Lectures
16
and
17
16.3
Fredholm
Operators
A nice way to think about compact operators is to show that set of compact op
erators is the closure of the set of finite rank operator in operator norm. In this
sense compact operator are similar to the finite dimensional case. One property of
finite rank operators that does not generalize to this setting is theorem from linear
algebra that if
T
:
X
→
Y
is a linear transformation of finite dimensional vector
spaces then
dim
(
ker
(
T
))
−
dim
(
Coker
(
T
))
=
dim
(
X
)
−
dim
(
Y
).
35
Of course if
X
or
Y
is infinite dimensional then the right hand side of equal
ity does not make sense however the stability property that the equality implies
could be generalized. This brings us to the study of Fredholm operators. It turns
out that many of the operators arising naturally in geometry, the Laplacian, the
Dirac operator etc give rise to Fredholm operators. The following is mainly from
H ¨ormander
Definition
16.13.
Let
X
and
Y
be Banach spaces and let
T
:
X
→
Y
be a bounded
linear operator.
T
is said to be
Fredholm
if the following hold.
1. ker
(
T
)
is finite dimensional.
2. Ran
(
T
)
is closed.
3. Coker
(
T
)
is finite dimensional.
If
T
is Fredholm define the
index
of
T
denoted Ind
(
T
)
to be the number dim
(
ker
(
T
))
−
dim
(
Coker
(
T
))
First let us show that the closed range condition is redundant.
Lemma
16.14.
Let
T
:
X
→
Y
be
a
operator
so
that
the
range
admits
a
closed
complementary
subspace.
Then
the
range
of
T
is
closed.
Proof:
C
be a closed complement for the range. We can assume that
T
is
injective since ker
(
T
)
is a closed subspace and hence
X
/
ker
(
T
)
is a Banach space
so we can replace
T
by the induced map from this quotient. Now consider the map
S
:
X
⊕
C
→
Y
defined by
S
(
x
,
c
)
=
T
(
x
)
+
c
.
S
is bounded linear isomorphism and hence by the open mapping theorem
S
is a
topological isomorphism. Thus Ran
(
T
)
=
S
(
X
⊕ {
0
}
)
is closed.
.
An important result that will be used over and over again is the openness of
invertibility in the operator norm.
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 Spring '10
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 Linear Algebra, Vector Space, Hilbert space, Fredholm, fredholm operators

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