On spectral properties of regular quasidefinite pencils F-λG

Results in Mathematics Vol. 19 (1991)

0378-6218/91/020089-21$1.50+0.20/0 (c) 1991 Birkhauser Verlag, Basel

o. Introduction

On spectral properties of regular

quasidefinite pencils F-AG

H. Langer and A. Schneider

Herrn Prof. Dr. Dr. h. c. Helmut Wielandt

zum 80. Geburtstag gewidmet

In the papers [SSl], [SS2] the eigenvalue problem

(0.1) Fy = AGy

with linear mappings F and G was studied under the assumption, that it is

"S-hermitian" . This means, that two forms F and q; associated with F and G,

respectively, are hermitian. Additionally it was supposed, that F or q; is

positive definite; in this case the problem (0.1) was called leftdefinite

or rightdefinite, respectively. This definiteness assumption allowed to

apply results of the spectral theory of (compact) hermitian operators in

Hilbert spaces, which led e.g. to completeness statements and expansions in

eigenfunctions. It is the aim of this note to generalize some of these

results to the case where both forms F and q; are indefinite, but at least

one of them has a finite number of negative squares . If F or q; has this

property, the problem (0.1) is called quasi-leftdefinite or quasi­

rightdefinite, respectively.

In the applications we have in mind F and G are differential boundary

expressions. In particular eigenvalue problems for ordinary formally self­

adjoint differential expressions with boundary conditions, depending on the

eigenvalue parameter A, are included. Quasi-rightdefinite problems of this

kind were studied e.g. in [0], [DLS]. Quasi-leftdefinite second order boundary

90 Langer and Schneider

eigenvalue problems with A-independent boundary conditions were considered

in [AEO], lOLl. In [IS] the eigenvalue problem for a regular Sturm­

Liouville operator with general A-linear boundary conditions was formulated

(see (3.1a-c) below), however, it was studied only in a special case

(~2 = 0, -~1 = ~1 = ~2 = 1 in (3.1c)), which led to a leftdefinite problem.

A well-known theorem of H. Wielandt [Wi] characterizes the compactness

of a hermitian operator in a not necessarily complete linear space with a

definite sesquilinear form by algebraic properties. In Section 1 of this note

we generalize this theorem to the situation where the hermitian sesquilinear

form has a finite number of negative squares. The main tool in Section 1

is Lemma (1.1); it is proved by extending a reasoning of F. W. Schafke [5]

to definitizable operators in Krein spaces. In Corollary (1.2) it is shown

that this lemma implies also a result of M. G. Krein about operators in

spaces with two norms.

In Section 2 the basic concepts of the theory of S-hermitian pencils

F-AG are introduced, following closely the lines of [551]. The generalization

to the quasi-leftdefinite or quasi-rightdefinite situation (instead of the

left- or rightdefiniteness in [551]) leads to some new features as now the

spectrum need not be real and Jordan chains of length> 1 can appear.

Nevertheless, the completenes of the root subspaces and expansion theorems

hold, if only A = 0 is not a singular critical point (or not an eigenvalue)

of the compact operator A associated with the problem (0.1).

In Section 3 the results of Section 2 are applied to the above

mentioned problem (3.1a-c) of [IS] in the general situation. It turns out,

that without restrictions to the signs of ~1' ~1' ~2' ~2 this problem is

quasi-leftdefinite, hence in a natural way it can be treated in a fix-space

setting. The essential step in this application is the description of the

completion of the linear space ~ (introduced for a general S-hermitian

pencil in Section 2) with respect to the quasi-definite form F. This

completion is a Hilbert or fi -space of functions, augmented by some copies x of (. It is therefore of the same structure as the basic spaces in similar

considerations in [0], [F], tWa], [018].

We mention that also singular quasi-leftdefinite boundary eigenvalue

problems of the form (0.1) can be treated in the same way. Then, however, the

expansion theorems become more complicated because of the possible presence

of singular critical points (com. tOLl). This will be done elsewhere.

Langer and Schneider

1. A generalization of a theorem of H. Wielandt

Recall that a Krein space is a linear space X, equipped with a nonde­

generate hermitian sesquilinear form [.,.], which admits a decomposition

X=X eX, + -

where (X±,±[.,.]) are Hilbert spaces and e means that the spaces X+ and X_

are orthogonal with respect to [.,.]: [X+,X_] = {O}. For the definition

of selfadjoint and definitizable operators in a Krein space X we refer to

[AI], [B], [L].

Lemma (1.1). Let A be a definitizable selfadjoint operator in the Krein

space X. If for some nonempty open interval II of the real axis there is a

dense subset:ft c X which is contained in (A-A):lI(A) for each A E ll, o then II c p(A).

Proof. Let f E :fto and A E ll. Then we have f = (A-A)hA with some hA E :lI(A),

or (A-A)-l f = hA. Let lloC II be an open subinterval, which is of definite

91

e.g. positive type. We consider a possibly smaller interval ll'c llo' such that

its boundary points are not critical points of A. Then for A E ll' we have

hence

J (t_A)-2 d[Etf,f] = [E(ll')~,hA] < 00.

ll'

This implies for ~ E ll', ~ # A:

or

hence

Therefore the function A ~ [EAf,f] is constant on ll' for each f of the

dense subset :fto ' which implies that EA is constant on ll'. As ll' comes

arbitrarily close to II , E~ is constant on II , which means that II c PIA). o ~ 0 0

The given interval II is, possibly up to a finite number of points,

the union of a finite number of intervals of definite type. Thus II n alA)

92 Langer and Schneider

consists only of at most finitely many points. As an isolated spectral

point A1 E 6 of a definitizable operator is an eigenvalue and hence has the

property that (A-A1)~(A) is not dence in X, the assumption that (A-A1)~(A)

is dense in X for all A E 6 implies that there is no such isolated spectral

point in 6. The lemma is proved.

Corollary (1.2). Let A be a selfadjoint operator in a Rx-space Ox. If for

some nonempty open interval 6 of the real axis there is a dense subset

~ c ° such that (A-A)~(A) J ~ for all A E 6, then 6 c p(A). o x 0

Indeed, a selfadjoint operator A in Ox is definitizable.

The following corollary is not used in the sequel. It shows, that a

result of H.G.Krein (which was independently later also proved by Reid,

Lax and Dieudonne) is an easy consequence of Lemma (1.1).

Corollary (1.3). Let A be a linear operator in the linear space (X, p) ,

where p is a semi norm on X, and suppose that

p(Ax) ~ Cp(x) (x E X)

with some C > o. Further, suppose that on X there is given a semidefinite

inner product (.,.) with the property l(x,y)1 ~ C1P(X)p(y) (x,y E X) with

some C1 > 0, and that (Ax,y) = (x,Ay) (x,y E X). Then

(1.4) 2 (Ax,Ax) ~ C (x,x) (x E X).

Proof. We can suppose that p is even a norm on X; let X be the (Banach­

space) completion of X with respect to this norm. The closure A of A in X is

a bounded linear operator in X such that

p(Ax) ~ Cp(x) (x E X).

It follows that for A with IAI > C the operator A - A maps X onto itself.

Also the inner product (.,.) can be extended by continuity to all of X, and

we consider the Hilbert space ~ generated by completion of (X,(.,.)). Then A generates a selfadjoint operator A in ~ with the property that all the real

points 1, IAI > C belong to pIA), as A-A for IAI > C contains the dense

subset X. As is well known, this implies (1.4).

Langer and Schneider

A linear space X, equipped with a hermitian nondegenerate sesquilinear

form [.,.] is said to be a nx-lineal, x an integer, if X contains a subspace

X_ of dimension x such that [x,x] < 0 if x E X_'x ~ 0, but no subspace of

dimension x+1 with this property. The linear subset ~ c X is said to be

dense in X, if for each

that if n ~ 00

X E X there exists a sequence (x ) c ~ such n

[x-xn,x-xn ] ~ 0 and [xn'y] ~ [x,y] for all y E X.

Each nx-lineal can be completed (in a unique way up to isomorphisms) to a

nx-space Ox. If A is a hermitian operator in X, mapping the dense set

~ c X into X (hermitian means that [Ax,y] = [x,Ay] for all x,y E ~), it

can be considered as a hermitian operator in Ox which allows a (hermitian)

closure A. The point A E ( is said to be an eigenvalue of the operator A

in X if A-A is not injective; the root subspace of A at A is the linear

span of all the kernels ker (A_A)V, v = 1,2,3, ... , the eigenspace of A at A

is ker (A-A).

Theorem (1.5). Let (:It,[.,.]) be a nx-lineal, and let A be a linear operator

from ~ c :It into :It such that

a) ~ is dense in :It,

b) A is hermitian,

c) the set of eigenvalues of A is bounded and 0 is its only possible

accumulation point; each eigenspace is of finite dimension.

d) If a E Q;\{O} then

dim ker (A-a) codim (A-a)~.

Then the closed hermitian operator A induced byA in the completion Ox of

:It is compact. If, moreover, the root subspace of A at an eigenvalue AO is

nondegenerate, then this root subspace coincides with the root subspace of

A at A • o

Proof. Let ° and A be defined as above. The condition d) implies that the x nonreal points, which are not eigenvalues of A, belong to pIA). Indeed, in

such a point A~i we have (A-A)~(A) J (A-A)~ = ~. As ~ is dense in Ox

and for the closed hermitian operator A the range (A-A)~(A) is closed,

we have (A-h)~(A) = nx' or A E pIA).

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If the real point AtO is not an eigenvalue of A, by c) this is true for

all the points of an interval 6 around A, and for all A'E 6 we have by d)

94 Langer and Schneider

hence, according to Lemma (1.1), 6 c ptA). Therefore the spectrum of A consists outside 0 of the eigenvalues of A in [\{O}.

Next we show, that each eigenvalue ~ ~O of A is of finite geometric o multiplicity. Otherwise, with F := (A-~ )~(A), the set Fi would be of o infinite dimension. With the isotropic part F' of F and a subspace F",

skewly linked with F', we have the decomposition

Ox = ,. Ell (F'-+- F") Ell K;

" i here F = F $ F', F = F' Ell K where dim K = 00. Now write

!lit = (A-~ )~ -+- 8 o

where 8 is some subspace of !lit with dim 8 = dim ker (A-~o) < 00 (see c) and d)).

If PK denotes the orthogonal projection onto K, then PK8 is of finite

dimension, therefore there exists a nonzero element Xo E K such that

[PK~'xo] = {OJ. Then

[xo,!lIt] = [x , (A-~ )~ -+- 8] o 0

(1. 6) [x , (A-~ )~] + [x ,8] = o 0 0

{O},

where we have used

= [Pxx o ,8] = [xo ,Px8] =

.1 that (A-~o)~ c F and KeF . As !lit is dense in ° , the

x .1 relation (1.6) yields Xo = 0, a contradiction. Thus K and hence also F is

of finite dimension, which implies that the geometric multiplicity of the

eigenvalue AO of A is finite. As A is selfadjoint in a rrx-space, also the

algebraic multiplicity of AO is finite. Thus, alA) is discrete outside 0,

hence A is compact.

In order to prove the last claim we consider the root subspace X~ of A at

~o and the decomposition o

Both subspaces on the right hand side are invariant under A, and

(A-~ )~ = (A-~ )X~ Ell (A-A )~ o 0 ~ 0 0 o

J. with ~o = ~ n X~ The space on the left hand side is of codimension

o P(AO) := dim ker(A-Ao)~ in !lit according to d), the first term on the right hand

side is also a space of codimension P(AO) in XA (as dim XA < 00 and for an o 0

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operator in a finite dimensional space the codimension of the range

the dimension of the kernel). Therefore the codimension of (A-A )~ o 0 is zero, or

(1. 7) .l

(A-A )~ =!f o 0 A

The completion Ox of ~ can be written as

(1.8) Ox = !fA Ell !f' , o

.l

o

equals .l

in !fA o

where !f' denotes the completion of !fA . Further, (1.7) implies that o

- - ~ (A1-Ao)~(A1) ~ !fA

o

where A1 denotes the restriction of A to D(A) n !f'. Therefore the range

of A1-AO is dense in !f'. As a deleted neighbourhood of AO belongs to

ptA) and hence also to P(A1 ), this range

the decomposition (1.8) implies that !fA

Ox' and the theorem is proved. o

must even coincide with !f'. Now

is also the root subspace of A in

If B is a compact selfadjoint operator in Ox its nonzero spectrum

consists of an at most finite number s of pairs of mutually different

nonreal eigenvalues A1, ... ,As and X1, ... XS' and of a (possibly finite)

sequence of (mutually different) real eigenvalues (A.)._ 1 ; only a ) )-s+ , .. finite number of these real eigenvalues can have an eigenvector x of

nonpositive type (that is [x,x] ~ 0). Denote these by AS+1,AS+2, ... ,At.

Then we have t s x, or, more exactly, s t L dim !fA + L X_(Ak ) s x,

j=l j k=s+l

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where the sign holds if A = 0 is not an eigenvalue of B with an eigenvector

of nonpositive type; here :tAo denotes the root subspace of B at Aj , j=1,2, ... , )

and x (A.) denotes - ) the number of negative squares of the inner product [.,.]

on:tA for real A. (observe that :tAo is nondegenerate). In each of the j) )

(positive type) eigenspaces :tA., j=t+1,t+2, ... we choose an orthonormal )

basis; the union of all these basis vectors we denote simply by

(e v ),v=1,2, ... . Further, let Po denote the selfadjoint, finite dimensional

Riesz projection corresponding to the set {A1, ... ,AS'X1' ... 'XS,AS+l, ... ,At}.

Then it holds (see [L], [B]):

If B is a compact selfadjoint and injective operator in Ox then we have

96 Langer and Schneider

for arbitrary x E n)( with the above notation

(1.9) x = p x + L [x,e ]e , o v v v

where the sum converges in the nonn of n)(.

2. S-hermitian linear pencils

Let ~ ,~,~ be complex linear spaces with ~ c ~ and let (.,.) be a o 0 0 0 0

hermitian sesquilinear form on ~o. We consider linear mappings

G,S: ~o ~ ~o

F: ~ ~~ o 0

with

(2.1) ~o := ker F n ker G = {O}.

This assumption is not an essential restriction as it can often be achieved

by considering the spaces ~o/~ and ~o/~ instead of ~o and ~o respectively. o 0

In this section we study spectral properties of the linear pencil

F-AG. We introduce the linear spaces

and

-1 ~1 := F (G~o)

XO := {O}l A

-1 k-1 (F-AG) ((F-AG)~l n G(XA )), kEN.

The spaces Xk, have the properties:

(1 ) xk Xk+1 A C A (kENu{O})l

k k +1 k k +j X 0 = X 0 0 = X 0 for all j E N; (2) if A A for koE N u {O}, then XA A

(3) if dim (Xl) A < 'l), then dim X~ < 00 for all kEN.

The point A E ( is an eigenvalue of the o pencil F-AG if X~ ¢ {O}, and X~

the eigenspace of the pencil F-AG at AO' The space XA o

is the root subspace. Finally we introduce the number

PIA) := dim x~,

o k 0 .= span{XA Ik E N}

o

and for an eigenvalue A this number is the geometric multiplicity.

Definition (2.2): The pencil F-AG is called S-hennitian if the bilinear

is

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form F(u,v) := (Fu,Sv) is hermitian on a subspace ~ with ~1 c ~ c ~o and

the form G(u,v) := (Gu,Sv) is hermitian on a subspace § with ~ c § c § . o

In the usual way one can show:

Lemma (2.3): If A1, A2 are eigenvalues of the S-hermi tian pencil F-AG wi th

A1 f. A2 , then

F(u,v) s G(u,v) z 0 for all u E XA ,v E XA . 1 2

Definition (2.4): The pencil F-AG is called reducible if 0 is not an eigen­

value and the inclusion G§ c F~ holds. o 0

A consequence of this property is the fact, that for each v E § there o

exists a unique u E ~o such that Fu z Gv. Hence we can define a mapping A

on ~o by

(2.5) Av :s U if and only if Fu = Gv.

For this mapping we get immediately:

(1) A is a linear mapping from §o onto ~1.

(2) The eigenvalue problem Fy = AGy is equivalent to y = AAy. Thus AO' AOf.O, -1 is an eigenvalue of the pencil F-AG if and only if AO is an

eigenvalue of A.

(3) For each subspace ~ with ~1 c ~ c §o' A defines an endomorphism of ~.

(4) If the pencil F-AG is S-hermitian, then A is hermitian with respect

to both forms F and G:

F(Au,v) = F(u,Av)

G(Au,v) = G(u,Av)

(u E ':f,v E §)

(u E §o,v E §)

Definition (2.6): The pencil F-AG is said to be of finite index if or all

A E ([ we have

(1) (p(A) :S) dim XA < 00 ; <S(A) := dim (i!o/(F_AG)~ ) < 00.

o (2) P(A) - <S(A) is independent of A.

The main result of this section is

Theorem (2.7): Let the pencil F-AG be reducible and of finite index. Then

for all A E ([ we have

(2.8) dim (G~o/(F_AG)~ ) :S pIA). 1

In this relation the equality holds, ifG is injective on ~1.

97

98 Langer and Schneider

Proof: The first statement of the theorem was proved in [551], p.16. In order

to prove the last statement, we can assume, that p{X)>O. Since dim ~X<OO, k

o there exists a ko E N such that ~X - KX • Below we shall show, that the

relation

(2.9) k k

dim KX 0 - dim «F-XG)~l n G(KX 0)) + dim K~

holds. Thus the injectivity of G yields

k k dim G{KX 0) = dim «F-XG)~l n G{KX 0)) + p{X).

k Hence there exists a subspace ~X c Kxo with

dim ~X '" p{X);

and this proves

dim (G(~o)/{F_XG)~ ) ~ p(X). 1

It remains to prove (2.9). We observe, that for an arbitrary linear

mapping B

(2.10)

~ ~ ~ and a finite-dimensional subspace A c ~ the equation

dim (B-1(B{~) n A)) .. dim (B(~) n A) + dim (ker B)

is valid. By virtue of the property (2) of the spaces ~ we conclude, that k +1 1

~X = Kxo and thus from (2.10) we get with KX - ker (F-XG),

k k +1 -1 ko dim Kxo = dim Kxo .. dim «F-XG) «F-XG)~l n G(Kx )) ..

k1 = dim «F-XG)~l n G(K ~ )) + dim Kx'

o and (2.9) is proved.

Theorem (2.11): Let the pencil F-XG be reducible, of finite index and let G be

injective on ~1' If ~ is an arbitrary linear space with ~1 c ~ c ~o and A

is the operator associated with the pencil, then for all X E [ the relations

p(X) .. dim (ker(I-XA) I~) = dim ( ~/(I-XA)~ ) hold.

Proof: The reducibility of the pencil implies p(O) = O. Therefore it is

sufficient to consider X~O. Obviously ker(I-XA)I~ = K~, which implies the

first equality. Choose a subspace Ax such that

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9t - (I-AA)9t';' .I.~: It is easy to see, that G is injective on .l.A, hence

dim (9t/(I_AA)9t) - dim.l.A - dim(G(.l.A))· Next we show, that

(F-AG)9t1 n G(.l.A) - {O}.

99

Indeed, if u E (F-AG)9t1 n G(.l.A), then there exist w E 9t1 and v E .l.A with (A;fO)

(F-AG)W = u .. G(AV).

This is equivalent to

(E-AA) (v+w) - v,

which implies v E (I-AA)9t n .l.A - {O}. Thus u - G(AV) .. 0 and therefore

dim (9t/(I_AA)9t) .. dim G(.l.A) ~ dim (G(~o)/(F_AG)9t) ~ PIA), 1

where the last inequality follows from (2.8). In the proof of Theorem (2.7)

it was shown, that there exists a subspace ~A c XA with the properties

dim ~A .. PIA), G(~A)n(F-AG)9t1 .. {O}.

The last relation implies

~An(I-AA)9t .. {O},

and since A is injective on ~A we conclude

dim ( 9t / (I-AA)9t) ~ dim A(~A) .. p(A).

The theorem is proved.

Definition (2.12): The S-hermitian pencil F-AG is called quasi-rightdefinite

(quasi-leftdefinite), if there exists a subspace 9t with 9t1 c 9t c ~

(9t1 c 9t c ~), such that the hermitian form F (G respectively) has an at most

finite number of negative squares on 9t.

Note that the number of negative squares of F or G may depend on the choice

of 9t.

Now, under the assumption that the pencil F-AG is S-hermitian, reducible

and quasi-leftdefinite or quasi-rightdefinite, the results of Section 1

and, in particular, Theorem (1.5) can be applied to the subspace 9t from

Definition (2.12) and the operator A from (2.5). We consider e.g. the

case where the pencil is quasi-leftdefinite, that is, the hermitian

sesquilinear form

[u,v] := F(u,v) .. (Fu,Sv) (u,v E 9t)

has a finite number x of negative squares. Then ~, equipped with this

100 Langer and Schneider

form, is a nx- lineal, and the operator A defined by (2.5), mapping ~ = ~ into ~, is hermitian. If the pencil F-XG is of finite index and G is

injective on ~1' the Theorem (2.11) implies that also the assumption d)

of Theorem (1.5) is satisfied with ~ = ~. Now this theorem yields

inunediately:

Theorem (2.13). Suppose that the pencil F-XG is S-hermitian, reducible, of

finite index and quasi-leftdefinite and that the operator G is injective on

~1. Further assume that the eigenvalues of the pencil are all isolated

(with the only possible accumulation point ro) and of finite geometric

mUltiplicity. Then the closed hermitian operator A, induced in the completion

nx of (~, [.,.]) by A, is compact. If additionally A is injective, then for

each x E n an expansion (1.9) holds. Moreover, if a root subspace X~ ,X ~o, X A 0

of the pencil F-XG is nondegenerate with respect to [.,.], then this ~oot - -1

subspace coincides with the root subspace of A at AO

Remark. The root subspace Xx ,Xo~o is nondegenerate with respect to o

[u,v] = F(u,v) (u,~ E ~), if it is nondegenerate with respect to the

hermitian sesquilinear form ~(u,v).

In order to see this we assume that Xx is degenerate with respect to the

form [u,v] = F(u,v) (u,v E ~), that is the ~ubspace

X~ :- {u E Xx [u,Xx] = {a}} o 0 0

contains nonzero elements. It is easy to see that X~ is invariant under A,

hence it contains an eigenvector Yo of A correspondiRg to the eigenvalue X~l. Now we get

~(YO,XA ) = (GYo'SXx ) o 0

1 = r- [Yo'Xx {O},

o 0 that is, also the form ~ degenerates on Xx ' which proves the claim.

o

3. A second order problem

1. In this section we consider the problem

(3.1a)

(3.1b)

(3.1c)

-(PD')' + qn = XrD on I = [a,b],

cosa D(a) - sina (PD')(a) = 0

-~lD(b) + ~2(PD')(b) = X(~iD(b) - ~2(PD')(b))

with some a E [O,n) and ~1' ~2' ~i' ~2 E R. We assume that

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-1 1 ro p ,q E L (I), r E L (I), p(t)>O, r(t)¢O a. e. on I,

and, moreover, that

(3.2)

otherwise the boundary condition (3.1c) can be written as a A-independent

one. Further, it is supposed that A-O is not an eigenvalue of (3.1), that

is the problem

-(pry')'+ qij = 0 on I,

cosa ry(a) - sina (pry')(a) = 0,

~lry(b) - ~2(Pry')(b) - 0,

101

has only the trivial solution ry - O. Introducing Yl 1= ry and Y2 := -(pry'), the

boundary value problem (3.1) is equivalent to the problem

, -1 Y1 + P Y2 - 0,

Y2 + QYl = Aryl' cosa y1(a) + sina Y2(a) = 0,

-[~lYl(b) + ~2Y2(b)1 s A[~iYl(b) + ~2Y2(b)1. 1)

On the set AC(I) of all absolutely continuous 2-vector functions on I we

define the mapping -1

F1Y:=Y'+(~ ~ )Y'

and on the space L2(I) of all 2-vector functions on I with square integrable

components the mappings

G1Y:=(~ g )Y' SlY := ( ~ -1 )Y' 0

It is easy to check that with the matrix

H = ( -~ 1 ) 0 for u,v E AC(I) the identity

holds a.e. on I. Further, we define for y E C(I) the "boundary mappings"

( COSo a F2y := Si~ a )y(a) + (-~1 -~2 )Y(b),

1) Here and in the sequel we denote spaces of scalar functions and vector

functions by the same symbol.

102 Langer and Schneider

S ... ( sin <X 2Y • 0 -coos <X )y(a) 0-1( 0 0) (b) - ~i ~2 y .

Then in analogy to (3.3) the identity

(3.4) (S2v )*((F2-XG2 )U) - ((F2-~G2)V)*(S2U) -

.. - [v(b)*Hu(b) - v(a)*Hu(a))

holds. Now choose

~o :- {y E AC(I) FlY E L2(I)},

~o := C(I),

~ := L2(I) e [2 o

and on ~ 0 the inner product

b (f,g) :~ J gl(t)*f1(t)dt + gif2

a

(f - ( :: ), 9 .. ( :: ), f 1,gl E L2(I), f 2,g2 E (2).

We define the mappings F,G and S as follows:

The pencil F-XG has the property (2.1). Indeed, Fy = Gy - 0 implies ry1=0 -1

and yi + P Y2 = 0, which yields Y1 .. 0 and Y2" 0 (observe r(t)¢O a.e.).

Further, we have for u,v E ~ o b

F(u,v) = (Fu,Sv) - J (Slv )*(F1u)dt + (S2v)*(F2u) a

and the relations (3.3),(3.4) imply

F(u,v) '" F(v,u) (u,v E ~o).

Similarly, if u,v E ~ , it holds o b

~(u,v) .. (Gu,Sv) .. J (SlV)*(G1u)dt + (S2V)*(G2U), a

~(u,v) = ~(v,u) (u,v E ~o).

Therefore the pencil F-XG is S-hermitian with ~ := ~ and ~ := ~ . o 0

As zero is not an eigenvalue of the problem (3.1), the mapping F is

injective. Well known results for two-point boundary value problems imply

Langer and Schneider

that in this case the corresponding inhomogeneous equation Fy - Gv has a

solution for each v E § , hence the pencil F-'A.G is reducible. Moreover, it o is of finite index (see [881], 8atz 3.9).

-1 The space ~1 - F (G§o) is the set of all solutions y E AC(I) of the

inhomogeneous boundary value problem Fy ,. Gv for arbitrary v E § . In o

particular, the vector functions y E ~1 satisfy

pyi. ,. -Y2 .

From this relation the injectivity of G on ~1 follows.

On the space § the form G becomes b C

G(u,v) = J u1v1r dt - D-1(~i.v1(b) + ~iv2(b))(~iu1(b) + ~iu2(b:). a

If r ~ 0, the pencil F-'A.G is evidently quasi-rightdefinite, and 1.. 0 < 0,

even iightdefinite on §o. We shall not consider this case here as it was

treated e.g. in [D],[FJ,[SSl,2],[Wa]. Instead in the following subsection,

we allow r to be positive and negative on sets of positive Lebesg .e

measure. Then the fonn G has infinitely many positive and infinit":!ly many

negative squares. However, as we shall see in the next subsectior, the

problem (3.1) is always quasi-Ieftdefinite.

2. Let ~ be the space I -1

~ :s {y E ~o yi. + p Y2 = 0, cosa Y1(a) + sina Y2(a) m O}.

The 2-vector functions y E ~ are represented by their first component Y1'

as the second component is given by Y2 = -pyi. These functions Yl are characterized by the following properties:

Y1 E AC(I), pyi E AC(I), -(pyi)'+ qyl E L2(I),

cosa Y1(a) - sina (pYi)(a) = o. Therefore the integrals

b 2 b 1 2 J plyil dt = J plpYil dt, a a

103

-1 1 exist as p ,q E L (I). A straightforward calculation shows that for y,z E ~

it holds

(3.5)

b F(y,z) = J (pyizi + qy1z1)dt + ctga Y1(a)zl(a)

a

where the second tenn on the righthand side is to be replaced by zero if

a = O. In the sequel we shall write F(Y1,Zl) instead of F(y,z), and also

104 Langer and Schneider

consider ~ just as the space of the first components of the elements of the

original ~.

It is well-known that the form given by the integral in (3.5) has at most

finitely many negative squares; obviously this form is positive definite if

q ~ o. As the other terms on the right hand side of (3.5) define a form with

at most three negative squares, the pencil F-AG is quasi-leftdefinite on ~.

It is easy to find the number of negative and positive squares of the

last term in (3.5), namely, this term has 2 positive squares if PIPio-1 > 0 -1 -1-1

and P1PiO < 0, 2 negative squares if P1PiO < 0 and P1PiO < 0, 1 positive -1 -1

and 1 negative square if P1PiO > 0 and P1PiO ~ O.

3. For the completeness and expansion theorems the completion of the

space ~ with respect to the norm topology defined by the form F from (3.5)

plays an essential role. In order to describe this norm topology we intro­

duce the Hilbert space ~ := L2 (I} $ ( $ (; here L2 (I) is the space of p b P

measurable functions f on I such that J plfl 2 dt < ro with the usual inner a-+ -+ T

product. Each y E ~ defines an element y E ~ : y:= (y',y(a},-(py'}(b})

Assume first that a ~ o. With the Gram operator fF in ~ = L~(I} $ ( $ (

a

a the form F can be written as

b P1Pi Jq(L}dL + ctga + -0--­a

(Y,z E ~).

Evidently rF is a compact perturbation of the unit operator in ~, hence fF is boundedly invertible if and only if it is injective.

For a given rF, the completion If of ~ with respect to the norm

topology defined by F can often be described more explicitely. To this end

it is useful to observe the following facts.

(i) The norm topology, defined on ~ by F, is given by the norm or seminorm

Langer and Schneider 105

(u E ~)

(see, e.g., [AIl).

(ii) If q E L2(I), then for each ~ = (V,Wl ,W2 )T E Z, there exists a sequence

(Un) c ~ such that

(n-+<D).

2 <D. Indeed, as pv E L _l(I) and the C -funct1ons are dense in this space, p <D

we can choose a sequence (vn ) C C (I) such. that

b 1 2 J - I pv-v I dt ---+ 0, vn(a) = wl ctga, vn(b) = -w2 • a p n

t 1 Define un(t) := wl + J -(-I V (L)d'L Then u E AC(I), pu~ = vn E AC(I),

a p L n n

2 -(pu~)'+ qun = -v~ + qun E L (I) and

cosa un(a) - sina (Pu~)(a) = cosa wl - sina vn(a) o. -+ ( l)T T Therefore un = p vn,wl ,w2 -----+ (V'Wl 'W2) in Z.

Now, if fF is invertible (hence ~2 ~ 0) and q E L2(I), then (i) and

(ii) imply that the completion ~ of ~ with respect to II . IIF coincides

with Z, equipped with the inner product

[y,il = (fFy,i)z (y,i E Z).

The embedding of ~ into ~ is again given by

~ 3 Y -----+ (y' ,y(a) ,-(py' )(b» T.

That is, convergence of a sequen~e (Yn) C ~ with respect to the norm of

the nx-space ~ to an element y E ~ means

y~ ---+ y'in L~(I), Yn(a) ---+ y(a), (py~)(b) ---+ (py')(b).

The closure A in ~ of the operator A in ~ can now easily be described.

To this end a slightly modified representation of ~ is more convenient.

If fF is invertible, the elements ~ E ~, ~ = (V,Wl ,W2)T can equivalently

be written as

(3.6) ~ = { wl + I v(L)dL, W2 }T =: { W'W2 }T,

where now w E AC(I) with w' E L~(I), w2 E (. Then an element f E ~

corresponds to {f,-(pf')(b)}. For y,f E ~ the equation y = Af is equivalent to

106 Langer and Schneider

-(py')' + qy = rf, cosa y(a) - sina (py')(a) = 0, (3.7)

As A = 0 is not an eigenvalue of (3.1), we can choose a fundamental system

v1,v2 of the homogeneous equation -(pv')'+ qv = 0 on I such that

cosa v1 (a) - sina(pvi) (a) = 0, - ~lv2(b) + ~2(pv2)(b) = 0

and

Ip:i p:;1 = -1.

Then the solution y of (3.7) can be written as

b t y(t) = V1(t)Jv2 (L)f(L)r(L)dL + v2 (t)Jv1(L)r(L)dL + Cv1(t)

t a with

With the representation (3.6) of the elements of a;.. the solution y is

given by

y = At =

This relation extends to the closure A of A as follows:

if t = {f, rp2}.

with

(3.9)

Langer and Schneider

But (3.8) implies for = 0 a.e. on I, hence fo a 0, and, using the formula

(3.9), ~o = 0 follows.

Thus if a ~ 0 and fF is invertible we have shown that the operators A

and A, associated with the problem (3.1), satisfy the conditions of

Theorem (2.13) and 0 ¢ apIA). Therefore the elements of ~, and, in

particular, the elements of ~ admit expansions in root functions of the

problem (3.1), which converge in the norm of the nx-space ~. According to

the definition of this norm for a sequence of functions (f ) c ~ this n

means, that the sequence of the derivatives (f~) converges in L~(I) and

the sequences fn(a) and (pf~)(b) converge in [. This implies in particular

the uniform convergence of the sequence (fn ).

If fF is not invertible, a suitable factor space of ~ has to be

considered. Finally suppose that a - O. Then the elements y E ~ satisfy

the condition y(a) = 0, and in the matrix representation of fF the second

row and the second column can be deleted. If, additionally, q = 0, ~' = 0 2

and ~2~1 ~ 0, then fF reduces to

(3.10) 1 ~1 b

fF = I - pIt) ~2 ~ . da,

and, if ~1 ~ 0, the nx - or Hilbert space ~ is L~(I), equipped with the

inner product given by the Gram operator in (3.10). In the particular case

-~1 = ~1 = ~2 = 1 this Gram operator is positive, hence x = O. This case

was considered in [IS].

107

108 Langer and Schneider

References.

[AEO] Atkinson, F. V., W. N. Everitt, K. S. Ong: On the m-coefficient of

Weyl for a differential equation with an indefinite weight function.

Proc. London Hath. Soc. 29, 368 - 384 (1974)

[AI] Azizov, T. Ya., I. S. Iokhvidov: Linear operators in spaces with an

indefinite metric. John Wiley & Sons, Chichester-New York-Brisbane­

Toronto-Singapure, 1989

[B] Bognar, J.: Indefinite inner product spaces. Springer Verlag, Berlin­

Heidelberg-New York, 1974

[D] Dijksma, A.: Eigenfunction expansion for a class of J-selfadjoint

ordinary differential operators with boundary conditions containing

the eigenvalue parameter. Proc. Roy. Soc. Edinburgh, 86A, 1 - 27

(1980)

[DL] Daho, K., H. Langer: Sturm-Liouville operators with an indefinite weight

function. Proc. Roy. Soc. Edinburgh, 78A, 161 - 191 (1977)

[DLS] Dijksma, A., H. Langer, H. de Snoo: Symmetric Sturm-Liouville operators

with eigenvalue depending boundary conditions. CMS Conference

Proceedings, Vol 8: Oscillation, Bifurcation, Chaos, 87 - 116 (1987)

[F] Fulton, C. T.: Two-point boundary value problems with eigenvalue

parameter contained in the boundary conditions. Proc. Roy. Soc.

Edingburgh, 77A, 293 - 308 (1977)

[IS] Ibrahim, R., B. D. Sleeman: A regular leftdefinite eigenvalue problem

with eigenvalue parameter in the boundary conditions. Lecture Notes

in Mathematics, 846, Proceedings, Dundee 1980, 158 - 167 (1981)

(Ll Langer, H.: Spectral functions of definitizable operators in Krein

spaces. Lecture Notes in Mathematics, 948, Proceedings, Dubrovnik 1981,

1 - 46 (1982)

Langer and Schneider 109

[S] Schafke, F. W.: Charakterisierung "kompakter" hermitescher Operatoren.

Hath. Ann. 262, 383 - 390 (1983)

[SSl] Schafke, F. W. und A. Schneider: S-hermitesche Rand-Eigenwert­

probleme. 1. Hath. Ann., 162, 9 - 26, (1965)

[SS2] Schafke, F. W. und A. Schneider; S-hermitesche Rand-Eigenwert­

probleme. II. Hath. Ann. 165, 236 - 260, (1966)

[Wa] Walter, J.: Regular eigenvalue problems with eigenvalue parameter in

the boundary conditions. Hath. Z. 133, 301 - 312 (1973)

[vii] \"/ielandt, H.: tiber die Eigenwertaufgaben mit reellen diskreten Eigen­

Vlerten. Hath. Nachr. i, 308 - 314 (1951)

Prof. Dr. Heinz Langer

Fachbereich Mathematik

Universitat Regensburg

D 8400 Regensburg

Prof. Dr. Albert Schneider

Fachbereich Mathematik

Universitat Dortmund

D 4600 Dortmund 50

Eingegangen am 27. November 1990