Review of Measure Theory
Let X be a nonempty set. We denote by P (X ) the set of all subsets of X .
Denition 1 (Algebras)
(a) An algebra is a nonempty collection A of subsets of X such that
i) A, B A = A B A
ii) A A = Ac = X \ A A
(b) A collection A of su
Compact Operators
In these notes we provide an introduction to compact linear operators on Banach and
Hilbert spaces. These operators behave very much like familiar nite dimensional matrices,
without necessarily having nite rank. For more thorough treatme
Review of Spectral Theory
Denition 1 Let H be a Hilbert space and A L(H).
(a) A is called selfadjoint if A = A .
(b) A is called unitary if A A = AA = 1 Equivalently, A is unitary if it is bijective (i.e.
l.
11 and onto) and preserves inner products.
(c)
The Spectrum of Periodic Schrdinger Operators
o
I The Physical Basis for Periodic Schrdinger Operators
o
Let d IN and let 1 , , d be a set of d linearly independent vectors in IRd .
Construct a crystal by xing identical particles at the points of the latt
Lattices and Periodic Functions
Denition L.1 Let f (x) be a function on IRd .
a) The vector IRd is said to be a period for f if
for all x IRd
f (x + ) = f (x)
b) Set
Pf =
IRd
is a period for f
If , Pf then + Pf and if Pf then Pf (sub x = z into f (x + )
An Elliptic Function The Weierstrass Function
Denition W.1 An elliptic function f (z ) is a non constant meromorphic function on C
that is doubly periodic. That is, there are two nonzero complex numbers 1 , 2 whose ratio
is not real, such that f (z + 1 )
Completion
Theorem 1 (Completion) If V , , V is any inner product space, then there exists a Hilbert
space H, , , H and a map U : V H such that
(i) U is 11
(ii) U is linear
(iii) U x, U y H = x, y V for all x, y V
(iv) U (V ) = U x x V is dense in H. If V
MATHEMATICS 511 Section 101
Operator Theory and Applications
PREREQUISITES:
A course in measure theory at the level of UBCs Math 420/Math 507.
It would be desirable to have also taken a course on Hilbert and/or Banach spaces like UBCs Math
421/Math 510,
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Review of Measurable Functions
Denition 1 (Measurable Functions) Let X and Y be nonempty sets and M and N
be algebras of subsets of X and Y respectively.
(a) A function f : X Y is said to be (M, N )measurable if
E N = f 1 (E )
xX
f (x) E
M
(b) A function
Review of Integration
Denition 1 (Integral) Let (X, M, ) be a measure space and E M.
(a) L+ (X, M) =
f is Mmeasurable . If f L+ (X, M), then
f : X [0, ]
n
ai (Ei E )
f (x) d(x) = sup
n IN, 0 ai < , Ei M for all 1 i n
i=1
E
n
ai Ei (x) f (x) for all x X
an
Review of Signed Measures and the RadonNikodym Theorem
Let X be a nonempty set and M P (X ) a algebra.
Denition 1 (Signed Measures)
(a) A signed measure on (X, M) is a function : M [, ] such that
(i) () = 0
(ii) assumes at most one of the values .
(iii) I
The Principle of Uniform Boundedness, and Friends
In these notes, unless otherwise stated, X and Y are Banach spaces and T : X Y
is linear and has domain X .
Theorem 1
(a) T is bounded if and only if
T 1 y Y
y
Y
1
=
xX
Tx
Y
1
has nonempty interior. (X , Y