TENTH ASSIGNMENT, DUE DECEMBER 4 IN CLASS18.155 FALL 2001RICHARD MELROSEProblem 1. (Poisson summation formula) As in class, let L ⊂ Rnbe an integrallattice of the formL =v =nXj=1kjvj, kj∈ Zwhere the vjform a basis of Rnand us ing the dual basis wj(so wj· vi= δijis 0 or1 as i 6= j or i = j) setL◦=w = 2πnXj=1kjwj, kj∈ Z.Recall that we defined(1) C∞(TL) = {u ∈ C∞(Rn); u(z + v) = u(z) ∀ z ∈ Rn, v ∈ L}.i) Show that summation over shifts by lattice points:(2) AL: S(Rn) 3 f 7−→ ALf(z) =Xv ∈Lf(z − v) ∈ C∞(TL).defines a map into smooth periodic functions.ii) Show that there exists f ∈ C∞c(Rn) such that ALf ≡ 1 is the costantfunction on Rn.iii) Show that the map (2) is surjective. Hint: Well obviously enough use thef in part ii) and show that if u is periodic then AL(uf) = u.iv) Show that the infinite sum(3) F =Xv ∈Lδ(· − v) ∈ S0(Rn)does indeed define a tempered distribution and that F is L-periodic andsatisfies exp(iw · z)F (z) = F (z) for each w ∈ L◦with equality in S0(Rn).v) Deduce thatˆF , the Fourier transform of F, is L◦periodic, conclude that itis of the form(4)ˆF (ξ) = cXw∈L◦δ(ξ − w)vi) Compute the constant c.vii) Show that AL(f) = F ? f.viii) Using this, or otherwise, show that AL(f) = 0 in C∞(TL) if and only ifˆf = 0 on L◦.12 RICHARD MELROSEProblem 2. For a measurable se t Ω ⊂ Rn, with non-zero measure, set H = L2(Ω)and let B = B(H) be the algebra of bounded linear operators on the Hilbert spaceH with the norm on B being(5) kBkB= sup{kBf kH; f ∈ H, kf kH= 1}.i) Show that B is complete with respect to this norm. Hint (probably notnecessary!) For a Cauchy sequence {Bn} observe that Bnf is Cauchy foreach f ∈ H.ii) If V ⊂ H is a finite-dimensional subspace and W ⊂ H is a closed subspacewith a finite-dimensional complement (that is W + U = H for some finite-dimensional subspace U ) show that there is a closed subspace Y ⊂ W withfinite-dimensional complement (in H) such that V ⊥ Y, that is hv, y i = 0for all v ∈ V and y ∈ Y.iii) If A ∈ B has finite rank (meaning AH is a finite-dimensional vector space)show that there is a finite-dimensional space V ⊂ H such that AV ⊂ Vand AV⊥= {0} whereV⊥= {f ∈ H; hf, vi = 0 ∀ v ∈ V }.Hint: Set R = AH, a finite dimensional subspace by hypothesis. Let N bethe null space of A, show that N⊥is finite dimensional. Try V = R + N⊥.iv) If A ∈ B has finite rank, show that (Id −zA)−1exists for all but a finite setof λ ∈ C (just quote some matrix theory). What might it mean to say inthis cas e that (Id −zA)−1is m eromorphic in z? (No marks for this secondpart).v) Recall that K ⊂ B is the algebra of compact operators, defined as theclosure of the space of finite rank operators. Show that K is an ideal in B.vi) If A ∈ K show thatId +A = (Id +B)(Id +A0)where B ∈ K, (Id +B)−1exists and A0has finite rank. Hint: Use theinvertibility of Id +B when kBkB< 1 proved in class.vii) Conclude that if A ∈ K then{f ∈ H; (Id +A)f = 0} and(Id +A)H⊥are finite dimensional.Department of Mathematics, Massachusetts Institute of TechnologyE-mail address: