## [uoi wfci11 uoiuiy u0n ufcin

is said to be a canonical set of Jordan chains of £ at Ao- By construction, kn > 1, and the following result is obtained (see, for example,

Rabier12).

Proposition 2.1. Take r € N U {oo} and £ e Cr(n,C(U,V)) such that £(Ao) is a non-invertible Fredholm operator of index zero. Assume that the length of all Jordan chains of £ at Ao is bounded from above by some integer less than or equal to r. Then, the integers ki,..., kn of the construction of a canonical set are the same for every canonical set of Jordan chains.

Proof. We will follow the construction and the notation of the canonical set of Jordan chains. Clearly, fci is the same for every canonical set of Jordan chains. Set

Then := U {0} is the linear subspace generated by K\. Call I dim ia and let {mo,i,... ,uo,n} be a basis of AT[£0] selected of a canonical set of Jordan chains. Then

This fact proves that k\ = • • • = k( are independent of the canonical set.

If £ = n, the proof is done. If t < n then kf+i is clearly independent of the canonical set. We repeat the reasoning with

Then E2 := K2 U {0} is the subspace generated by K2, one has E2 D Ei and

So fc^+i = • • ■ = fcdim e-2 are independent of the canonical set. Repeating this procedure as many times as necessary concludes the proof. □

Proposition 2.1 proves the consistency of the following concept.

Definition 2.3. Take r G Nu{oo}. Let £ € Cr(fi, C(U, V)) satisfy £(A0) G \$o {U, V). Suppose that the length of all Jordan chains of £ at Ao is bounded from above by some integer less than or equal to r. Following the above notation, the integers fci,..., kn are called the partial multiplicities of £ at Aq, and

is known as the multiplicity of £ at Ao.

If r = oo and there is some eigenvector of infinite rank, we define the multiplicity of £ at Ao as M[£; Ao] := oo.

We leave the partial multiplicities and the multiplicity M of £ at Ao undefined if £ is of class Cr but it is not of class Cr+1 in any neighborhood of Ao and there are Jordan chains of length r + 1 of £ at Ao- We also leave the partial multiplicities undefined when r = oo and there is some eigenvector of infinite rank.

We now introduce some notation. Let n > 1 be an integer and consider for each 1 < j < n a vector space Xj and a vector Xj £ Xj. We define col[a;i,...,x„

Now take n,k > 1 two natural numbers, and for each integer 1 < j < n, consider two vector spaces Uj and Vj, and an operator Bj g C(Uj,Vj). We define trng{B1,B2,...,Bn} :=

IBi B2