2.1. Fredholm operators
LECTURE 2
Fredholm Theory
Let X and Y be Banach spaces. A bounded linear operator D : X ----t Y is called
a Fredholm operator if it has a closed range and the kernel and cokernel of D
are both finite dimensional. Throughout we think of the cokernel as the quotient
coker D = Y / im D. The index of a Fredholm operator D is defined as the difference
of the dimensions of kernel and cokernel:
index D = dim ker D - dim coker D.The Fredholm property and the index are stable under perturbations. In particular,
the set of Fredholm operators is open with respect to the norm topology, and the
index is constant on each component. Moreover, if Dis Fredholm and K: X ----t Y
is a compact linear operator, then D + K is again a Fredholm operator and it has
the same index as D.
Exercise 2.1. Let X , Y, Z be Banach spaces, D : X ----t Y be a bounded linear
operator, and K: X ----t Z be a compact linear operator. Suppose that there exists
a constant c > 0 such that the following inequality holds for all x E X
(12) llxllx :::; c (llDxJJy + JJKxJJz) ·
Prove that D has a closed range and a finite dimensional kernel. Use this to prove
that the Fredholm property of D is invariant under small perturbations. D
A smooth (C^00 ) map l : X ----t Y is called a Fredholm map if its differential
dl(x ) : X ----t Y is a (linear) Fredholm operator for every x E X. In this case it
follows from the stability of the Fredholm index that the index of dl(x) is indepen-
dent of x <J,nd we write index(!)= indexdl(x). A vector y E Y is called a regular
value of l if dl(x) : X ----t Y is onto for every x E l-^1 (y). If y is a regular value
then, as in the finite dimensional case, the implicit function theorem asserts that
M = l-^1 (y)
is a smooth finite dimensional manifold. Its tangent space at x E M is given by
TxM = ker dl(x)
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