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Therefore, it consists of exactly Jordan chains of length k, ..., and t\ Jordan chains of length 1. Proposition 2.1 concludes the proof. □

As a consequence of Propositions 3.1 and 6.2, the following result on the existence of local Smith form is obtained.

Corollary 6.1. Let m >1 be an integer and take r g {oo,w}. Consider £ g Cr(ft, ,Mm(K)). Suppose Ao is a k-transversal eigenvalue of £, for some integer k > 1. Set (21). Then, there exist an open neighborhood ft' of Ao and families

€ Cr(ft',M„(K)), 3) g C"(ft',A4m(]K)), such that <£(Ao) and 5(Ao) are invertible,

!D(A) = diag { (A - A0)fc,. ., (A - X0)k,..., A - A0,. ., A - A0,1,. , 1 }, ek m-n

where n = dimJV[£o]. Consequently, the partial multiplicities of £ at Ao ore given by (22), and det£ has a zero of order k

The next result relates the invariants of the constructions of the multiplicity based on a transversalization and based on Jordan chains. In particular, it proves that both multiplicities coincide.

Proposition 6.3. Take r g n* u {oo} and consider an £ g Cr(Q, £(U, 7)) satisfying £(Ao) g $o (U,V). Suppose Ao is a k-algebraic eigenvalue of 2, for some integer 1 < k < r, and let $ g Cr(fi, C(U)) satisfy that $(Ao) = Iu and that Ao is a k-transversal eigenvalue of £* := £$. Then, the lengths of the Jordan chains of £* at Ao in every canonical set equal those of £ at Ao- When we set tr.= dim£*{N[£$]n---nN[£*_1]), 1 < j < k, (24)

every canonical set of Jordan chains of £ at Ao possesses exactly £j chains of length j, for 1 < j < k.

If, in addition, U and V have finite dimension m and r G {oo,w}, then there exist an open neighborhood ft' of Ao and families

<SeCr(Q',£(U,V)), D g c1"(ft', £(£/)), ff g cr(ft', £([/)), such that £(Ao) and S'(Ao) are isomorphisms,

£= «Off and, for all A € SI', equality (23) holds forn = dimiV[£o] and for a certain basis in U.

Proof. Thanks to Proposition 6.2, every canonical set of Jordan chains of £* at Ao consists of exactly £j Jordan chains of length j for each 1 < j < k. Thanks to Propositions 2.1 and 2.2, every canonical set of Jordan chains of £ at Ao consists of lj Jordan chains of length j for each 1 < j < k. This completes the proof of the first part of the statement.

Now, suppose dim U = dim V = m < oo and r e {oo,w}. Then, thanks to Corollary 6.1, there exist an open neighborhood CI' of A0 and families

<*£Cr(Sl',C{U,V)), S £C"(Sl\ £([/)), 3&Cr(n',£(U)), such that <E(Ao) and i?(Ao) are isomorphisms, £$ = CDS' and equality (23) holds for all A £ SI', where n = dim7V[£0]. The fact that $(A0) is an isomorphism concludes the proof. □

In the next result we explain another relevant facet of the equivalence between the constructions of the multiplicity of Sections 2 and 5.

Proposition 6.4. Take r e N* U {oo} and consider an £ e Cr(Sl, £(U, V)) satisfying £(Ao) £ $o(U, V). Let Ao be a k-algebraic eigenvalue of £ for some integer 1 < k < r. Let $ € Cr(Sl,C(U)) be such that $(A0) is an isomorphism and Ao is a k-transversal eigenvalue of £$ := £$. Denote (24). Then, for alll<i<k, i-1 k dim AT[trng{£0,... ,£¿-1}] = ^jij j=1 j=i

7V[trng{£*,..., S*^}} = ( f] N[£*]) x • • • x N[£*].

Call rij := dim N[£$} fl • • • n N[£f], 0 <j<k.

Then i-1

By Lemma 4.1, tij = ^ for all 0 < j < k, and, hence, i—1 k i—1 k dimiV[tmg{£0$,...,£f_1}] = ^ £ ^ = +

Note finally that for all 0 < s < k, trng{£o > = trng{£0,. •., £s} trng{$0, and, since trng{<£>o,..., $s} is an isomorphism, dimjV[trng{£*,.. .,£?}] = dim jV[trng{£0,.. .,£,}]. This concludes the proof. □

7. Characterization of the existence of the local Smith form

In this section we use the two theories of algebraic multiplicity that we have analyzed so far to characterize the existence of the local Smith form of a family at a point in terms of the invariants of the multiplicity.

Proposition 7.1. Take r 6 N U {oo,u>}, an open set ii c K and Ao g f2. Let £, 971 € Cr(£l,C(U,V)). Suppose that Ao is a k-algebraic eigenvalue of £ for some integer 0 < k < r, and that 9Jtn = £n for all 0 < n < k. Then, there exists an open subset f2' of fl containing Ao and a family $ g Cr~k(W,£(V)) such that

Moreover, Ao is a k-algebraic eigenvalue offJJl.

Proof. Let D be an open neighborhood of Ao contained in il such that £(A) is an isomorphism for all A g D \ {Ao}. Now, consider the family ©:£>-> £(V) defined by

©(A0) := lim ©(A) = lim [£(A) + o((A - A0)fe)l £(A)_1 = Iv.

Note that

6(A) =IV + [971(A) - £(A)] £(A)-1, A € D \ {A0}.

The function A«(A- Ao)_fc[0K(A) - £(A)] is of class Cr~k in £1, and so is the function A i—> (A — Ao)fc£(A)_1 in D, by Lemma 5.1. Therefore, <8 is of class CT~k and takes invertible values for A in some neighborhood Cl' of Ao-Define, for A € Cl',

The main finite-dimensional result of this chapter reads as follows. Note that conditions CI and D1 below give us the local Smith form.

Theorem 7.1. Let m > 1 be an integer, take r £ N U {oo,w}, a point Ao £ K, a neighborhood Cl of Ao, and a family £ £ Cr(Cl, £(Km)).

If r £ {oo, u>}, then the following conditions are equivalent:

Cl. There exist an open set CI' containing Ao and contained in CI, two families £,3 £ Cr(Cl', £(Km)) with <E(A0) and 3(A0) invertible, and a factorization (6) holds with (7) for some integers ki,...,kn > 1 where n = dim AT[£o]. C2. The order of det £ at Ao is finite. CS. Ao is an algebraic eigenvalue of £.

C4- The lengths of all Jordan chains of £ at Ao are uniformly bounded from above.

C5. No Jordan chain of £ at Ao can be continued indefinitely. C6. The multiplicity x[£; Ao] is finite.

If r > 0, then the following conditions are equivalent:

Dl. There exist an open set Cl' containing Ao and contained in Cl, two families (S,3 £ C(fi',£(Km)) with <E(Ao) and 3(Ao) isomorphisms, and equalities (6) and (7) hold for some integers 1 < ki,...,kn < r with n = dim AT[£q]. D2. Ao is an algebraic eigenvalue of £ of order k <r. D3. The lengths of all Jordan chains of £ at Ao are bounded above by k <r. D4. The multiplicity x[£; Ao] exists and is finite.

Moreover, in this case, the families <£ and 3 of Dl can be chosen so that £ is of class Cr~k and 3 is analytic.

Proof. First we assume that r £ {oo, u>} and prove the equivalence C1-C6. CI implies C2: Equations (6) and (7) imply det £(A) = (A - \o)kl+'"+kn det C(A) det ff(A), A eO', with det<E(A0)detff(A0) ¿0.

C2 implies C3: Suppose det £ has a zero of order s € N at Ao- Then, there exists a e Cr(Sl) such that a(A0) ^ 0 and det£(A) = (A - Ao)sa(A) for X G SI. Now we apply an elementary result by Kato8 which asserts that there exists a constant 7 > 0 such that for each invertible operator A e £(Km) the following estimation holds:

Thanks to this inequality we conclude that ||£(A)-11| < C|A — Ao |s for some C > 0 and for all A in a perforated neighborhood of Ao- Therefore, Ao is an algebraic eigenvalue of £.

C3 implies C4: Suppose Ao is a fc-algebraic eigenvalue of £. Thanks to Proposition 6.3, the length of every Jordan chain of £ at Ao is bounded from above by k.

C4 implies C5 is obvious, C5 implies CI is Proposition 3.1, and C3 is equivalent to C6 is Definition 5.2. This completes the proof of the theorem when r e {00, a>}.

The statement of the theorem is obvious if r = 0. Now suppose that r > 1 is an integer.

D1 implies D2: If equations (6) and (7) hold for some neighborhood SI' of Ao, and some continuous families <£ and 3" defined in SI' with values in the set of invertible mx m matrices, then

S(A)-1 = diag {(A - Ao)"*1, • ■ ■, (A - A0)-fc",l,..., l}, A G Si'\ {A0}, and it is easily seen that u^Am^qA-Aor*

for all A in a perforated neighborhood of Ao, for some constant C > 0, and k := max{fci,..., kn}.

D2 implies Dl: Let us define the family £

By Proposition 7.1, Ao is a fc-algebraic eigenvalue of the analytic function £, there exists a function $ of class CT~k in a neighborhood of Ao with values in C(V) such that ££ = £ and £(A0) = Iv- By Proposition 6.3, there exist two analytic families <£,5 defined on a neighborhood of A0 whose values are m x m matrices such that <£(Ao) and 3"(Ao) are isomorphisms, £(A) = ¿(A)3)(A)ff(A) for A ~ A0, and (7). Then, £ = S^&Dff.

D3 implies D2: We define the family £ through (25). Then, £ is analytic and k is an upper bound for the length of its Jordan chains at Ao- Let k\<k be the optimal bound. By the equivalence of C1-C6 of the first part of the theorem and by Proposition 6.3, Ao is a fci-algebraic eigenvalue of £. By Proposition 7.1, Ao is a fci-algebraic eigenvalue of £.

D2 implies D3: We define the family £ through (25). By Proposition 7.1, Ao is a fc-algebraic eigenvalue of £, and, by Proposition 6.3, the length of the Jordan chains of £ at Ao are bounded from above by k. Then, by Propositions 2.3 and 7.1, £ and £ have the same Jordan chains at A0. Therefore, the lengths of the Jordan chains of £ at Ao are bounded from above by k.

D2 is equivalent to DJ: This is Definition 5.2. □

The following result allows us to reduce an infinite-dimensional problem with the Fredholm condition to a finite-dimensional setting; it is taken from Gohberg et al.5 (see also Gohberg and Sigal7).

Lemma 7.1. Suppose r € N U {oo,w} and £ e Cr(i2, C(U, V)) satisfies £(Ao) € $o(£/, V). Then, there exist an open neighborhood i!' of Ao contained in a decomposition U = Uq © t/i with dim Uq < oo and Ui a closed subspace of U, and three families

€€Cr(il',C(U,V)), $eCr{il',£(U)) and m e Cr(Sl', C(U0)) such that <£(A0) and 5(Ao) are isomorphisms and

Proof. Since £(Ao) is Fredholm of index zero, there exists a finite-rank operator F £ £({/, V) for which the family <E defined by

<£(A) := £(A) + F, A G fi, takes invertible values for A in a neighborhood of Ao- As F has finite rank, N[F] has a topological finite-dimensional complement Uq in U, so

Let P £ £(U) be the projection onto Uo with kernel N\F\. Then P has finite rank and FP = F, because Iu — P is a projection onto N[F], Thus,

Iu - ¿(X^F = [Iu - P<£(A)-1FP] [Iu - (Iu - P) <B(X^FP]

for A ~ Ao- Set, for each A sufficiently close to Ao,

This family # is of class C and invertible in a neighborhood of Ao- In fact,

Thus,

£(A) = e(A)[/c/ - <£(A)-1.F] = €(A) [lv - P€(X)~1FP] 3(A), A ~ A0. Define

for A ~ A0. With respect to the decomposition U = i?[P] © N[P], the family 0 can be expressed in the form

0(A) = diag{0(A)\u0,In[f]}, A ~ A0, since 0(A)([/o) C Uo and

®W|jV[F] = -f|iV[F] , A ~ A0 . This concludes the proof. □

Lemma 7.1 allows us to generalize Theorem 7.1 to the infinite-dimensional case. The following characterization is thus obtained. Again, CI and D1 below give us the local Smith form.

Theorem 7.2. Take r G NU{oo,o;}, Ao G K, a neighborhood Q. of X0, and a family £ € Cr(9., C(U, V)) with £(A0) G (U, V).

If r £ {oo,a;}, then the following conditions are equivalent:

CI. There exist an open neighborhood fi' of Ao contained in fI, two families <B G Cr(W,£(U,V)) and 3 G Cr(W,£(U)) such that <£(A0) and 3(A0) are isomorphisms, and a decomposition U = Uo®U\ with dim Uo < oo and U\ a closed subspace of U such that

2>(A) = diag{(A-Ao)'c\...,(A-Ao)fe"}, A G ft', (27)

/or some integers k\,...,kn > 1, and n = dim7V[£0]. C2. For every neighborhood ft' of Ao, any Banach spaces U and V, any topological decompositions U = U\ © U2 and V = V\ © V2 such that dim Ui = dimVi < 00, any families

<£ G C°°(ft', £(V, V)), ff G C°°(ft', £(U, U))

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