1.1. DEFINITIONS AND PRELIMINARIES 9
Next, we define the index of A ∈ Fred(B, B ) to be
ind A = dim Ker A − codim Im A.
In finite dimensions, the index depends only on the spaces and not on the operator.
The following proposition shows that the index is stable under compact per-
turbations [164].
Proposition 1.4. If A : B → B is Fredholm and K : B → B is compact, then
their sum A + K is Fredholm, and
ind (A + K) = ind A.
Proposition 1.4 is a consequence of the following fundamental result about the
index of Fredholm operators.
Proposition 1.5. The mapping A → ind A is continuous in Fred(B, B ); i.e.,
ind is constant on each connected component of Fred(B, B ).
1.1.3. Characteristic Value and Multiplicity. We now introduce the no-
tions of characteristic values and root functions of analytic operator-valued func-
tions, with which the readers might not be familiar. We refer, for instance, to the
book by Markus [175] for the details.
Let U(z0) be the set of all operator-valued functions with values in L(B,B )
which are holomorphic in some neighborhood of z0, except possibly at z0.
The point z0 is called a characteristic value of A(z) ∈ U(z0) if there exists a
vector-valued function φ(z) with values in B such that
(i) φ(z) is holomorphic at z0 and φ(z0) = 0,
(ii) A(z)φ(z) is holomorphic at z0 and vanishes at this point.
Here, φ(z) is called a root function of A(z) associated with the characteristic value
z0. The vector φ0 = φ(z0) is called an eigenvector. The closure of the linear set of
eigenvectors corresponding to z0 is denoted by KerA(z0).
Suppose that z0 is a characteristic value of the function A(z) and φ(z) is an
associated root function. Then there exists a number m(φ) ≥ 1 and a vector-valued
function ψ(z) with values in B , holomorphic at z0, such that
A(z)φ(z) = (z −
z0)m(φ)ψ(z),
ψ(z0) = 0.
The number m(φ) is called the multiplicity of the root function φ(z).
For φ0 ∈ KerA(z0), we define the rank of φ0, denoted by rank(φ0), to be the
maximum of the multiplicities of all root functions φ(z) with φ(z0) = φ0.
Suppose that n = dim KerA(z0) +∞ and that the ranks of all vectors in
KerA(z0) are finite. A system of eigenvectors φ0,
j
j = 1, . . . , n, is called a canonical
system of eigenvectors of A(z) associated to z0 if their ranks possess the following
property: for j = 1, . . . , n, rank(φ0)
j
is the maximum of the ranks of all eigenvectors
in the direct complement in KerA(z0) of the linear span of the vectors φ0,
1
. . . ,
φ0−1.j
We call
N(A(z0)) :=
n
j=1
rank(φ0)j
the null multiplicity of the characteristic value z0 of A(z). If z0 is not a characteristic
value of A(z), we put N(A(z0)) = 0.
