PDE LECTURE NOTES, MATH 237AB
157
11.
Introduction to the Spectral Theorem
The following spectral theorem is a minor variant of the usual spectral theo
rem for matrices.
This reformulation has the virtue of carrying over to general
(unbounded) self adjoint operators on in
fi
nite dimensional Hilbert spaces.
Theorem 11.1.
Suppose
A
is an
n
×
n
complex self adjoint matrix, i.e.
A
∗
=
A
or equivalently
A
ji
=
¯
A
ij
and let
µ
be counting measure on
{
1
,
2
, . . . , n
}
.
Then
there exists a unitary map
U
:
C
n
→
L
2
(
{
1
,
2
, . . . , n
}
, dµ
)
and a real function
λ
:
{
1
,
2
, . . . , n
}
→
R
such that
UAξ
=
λ
·
Uξ
for all
ξ
∈
C
n
.
We summarize this
equation by writing
UAU
−
1
=
M
λ
where
M
λ
:
L
2
(
{
1
,
2
, . . . , n
}
, dµ
)
→
L
2
(
{
1
,
2
, . . . , n
}
, dµ
)
is the linear operator,
g
∈
L
2
(
{
1
,
2
, . . . , n
}
, dµ
)
→
λ
·
g
∈
L
2
(
{
1
,
2
, . . . , n
}
, dµ
)
.
Proof.
By the usual form of the spectral theorem for selfadjoint matrices, there
exists an orthonormal basis
{
e
i
}
n
i
=1
of eigenvectors of
A,
say
Ae
i
=
λ
i
e
i
with
λ
i
∈
R
.
De
fi
ne
U
:
C
n
→
L
2
(
{
1
,
2
, . . . , n
}
, dµ
)
to be the unique (unitary) map determined
by
Ue
i
=
δ
i
where
δ
i
(
j
) =
½
1
if
i
=
j
0
if
i
6
=
j
and let
λ
:
{
1
,
2
, . . . , n
}
→
R
be de
fi
ned by
λ
(
i
) :=
λ
i
.
De
fi
nition 11.2.
Let
A
:
H
→
H
be a possibly unbounded operator on
H.
We let
D
(
A
∗
) =
{
y
∈
H
:
∃
z
∈
H
3
(
Ax, y
) = (
x, z
)
∀
x
∈
D
(
A
)
}
and for
y
∈
D
(
A
∗
)
set
A
∗
y
=
z
.
De
fi
nition 11.3.
If
A
=
A
∗
the
A
is self adjoint.
Proposition 11.4.
Let
(
X, µ
)
be
σ
—
fi
nite measure space,
H
=
L
2
(
X, dµ
)
and
f
:
X
→
C
be a measurable function. Set
Ag
=
fg
=
M
f
g
for all
g
∈
D
(
M
f
) =
{
g
∈
H
:
fg
∈
H
}
.
Then
D
(
M
f
)
is a dense subspace of
H
and
M
∗
f
=
M
¯
f
.
Proof.
For any
g
∈
H
=
L
2
(
X, dµ
)
and
m
∈
N
,
let
g
m
:=
g
1

f

≤
m
.
Since

fg
m

≤
m

g

it follows that
fg
m
∈
H
and hence
g
m
∈
D
(
M
f
)
.
By the dominated
convergence theorem, it follows that
g
m
→
g
in
H
as
m
→ ∞
,
hence
D
(
M
f
)
is
dense in
H.
Suppose
h
∈
D
(
M
∗
f
)
then there exists
k
∈
L
2
such that
(
M
f
g, h
) = (
g, k
)
for all
g
∈
D
(
M
f
)
,
i.e.
Z
X
fg
h dµ
=
Z
X
g
k dµ
for all
g
∈
D
(
M
f
)
or equivalently
(11.1)
Z
X
g
(
fh
−
k
)
dµ
= 0
for all
g
∈
D
(
M
f
)
.
Choose
X
n
⊂
X
such that
X
n
↑
X
and
µ
(
X
n
)
<
∞
for all
n.
It is easily checked
that
g
n
:= 1
X
n
fh
−
k
¯
¯
fh
−
k
¯
¯
1

f

≤
n
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158
BRUCE K. DRIVER
†
is in
D
(
M
f
)
and putting this function into Eq. (11.1) shows
Z
X
¯
¯
fh
−
k
¯
¯
1

f

≤
n
dµ
= 0
for all
n.
Using the monotone convergence theorem, we may let
n
→ ∞
in this equation to
fi
nd
R
X
¯
¯
fh
−
k
¯
¯
dµ
= 0
and hence that
¯
fh
=
k
∈
L
2
.
This shows
h
∈
D
(
M
¯
f
)
and
M
∗
f
h
=
fh.
Theorem 11.5
(Spectral Theorem)
.
Suppose
A
∗
=
A
then there exists
(
X, µ
)
a
σ
—
fi
nite measure space,
f
:
X
→
R
measurable, and
U
:
H
→
L
2
(
x, µ
)
unitary
such that
UAU
−
1
=
M
f
.
Note this is a statement about domains as well, i.e.
UD
(
M
f
) =
D
(
A
)
.
I would like to give some examples of computing
A
∗
and Theorem 11.5 as
well.
We will consider here the case of constant coe
ﬃ
cient di
ff
erential operators
on
L
2
(
R
n
)
.
First we need the following de
fi
nition.
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 Matrices, Continuous function, Dominated convergence theorem, dτ, Monotone convergence theorem

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