LECTURE 14, 18.155, 24 OCTOBER 2013
Operators on Hilbert space, continued.
(1) For a compact operator the spectrum is of the form D cfw_0
where D C \ cfw_0 is a discrete set (possibly empty) consisting
of eigenvalues; cfw_0 may or may not be an eigenvalue
LECTURE 15, 18.155, 29 OCTOBER 2013
Fredholm and Trace class operators
Parial isometries
A unitary operator is an inner-product preserving bijection possibly
between two dierent Hilbert spaces. More generally a partial isometry
is a bounded operator V : H
LECTURE 16, 18.155, 31 OCTOBER 2013
Trace class operators Cont
Last time I dene and operator to be of trace class if it is a nite
sum of products of Hilbert-Schmidt operators
N
(1)
Ai Bi , Ai , Bi HS(H ).
T=
i=1
from which it follows that
(2)
T
TC
| T ej
LECTURE 17, 18.155, 5 NOVEMBER 2013
The rst thing I want to talk about in relation to half-spaces and
bounded domains is the restriction theorem for Sobolev spaces. So
consider the embedding map
E : Rn1
(1)
x (x , 0) Rn .
Pullback under this is the restri
LECTURE 18, 18.155, 7 NOVEMBER 2013
This was partly reconstructed later.
Smoothly bounded domains
(1) B Rn is bounded
(2) B = int(B ) is the closure of its interior
(3) For each p B there exists f C (Rn ; R) such that df (p) = 0
(so f /paxj (p) = 0 for so
18.155 LECTURE 19: 12 NOVEMBER, 2013
Let B Rn be a smoothly bounded domain.
Let me recall some Sobolev spaces associated to B. In fact it is convenient
to consider the unbounded domain B = Rn \ B which has B B = B =
as well. Then for each m R we can de
34
RICHARD B. MELROSE
6. Test functions
So far we have largely been dealing with integration. One thing we
have seen is that, by considering dual spaces, we can think of functions
as functionals. Let me briey review this idea.
Consider the unit ball in Rn
42
RICHARD B. MELROSE
7. Tempered distributions
A good rst reference for distributions is [2], [4] gives a more exhaus
tive treatment.
The complete metric topology on S (Rn ) is described above. Next I
want to try to convice you that elements of its dual
LECTURE NOTES FOR 18.155, FALL 2004
63
10. Sobolev embedding
The properties of Sobolev spaces are briey discussed above. If m
is a p ositive integer then u H m (Rn ) means that u has up to m
derivatives in L2 (Rn ). The question naturally arises as to the
LECTURE NOTES FOR 18.155, FALL 2004
83
12. Cone support and wavefront set
In discussing the singular support of a tempered distibution above,
notice that
singsupp(u) =
only implies that u C (Rn ), not as one might want, that u S (Rn ).
We can however ren
LECTURE NOTES FOR 18.155, FALL 2004
99
16. Spectral theorem
For a b ounded operator T on a Hilbert space we dene the spectrum
as the set
(16.1)
spec(T ) = cfw_z C; T z Id is not invertible.
Proposition 16.1. For any bounded linear operator on a Hilbert sp
LECTURE NOTES FOR 18.155, FALL 2004
67
11. Differential operators.
In the last third of the course we will apply what we have learned
about distributions, and a little more, to understand properties of differential operators with constant coecients. Befor
LECTURE NOTES FOR 18.155, FALL 2004
47
8. Convolution and density
We have dened an inclusion map
(8.1)
n
n
S (R ) u S (R ), u ( ) =
(x) (x) dx S (Rn ).
Rn
This allows us to think of S (Rn ) as a subspace of S (Rn ); that is we
habitually identify u with .
58
RICHARD B. MELROSE
9. Fourier inversion
It is shown above that the Fourier transform satises the identity
(9.1)
(0) = (2 )
n
( ) d S (Rn ) .
Rn
If y Rn and S (Rn ) set (x) = (x + y ). The translationinvariance of Lebesgue measure shows that
( ) = eix
130
RICHARD B. MELROSE
18. Solutions to (some of) the problems
Solution 18.1 (To Problem 10). (by Matja Konvalinka).
z
Since the topology on N, inherited from R, is discrete, a set is com
pact if and only if it is nite. If a sequence cfw_xn (i.e. a func
LECTURE 13, 18.155, 22 OCTOBER 2013
Operators on Hilbert space.
In the next three lectures I will go through the basic results on operators on Hilbert space before returning to the behaviour of particular
dierential operators. I am assuming that everyone
18.155 LECTURE 11, 10 OCTOBER, 2013
Last time I talked about hypoellipticity, today I want to talk about ellipticity for
constant coecient operators. By denition (from last week) a polynomial P ( ) of
degree m in n variables is elliptic if its leading par
18.155 LECTURE 10, 3 OCTOBER, 2013
This week I will talk about hypoellipticity and ellipticity.
First let me recall properties of convolution which we will use below. We started
by investigating the integration involved in
(1)
u v ( x) =
u(x y )v (y )dy.
CHAPTER 2
Hilbert spaces and operators
1. Hilbert space
We have shown that Lp (X, ) is a Banach space a complete
normed space. I shall next discuss the class of Hilbert spaces, a special class of Banach spaces, of which L2 (X, ) is a standard example,
in
CHAPTER 3
Distributions
1. Test functions
So far we have largely been dealing with integration. One thing we
have seen is that, by considering dual spaces, we can think of functions
as functionals. Let me briey review this idea.
Consider the unit ball in
CHAPTER 4
Elliptic Regularity
Includes some corrections noted by Tim Nguyen and corrections by,
and some suggestions from, Jacob Bernstein.
1. Constant coecient operators
A linear, constant coecient dierential operator can be thought
of as a map
(1.1)
P (
CHAPTER 5
Coordinate invariance and manifolds
For the geometric applications we wish to make later (and of course
many others) it is important to understand how the objects discussed
above behave under coordinate transformations, so that they can be
trans
CHAPTER 6
Invertibility of elliptic operators
Next we will use the local elliptic estimates obtained earlier on open
sets in Rn to analyse the global invertibility properties of elliptic operators on compact manifolds. This includes at least a brief discu
CHAPTER 7
Suspended families and the resolvent
For a compact manifold, M, the Sobolev spaces H s (M ; E ) (of sections of a vector bundle E ) are dened above by reference to local
coordinates and local trivializations of E. If M is not compact (but is
par
The material here can be found in Hrmanders Volume 1, Chapter VII but
o
he has already done almost all of distribution theory by this point(!) Joshi and
Friedlander Chapter 8.
Recall that S (Rn ) is a complete metric space.
We know that convergence with
18.155 LECTURE 3: 12 SEPTEMBER, 2013
We showed F : S (Rn ) S (Rn ) is continuous, in fact
N C N +n+1 .
To prove it is an isomorphism we start with two Lemmas the rst is very
standard
Lemma 1. There exists S (Rn ), (x) 0, (x) = 1 in |x| <
(x) = 0 in |x
18.155 LECTURE 4: 17 SEPTEMBER, 2013
Last week I went through the proof that the Fourier transform
(1)
=
F : S (Rn )
eix (x)dx S (Rn )
is an isomorphism and its basic properties and extension to a bijection on S (Rn )
and on L2 (Rn ). The latter was base
18.155 LECTURE 5, 19 SEPTEMBER, 2013
So, today I rst want to prove the Schwartz structure theorem. Let me rst
remind you of the Sobolev embedding theorem. What we notices is that
For v S (Rn ), (1 + | |2 )s/2 v L2 (Rn ) = v L1 (Rn ) if s > n/2.
Applying t