Riemann–Hilbert problem 191
If we assume that t is a parameter we have ∂
∂ z
j
= ∂
y
j
∂ z
j
∂ ∂
y
j
= t ∂
∂ y
j
. Noting that
|α| = |β| we obtain y
α
∂
β y
= z
α
t
| α|
t
−| β|
∂
β z
= z
α
∂
β z
. Hence P is transformed to the following operator on the torus
ˆ P
= X
| α|=|β|
a
αβ
z
α
∂ ∂
z
β
. This is identical with the operator introduced in the previous section if we set z
j
= e
iθ
j
.
4. Ordinary differential operators
Consider the following ordinary differential operator pt, ∂
t
=
m
X
k=0
a
k
t∂
k t
, where ∂
t
= ∂∂t and a
k
t is holomorphic in ⊂ C. For the sake of simplicity, we
assume = {|t| r}, where r 0 is a small constant. We consider the following
map p : O
7→ O. The operator p is singular at t
= 0. Therefore, instead of considering at the origin directly we lift p onto the torus T
= {|t| = r}. In the following we assume that r = 1 for the sake of simplicity. The case r
6= 1 can be treated similarly if we consider the weighted space.
Let L
2
T be the set of square integrable functions on the torus, and define the
Hardy space H
2
T by
H
2
T :
= {u =
∞
X
−∞
u
n
e
inθ
∈ L
2
; u
n
= 0 for n 0}. H
2
T is closed subspace of L
2
T . Let π be the projection on L
2
T to H
2
T .
Namely, π
∞
X
−∞
u
n
e
inθ
=
∞
X u
n
e
inθ
. In this situation, the correspondence between functions on the torus and holomorphic
functions in the disk is given by O
∋
∞
X u
n
z
n
←→
∞
X u
n
e
inθ
∈ H
2
T.
192 M. Yoshino
By the relation t∂
t
7→ D
θ
the lifted operator on the torus is given by ˆp =
X
k
a
k
e
iθ
e
− ikθ
D
θ
D
θ
− 1 · · · D
θ
− k + 1, where we used t
k
∂
k t
= t∂
t
t∂
t
− 1 · · · t∂
t
− k + 1. By definition we can easily see that π
ˆp = ˆp. For a given equation Pu
= f in some neighborhood of the origin we consider ˆp ˆu = ˆf on the torus, where ˆfθ = f e
iθ
. If we obtain a solution ˆu =
P
∞
u
n
e
inθ
∈ H
2
T of
ˆp ˆu = ˆf, u := P
∞
u
n
t
n
is a holomorphic extension of ˆu into |t| ≤ 1. The
function Pu − f is holomorphic in the disk |t| ≤ 1, and vanishes on its boundary since
ˆp ˆu = ˆf. Maximal principle implies that Pu = f in the disk, i.e, u is a solution of a given equation. Clearly, the maximal principle also implies that if the solution on
the torus is unique, the analytic solution inside is also unique. Hence it is sufficient to study the solvability of the equation on the torus.
Reduced equation on the torus
Define hD
θ
i by the following hD
θ
iu := X
n
u
n
hnie
inθ
, hni = 1 + n
2 12
. This operator also operates on the set of holomorphic functions in the following way
ht∂
t
iu := 1 + t∂∂t
2 12
u =
X u
n
hniz
n
. We can easily see that
D
θ
D
θ
− 1 · · · D
θ
− k + 1hD
θ
i
− k
= I d + K, where K is a compact operator on H
2
. It follows that since
hD
θ
i
− m
is an invertible operator we may consider ˆphD
θ
i
− m
instead of ˆp. Note that ˆphD
θ
i
− m
= π ˆphD
θ
i
− m
, and the principal part of ˆphD
θ
i
− m
is a
m
e
iθ
e
− imθ
. Hence, modulo compact operators we are lead to the following operator ∗
π a
m
e
iθ
e
− imθ
: H
2
7→ H
2
. Indeed, the part with order m is a compact operator if
hD
θ
i
− m
is multiplied. The last operator contains no differentiation, and the coefficients are smooth. It
should be noted that although a
m
t vanishes at t = 0, a
m
e
iθ
does not vanish on the torus.
D
EFINITION
2. We call the operator ∗ on H
2
T a Toeplitz operator. The func-
tion a
m
e
iθ
is called the symbol of a Toeplitz operator.
Riemann–Hilbert problem 193
5. Riemann-Hilbert problem and solvability
