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Note that <£ and 3 take invertible values. The product ¿£5" has the form

= diag{0(A), (A - A0)fc} diag{/Km-i, bmm(X)}, A G ft', where 0 G Cr(fl, ,Mm_i(IK)) is the matrix whose entry occupying the place (i,j), for 1 < i,j < m - 1, is n n bijn

Now, suppose (uo,... ,us) G (Rm 1)s+1 is a Jordan chain of © at Ao-Then,

is a Jordan chain of diag{<5,£mm} at Ao- Thus, by (10) and Proposition 2.2, it follows from our assumption that no Jordan chain of © at Ao can be continued indefinitely.

Thanks to the induction hypothesis applied to ©, we can express © in the form (8), for some open neighborhood Q' of Ao contained in Q, some families € Cr(Q', A4m_i(K)) with 2tt(A0) and <tt(A0) invertible, and some J) G Cr(f2', A/im_i(K)) diagonal of the form (9) for some integers Pi) • • • ,Pm-1 > 0. Then, for A G ii', diag{0(A), (A - A0)fc} = diag{9H(A), 1}

Recovering the initial permutation matrices, which were not written explicitly, what we have actually proved is that there exist three families <£, 2), belonging to Cr(fi', A1m(K)) such that ¿(Ao) and §(Ao) are invertible, and

£(A) = diag{(A - A0)P1,..., (A - A0)p-}, A G V, for some integers p\,... ,pm > 0. Multiplying by appropriate permutation matrices, we can assume that pi > • ■• > pm. Thus, the family <E of the statement is the product of <£ by some permutation matrix, 3 is the product of # by another permutation matrix, whereas

2(A) = diag{(A - A0)p\..., (A - Ao)3""}, A G ft', for some integers p\ > ■ • ■ > pm > 0. Therefore, (6) is satisfied. Now, it is easy to see, from the diagonal structure of 53, that p\ > • • ■ > pn equal the partial multiplicities of 33 at Ao, and pn+i = ■ ■ ■ = pm = 0; here n = dim iV[£o]- Proposition 2.2 finishes the proof. □

Let £ G Cr(f2, £({/, V)). It is said that £ possesses a local Smith form at Ao if there exist

• families <E G C(Q',C(U,V)) and £ G C(ii', £(£/)) such that £(Ao) and 3"(Ao) are isomorphisms,

• a decomposition U = Uq®U\ with dim Uq < oo and U\ a closed subspace of U,

• integers n > 0 and k\,... ,kn > 1 such that for all for A 6 Q',

Necessarily, n = dim7V[£0] and £(A0) € $0(U,V). The family D is called the local Smith form of £ at Ao- We will see in Proposition 8.2 that it is unique, up to a rearrangement of the exponents fci,..., kn. Of course, the interesting case appears when £o is not an isomorphism; otherwise, one may take U\ = U and the families <£ = £ and 3(A) = Ijj for X efl.

Theorem 3.1 states that every smooth family of square matrices such that no Jordan chain at Ao can be continued indefinitely possesses a local Smith form. In Theorem 7.1 we will see that condition on the Jordan chains is not only sufficient for the existence of local Smith form for smooth families of square matrices, but also necessary. Necessary and sufficient conditions for the existence of local Smith form for operator-valued families of class Cr will be established in Theorem 7.2.

4. Transversal eigenvalues

This and the next sections collect the main results concerning the algebraic multiplicity introduced by Esquinas and Lopez-Gomez2 and further extended by Lopez-Gomez9. In the latter, one can find all the detailed proofs; although they are stated for the case K = M, the same proofs are valid for K = C.

As in the previous sections, r > 0 is an integer number or infinity, £ g Cr(n,£(U, V)) and £(A0) g $0(U, V) for some point A0 g CI. Recall the notation (1) for the Taylor coefficients of £ at Ao. We define the concept of transversal eigenvalue.

Definition 4.1. Take r > 1. Let £ g Cr(fi, £(£/, V)) satisfy £(A0) g $o(U,V). Suppose Ao g ii is an eigenvalue of £, and take an integer 1 < k < r. It is said that Ao is a k-transversal eigenvalue of the family £ if the following topological direct sum decomposition is satisfied:

0£j(AT[£o] n • • • n AT^-i]) © /?[£0] - V, j=l with

The integer k is called the order of transversality of £ at Ao.

Condition (11) makes the integer k uniquely determined. As it will be shown in Section 6, the structure of canonical sets of Jordan chains of a family at a transversal eigenvalue is extremely simple. Subsequently, we present a construction that shows the relationships between the dimensions of some vector spaces related to the transversality condition.

Lemma 4.1. Take r > 1. Let £ £ Cr{n,C(U,V)) be such that £(A0) € (U,V). Suppose Ao £ SI is a k-transversal eigenvalue of Z, for some integer 1 < k < r. Set i

and i-1

Pi Nfa] c p| C Ar[£0] j=o j=o and AT[£0] has finite dimension, then there exists a finite-dimensional complement Ui of iV[£o] n ■ • • (~1 A^[£j] in iV[£0] n • • • n iV[£j-i]. Thus,

N[£o] n • • • n iV[£i_i] = Ui® (JVfjCo] n ■ ■ • n Nfa]), 1 <» < k, and, hence,

N[£o] = U!®---®Uk® (iV[£0] n • ■ • n N[£k]).

Moreover, by construction,

£i{N{2,0] n • • • n N[Sii-1]) = £i(Ui), l<i<k, and for each 1 < i < k we have dimUi = dim£,({/j), since the restriction of £i from Ui to £i(Ui) is an isomorphism.

On the other hand, since Ao is a fc-transversal eigenvalue of £, we have

£i(£/i) © ■ • • © £fc(C/fe) © R[£o] = V

and, hence, k k dimJV[£0] = codimfl[£0] = ^ dim £<(C/i) = ^ dim 17,,

because £o is Fredholm of index zero. Thus,

N[Z0] = Ui © • • ■ © Uk and, in particular, iV[£o] n ■ • ■ n N[Zk] = {0} .

Therefore, nk = 0. Finally, applying the rank-nullity theorem to each of the operators i-1

£* : p| N[£j] V, 1 <i<k, j=o gives (12). Applying induction in (12) and using nk = 0 give (13). Finally lk is nonzero because of condition (11) of the definition of order of transversality. □

5. Algebraic eigenvalues and transversalization

This section explain the concept of algebraic eigenvalue and shows that families having an algebraic eigenvalue can be transversalized. The following concept is due to Lopez-Gomez9.

Definition 5.1. Consider a function £ : fi —> C(U, V) and suppose A0 G O is an eigenvalue of £. It is said that Ao is an algebraic eigenvalue of £ if £(Ao) is a Fredholm operator of index zero and there are a natural number v > 1 and two real numbers 5 > 0 and C > 0 such that for each 0 < |A — Ao| < 5, the operator £(A) is an isomorphism and

The least natural number v > 1 for which (14) holds true for some C, 5 > 0 and all 0 < |A — Ao| < <5 is called the order of Ao-

Given an integer k > 0 and a Ao g f2, it is said that Ao is a fc-algebraic eigenvalue if

• or k > 1 and Ao is an algebraic eigenvalue of order k of £.

When the family £ is (real or complex) analytic, the order of Ao as an algebraic eigenvalue of £ is nothing but the order of Ao as a pole of £_1. If £ is not analytic there is no concept of pole, but in the authors10 paper it is shown that the right counterpart of the concept of pole of £_1 is that of the order as an algebraic eigenvalue of £. Actually, such concept of pole shares many of the local properties characteristic of the analytic case, among them the existence of (finite) Laurent expansions. We will not explain that theory here, but only present a corollary that will be useful later. The following result was proved by Lopez-Gomez9 by using the theory of multiplicity explained in this section and the one introduced by Magnus11.

Lemma 5.1. Take r g NU {oo}. Suppose £ g Cr(n,C(U,V)) satisfies £(Ao) g V), and (A — Ao)fe£(A)_1 exists and is bounded for A in a perforated neighborhood of Ao, and for some integer k < r. Then, the function A i—> (A — Ao)fc£(A)~1 is of class Cr~k in a neighborhood of Ao.

The main result of this section establishes that if Ao is a ^-algebraic eigenvalue of £ for some integer 1 < u < r, then there exists a polynomial $ : K —» C(U) such that $(Ao) is an isomorphism, and Ao is a ^-transversal eigenvalue of the product family £* := £$. Moreover, the dimensions of £,f(N[£$] n • • • n AT[£f_i]) for 0 < j < r + 1 do not depend on the family of isomorphisms

Of course, a polynomial $ : K —> C(U) is a family of the form n

$(A) = ^TnA", AGK, j=o for some integer n > 0 and some operators To,... ,Tn e C(U). Also, the product of two families is defined in the natural way. Namely, if W is another Banach space, Q. and SI' are two neighborhoods of Ao, and

£: Q C(U,V), Tt-M'^£(W,U) are two maps, then the product map £9Jl: Q. fl —» £(W, V) is defined as £9Jt(A) := £(A)9Ji(A), A e fi D fi'.

The next theorem is attributable to Esquinas1 and Lopez-Gomez9.

Theorem 5.1. Take r € N U {00} and let £ e Cr(ft, C(U, V)) be such that £(Ao) is a non-invertible Fredholm operator of index zero.

(a) If Ao is an algebraic eigenvalue of £ of order 1 < v < r, then there exists a polynomial $ : K —+ C(U) such that <3?(Ao) = Iu and Ao is a v-transversal eigenvalue of £$.

• Ao is a ki-transversal eigenvalue of := £$ for some 1 < fci < r,

• Ao is a k2-transversal eigenvalue of £* := for some 1 < k2 < r.

Then k\ = k2, the point Ao is an algebraic eigenvalue of £ of order k\ and, for each 1 < j < k\, dim£*(P| iV[£f]) = dim£*(P| iV[£f]). ¿=0 i=0

Theorem 5.1 makes the following concept of multiplicity consistent (see Esquinas1 and Lopez-Gomez9).

Definition 5.2. Take r € NU{oo} and let £ € Cr(ft, C(U, V)) be such that £(Ao) is a Fredholm operator of index zero for some Ao € ft. If Ao is an algebraic eigenvalue of £ of order 1 < v < r, then the algebraic multiplicity of £ at Ao, denoted by x[£; Ao], is defined by k j —1 Ao] = ¿j ■ dim£*(n j—0 ¿=0

where $ € Cr(ft, £({/)) is such that $(Ao) is an isomorphism, and Ao is a fc-transversal eigenvalue of the family

If £ is of class C°° and Ao is a non-algebraic eigenvalue of £ we define x[£; A0] = 00.

The multiplicity of £ at Ao is said to be defined when some of the following options holds:

(b) £ is of class C in a neighborhood of Ao, for some integer r > 0, and Ao is a fc-algebraic eigenvalue of £, for some 0 < k < r.

The multiplicity of £ at Ao is said to be finite when option (b) above holds.

The multiplicity x[£; Ao] is left undefined when Ao is a non-algebraic eigenvalue of £ and £ is not of class C°° in any neighborhood of Ao- Also, it is left undefined when Ao is a ¿-algebraic eigenvalue of £, but £ is not of class Ck in any neighborhood of Ao- By the results of Section 6, x[£; Ao] is equivalent to the concept of multiplicity reviewed in Section 2.

As an example, consider the family £ g C°°(M) defined by (3). That family is real analytic in R \ {0}, but zero is not an algebraic eigenvalue of £. Moreover, £j = 0 for all j > 0, and, therefore, 0 is not a fc-transversal eigenvalue of £$ for any integer k > 0 and any $ g C°°(R), as Theorem 5.1 predicts.

6. Canonical sets at transversal eigenvalues

In this section we prove that the theories of multiplicity according to Jordan chains (reviewed in Section 2) and through transversalization (reviewed in Section 5) are equivalent.

Lemma 6.1. Let k > 0 be an integer, consider operators £o,...,£fc g C{U, V), and suppose the following spaces conform a direct sum k-1

R[Z0], £i(W[£O]), • ■ ■, £fc(f| ^i^'])- (15)

Proof. Obviously, (17) implies (16). Suppose (16). We will prove by induction on k that (17) holds. It follows from (16) that 2qu0 = 0, and, hence, (17) holds true for k = 0. Assume that, for every (yo,.yk-i) € Uk such that i

By the induction hypothesis applied to (uq,..., uk~i), we find that k-i-j

On the other hand, particularizing (16) at j= k gives k

Thanks to (18) we obtain

£o«*Gie[£o], £iU*_i G ft n • • ■ n iV[£i_i]), 1 <i<k.

Thus, since the spaces (15) are in direct sum, (19) implies £ous = 0, and £iUk-i = 0 for 1 < i < k. In other words, £k-jUj = 0 for 0 < j < k. Combining these relations with (18) gives k—j

for 0 < j < k, and, therefore, (16) is true. □

The following result completely describes the Jordan chains at transversal eigenvalues.

Proposition 6.1. Take r G N* U {oo} and consider an £ £ Cr(f2, £{U, V)) satisfying £(Ao) G $o(U,V). Suppose Ao is a k-transversal eigenvalue of £ for some integer 1 < k < r. Then, for each integer 0 < s <r, the ordered set (uo,... ,us) G Us+1 is a Jordan chain of £ at Ao if and only if uo ^ 0 and s-i

Moreover, there are no Jordan chains of length greater than k.

Proof. Take an integer 0 < s < r. It is a consequence of Lemma 6.1 that the ordered set (uo,..., us) € Us+1 is a Jordan chain of £ at Ao if and only if Uo 0 and (20). Moreover, (20) and Lemma 4.1 imply that there are no Jordan chains of length greater than k. □

The remaining results of this section establish some very deep, and a priori rather hidden, connections between the algebraic invariants introduced in Sections 5 and 2.

Proposition 6.2. Suppose r € N* U {oo}, £ £ Cr(Q, C(U, V)) satisfies £(Ao) £ {U, V), and Ao is a k-transversal eigenvalue of £ for some integer 1 <k<r. Set tj :=dim£,j(N[£0}n---nN[£,j_1]), 1 <j<k. (21)

Then, every canonical set of Jordan chains of £ at Ao is formed by exactly £j Jordan chains of length j, for each j € {1,..., k}. Moreover, the partial multiplicities of £ at Ao are given by

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