ha 1. A = α 0 < 1. A : X k X k boundedly for each k K 3. For each k K here are consans α k < 1 and C k < so ha, for u X k, we have he esimae Au k α k u k + C k u k 1. (5) If v X k, for a k K, hen he sequence {u j } defined in (3) converges o u in he X k -opology and herefore he soluion o (1) belongs o X k. Remark 1. The esimae in (5) is naural from he perspecive of pseudodifferenial operaors. Suppose ha M is a compac manifold and X k = H k (M). If A is a pseudodifferenial operaor on M, of order zero, whose principal symbol saisfies sup x M lim sup σ 0 (A)(x, ξ) = α < 1, (6) ξ hen A saisfies he esimaes in (5), wih α k = α. See []. Proof. The proof of he heorem is a small exension of he proof for he k = 0 case. By inducion assume ha we have shown, for l < k, ha here are consans, {β l } less han 1, and consans {C l } so ha, for all j, we have: u j+1 u j l C lβ j l u 1 u 0 l. (7) The esimae in (5), and he definiion of he sequence, {u j }, imply ha u j+1 u j k α k u j u j 1 k + C k u j u j 1 k 1 (8) Applying (7) gives he esimae u j+1 u j k α k u j u j 1 k + C k C k 1β j 1 k 1 u 1 u 0 k 1 (9) Using his esimae recursively we obain u j+1 u j k α j k u 1 u 0 k + C k C k 1 [ j m=1 β j m k 1 αm 1 k ] u 1 u 0 k 1. (10) From his esimae i is immediae ha here exiss a consan β k < 1 and C k < so ha u j+1 u j k C kβ j k u 1 u 0 k. (11) This complees he proof of he inducion hypohesis. The proof of he heorem follows easily from (11) via he k = 0 argumen.

3 Posiive self adjoin operaors 3 Posiive self adjoin operaors In his secion we assume ha X 0 is a Hilber space and ha B : X 0 bounded, posiive, self adjoin operaor. For every v X 0 he equaion X 0 is a (Id +B)u = v (1) has a unique soluion. By slighly modifying he equaion, he soluion can be found by summing a Neumann series. In paricular, we observe ha, if γ is a consan, hen u is a soluion o (1) if and only if u is a soluion o [Id + + γ (B γ Id)]u = v. (13) + γ Lemma 1. Le B : X 0 X 0 be a bounded, posiive, self adjoin operaor. If γ = B hen he Neumann sequence for equaion (13): u 0 = + γ v, u j+1 = + γ v + γ (B γ Id)u j. (14) converges, in he X 0 -opology, o he unique soluion of (1). Proof. Since B is a posiive operaor, he specrum of B lies in he inerval [0, γ], and, herefore + γ (B γ Id) = γ < 1. (15) + γ This in urn implies ha he sequence defined in (14) converges in he X 0 -opology o he soluion of equaion (13). To apply Theorem 1 o equaion (1) requires furher hypoheses on B. Firs we assume ha B : X k X k is bounded for every k K. A simple naural assumpion is ha B is a smoohing operaor, so ha here are consans {C k } such ha, we have he esimaes Bu k C k u k 1. (16) In fac somewha less is needed o apply he heorem. I suffices o assume ha here are consans {α k } and {C k } so ha, for every k, we have α k < 1, Bu k α k u k + C k u k 1. (17) Proposiion 1. Le K be a posiive ineger, or infiniy. Assume ha B : X k X k is a bounded operaor for k K, which is moreover posiive and self adjoin, when k = 0. Suppose here are consans {α k } and {C k } so ha he esimaes in (17) hold for k K. If v X k, for a k K, hen he ieraes defined in (14) converge o u in he X k -opology.

4 3 A Marchenko equaion 4 Proof. We only need o verify he hird hypohesis of Theorem 1 for he operaor +γ (B γ Id). Le u X k, hen he riangle inequaliy, and our assumpions on B imply ha + γ (B γ Id)u k + γ Bu k + γ + γ u k [ ] αk + γ u k + C k + γ + γ Bu k 1. (18) The assumpion α k < 1 implies ha α k + γ + γ < 1. (19) Hence we can apply Theorem 1 o complee he proof of he Proposiion. Remark. Suppose ha B is a pseudodifferenial operaor of order zero, on a compac manifold M wih a nonnegaive principal symbol, σ 0 (B). If hen + γ 1 (B γ1 sup x M lim sup σ 0 (B)(x, ξ) = γ 1, (0) ξ Id) is an order zero pseudodifferenial operaor wih principal symbol + γ 1 (σ 0 (B) γ1 γ1 ). The sup-norm of his symbol is bounded by + γ 1. Using for {X k } he sandard Sobolev spaces {H k (M)}, Garding s inequaliy implies ha, for each k > 0, here is a consan C k, so ha: (B γ 1 + γ 1 Id)u k γ 1 u k + C k u k 1. (1) + γ 1 Suppose ha B is also a posiive self adjoin operaor, wih L -norm γ. If γ = max{ γ 1, γ } hen, for v L (M), Lemma 1 applies o show ha he sequence in (14) converges in L (M) o he soluion, u, of (Id +B)u = v. If v H k (M), hen he esimaes in (1), wih γ 1 replaced by γ allow Theorem 1 o be applied o conclude ha his sequence also converges in H k (M). Remark 3. If B is a posiive operaor hen here are many oher ieraion schemes which converge o he soluion of (1). For example, he conjugae gradien mehod provides a differen sequence. I seems quie an ineresing quesion wheher, under hypoheses like hose in he proposiion, hese schemes provide sequences which converge in he X k -opology. 3 A Marchenko equaion We finish by considering a concree example which arises in he inverse scaering heory of he Zakharov-Shaba -sysem. Suppose ha f is funcion defined on R

5 3 A Marchenko equaion 5 which belongs o L 1 ([, )), for every finie. For each such, define he operaor, F, on L ([, )) F h(s) = f(s + y)h(y)dy. () Lemma. If f belongs o L 1 ([, )), hen he F is a bounded operaor on L ([, )) wih F = f(x) dx. (3) Proof. The proof is a sraighforward applicaion of he Cauchy-Schwarz inequaliy, which we leave o he ineresed reader. A simple calculaion shows ha he adjoin of F, as a operaor on L ([, )), is given by F h(s) = f(s + y)h(y)dy. (4) The Marchenko equaion for he ZS- sysem can be wrien [(Id +F F )k ](s) = f(s + ). (5) For each, his evidenly saisfies he hypoheses of Lemma 1 wih γ = F F, where γ = f(x) dx. For applicaions, i is of considerable ineres o know when a sequence converging o a soluion of he Marchenko equaion also converges in a sronger opology. For his case we ake X k = H k ([, )), wih he norms h k = k xh j L ([, )). j=0 Lemma 3. Suppose ha he funcions {f, x f,..., k xf} belong o L 1 ([, )). Then F defines a bounded map from X 0 o X k, for j = 1,..., k. Proof. This follows immediaely from Lemma and he observaion ha j x[f h](s) = ( j xf)(s + y)h(y)dy. (6) Using inegraion by pars one easily proves he following:

6 3 A Marchenko equaion 6 Lemma 4. Suppose ha he funcions {f, x f,..., x k 1 f} belong o L 1 ([, )) L ([, )). Then F defines a bounded map from X j o X j for each j k. Proof. We skech he k = 1 case. Inegraing by pars we see ha x [F h](s) = f(s + )h() f(s + y)h x (y)dy. (7) The claim follows from he hypoheses of he lemma, Lemma and he elemenary esimae: h() h x L ([, )) + h L ([, )). (8) The general case follows by repeaedly differeniaing (7) and inegraing by pars. As F F is a posiive self adjoin operaor on L ([, )), we can apply Proposiion 1 o he Marchenko equaion o obain: Proposiion. Suppose ha he funcions {f, x f,..., xf} k belong o L 1 ([, )) and {f, x f,..., x k 1 f} o L ([, )). If v H k ([, )) hen he sequence defined by h 0 = + γ v, h j+1 = + γ v + γ [F F γ Id]h j (9) converges in H k ([, )) o he unique soluion of [Id +F F ]h = v. (30) Proof. Lemma 3, and he hypoheses imply ha for each j k, here is a consan C j so ha, for all u L ([, )), we have he esimae F F u j C j F u 0 γ C j u 0. (31) Hence F F saisfies saisfies he hypoheses of Proposiion 1 wih α k 0. The k = 1 case is of paricular ineres in applicaions. In his case, he image of he uni ball in L ([, )) under F F consiss of uniformly bounded, uniformly equiconinuous funcions. Hence he ieraes defined in (18) are also uniformly bounded and uniformly equiconinuous. I is herefore an easy consequence of he Arzela-Ascoli heorem ha hey converge locally uniformly o he soluion of he Marchenko equaion.

7 References 7 Acknowledgmen I would like o ha Carlos Tomei And Jeremy Magland for helpful remarks and correcions. References [1] F. RIESZ AND B. SZ.-NAGY, Funcional Analysis, Frederick Ungar Publishing Co., New York, Translaed from he nd French Ediion by Leo F. Boron. [] M. E. TAYLOR, Pseudodiferenial Operaors, Princeon Univ. Press, Princeon, 1981.