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.