PROBLEM SET 6, 18.155
DUE 30 OCTOBER, 2015
I think you should do the first 7 questions, but that is where it starts
to get a bit interesting.
The circle as a manifold. Define the circle as the quotient S = R/2πZ
in terms of the equivalence relation x ≡ x + 2πk, k ∈ Z with the
projection map
π : R −→ S, f : S −→ C =⇒ π ∗ f = f ◦ π
so π ∗ f is periodic of period 2π.
(1) Define C ∞ (S) = {u : S −→ C; π ∗ u ∈ C ∞ (R)} with the (metric)
topology of uniform convergence of all derivativies on [0, 2π] (or
a larger interval) and show that there is a continuous averaging
map
X
(1) π∗ : S(R) −→ C ∞ (S), s.t. π ∗ (π∗ (ψ))(x) =
ψ(x + 2πk).
k∈Z
(2) Construct a partition of unity 0 ≤ ψi = π∗ (ψ̃i ) ∈ C ∞ (S), i =
0, 1, supp ψ̃i ⊂ (i, 2π + i), ψ0 + ψ1 = 1 and show that
X
C ∞ (S) 3 v −→
π∗ (ψ̃i π ∗ u) ∈ C ∞ (S)
i=1,2
is an isomorphism.
(3) Define C −∞ (S) = (C ∞ (S))0 and show that C ∞ (S) ,→ C −∞ (S) by
Z
Z
∗
∗
f 7−→ Uf , Uf (v) =
(π f )(π v) = (ψ̃0 + ψ̃1 )(π ∗ f )(π ∗ v)
(0,2π)
R
is an injection which is independent of choice of the partition
of unity (and which we subsequently treat as an identification).
(4) Define π ∗ : C −∞ (S) −→ S 0 (R) (aggressive notation) by π ∗ u(φ) =
u(π∗ φ) and show that this is a bijection to the space of 2πperiodic distributions with the pairing
(2)
u(v) = π ∗ u((ψ̃0 + ψ̃1 )π ∗ v), u ∈ C −∞ (S), v ∈ C ∞ (S)
which is consistent with the identification of C ∞ (S) as a subspace.
1
2
PROBLEM SET 6, 18.155 DUE 30 OCTOBER, 2015
(5) Show that if u ∈ C −∞ (R) then the Fourier transform
X
∗ u(ξ) = 2π
d
π
ck δ(ξ − k) in S 0 (R), |ck | ≤ C(1 + |k|)N
k∈Z
where C and N depend on u and conversely. These are (by
definition here) the Fourier coefficients of u.
∗u = π
∗ u and think about where
d
d
Hint: Check that exp(2πiξ)π
and how exp(2πiξ) − 1 vanishes.
(6) Define L2 (S) = {u ∈ C −∞ (S); π ∗ u ∈ L2loc (R)} and show that it
is a Hilbert space with norm
21
Z
∗ 2
(ψ̃0 + ψ̃1 )|π v|
(3)
= kπ ∗ vkL2 (0,2π)
R
and that for appropriate constants the ek = ck π∗ exp(ikx) form
a complete orthonormal basis.
(7) Show that u ∈ C ∞ (S) if and only if the ‘Fourier coefficients’ ck
above form a rapidly decreasing sequence, |ck | ≤ CN (1 + |k|)−N
for all N and that
X
ck exp(ikx) in C ∞ (S).
(4)
u=
k
(8) Define the Hardy space H as the subspace of L2 (S) spanned
by the ek with k ≥ 0 and show that if S is the orthogonal
projection onto H then
H 3 u 7−→ S exp(ikx)v ∈ H, k ∈ Z
is a Fredholm operator of index −k.
(9) For a ∈ C ∞ (S) denote by Ta : H −→ H the ‘Toeplitz operator’
S ◦ (a×) and show that the Fourier coefficients of Ta u are
X
dk =
ak−l cl
l≥0
where the aj are the Fourier coefficients of a.
(10) Show that if a, b ∈ C ∞ (S) then
Tab − Ta Tb
is a compact operator on H.
(11) Show that if 0 6= a ∈ C ∞ (S) then Ta is a Fredholm operator on
H.
Hint: Show that T1/a is a 2-sided parameterix.
(12) Prove Toeplitz’ index formula that if 0 6= a ∈ C ∞ (S) then
ind(Ta ) = − wn(a)
PROBLEM SET 6, 18.155
DUE 30 OCTOBER, 2015
3
where the winding number is
l(2π) − l(0) where π ∗ a(x) = |π ∗ a| exp(2πil(x))
with l(x) continuous.
Hint: Show that there is a continous map [0, 1] 3 t 7−→
at (x) ∈ C ∞ (S) such that at 6= 0, a0 = 1 and a1 is the integer
wn(a). Show that Tat has constant index as a function of t.