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Integr. equ. oper. theory 39 (2001) 127-152 0378-620X/01/020127-26 $1.50+0.20/0 9 Birkh~iuser Verlag, Basel, 2001

I IntegralEquations and OperatorTheory

COMPACT PERTURBATION OF DEFINITE TYPE SPECTRA SELF-ADJOINT QUADRATIC OPERATOR PENCILS

V. ADAM

JAN, H. LANGER,

OF

M. M()LLER

Self-adjoint quadratic operator pencils L(A) -- A2A § AB § C with a noninvertible leading operator A are considered. In particular, a characterization of the spectral points of positive a n d of negative type of L is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.

1. INTRODUCTION

It was shown in [LMMa 2] that for a self-adjoint analytic operator function L with compact spectrum spectral points of positive type or of negative type are in some sense stable under a compact additive perturbation. E.g. if A is a spectral point of positive type of L and Lt denotes the perturbed analytic self-adjoint operator function, then A is a spectral point of positive type or a point of the resolvent set of L1, or it becomes an eigenvalue of L1 for which the inner product (L~ (A)., 9) has a finite number of nonpositive squares on the algebraic eigenspace of L1 at A. In particular, A can never become an accumulation point of the nonreal spectrum of L1. In the present paper these results are applied to self-adjoint quadratic operator pencils

(1.1)

L(A) = A2A + AB + C.

To this end we first find assumptions which ensure the existence of spectral points of positive or negative type. It has been known for a long time, see e.g. [KL 1], that for a monic strongly damped (or weakly hyperbolic) quadratic pencil (1.1) all spectral points with the exception of at most one point have this property. Here we show that also for not strongly damped quadratic pencils such points often can easily be found by means of the numerical range of the pencil L.

The authors thank Alexander Markus for reading the manuscript and valuable suggestions. The third author acknowledgespartial support by the John Knopfmacher Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand and by the NRF of South Africa.

128

Adamjan, Langer, MSller

In the second part of this paper we consider a more special nonmonic pencil (1.1) which arises in magnetohydrodynamics, see [Li, Section 7.10]. Since it has in general unbounded spectrum, we first linearize it and transform it into a problem with bounded spectrum. This Iinearization is not a standard one because the leading operator A has a nontrivial (infinite-dimensional) kernel. In Section 2 we introduce the necessary definitions and recall some known results which will be used later. In Section 3 we consider operator pencils (1.1) under a minimal assumption (see (3.2)) and show that spectral points of positive or negative type can be found by means of the numerical range of L. In fact, we consider all elements x ~ 0 for which the quadratic polynomial ~x(A) : (L(A)x, x) has two real zeros, and denote by A+ (A_, respectively) the set of all such zeros for which the derivative ~ at this zero is positive (negative, respectively). Then the inner points of A+ \ A_ are spectral points of positive type or belong to the resolvent set of L. In Section 4 a nonmonic pencil is considered for which the leading operator is a projection with in general infinite-dimensional range and kernel, which arises from the problem in magnetohydrodynamics mentioned above. We linearize this pencil and associate with it a linear self-adjoint (in general nonmonic) pencil in a Hilbert space and a linear monic self-adjoint pencil in a Krein space. In Sections 3 and 4 also statements on the behavior of the spectral properties under compact perturbations are obtained; they immediately follow from the results of [LMMa 1], [LMMa 2]. In the last section the example from magnetohydrodynamics is considered in more detail. We finally mention that spectral points of positive and of negative type for arbitrary, not necessarily weakly hyperbolic self-adjoint operator pencils were first introduced by P. Lancaster, A. Markus and V. Matsaev in [LaMMa].

2. PI~ELIMINARIES

2.1. S p e c t r u m of a n a n a l y t i c o p e r a t o r f u n c t i o n . The following definitions for analytic operator functions will be applied later to A-linear and A-quadratic functions. For an operator function K which is analytic in a domain ~DK C C and the values of which are bounded linear operators in the Hilbert space ~ the resoIvent set p(K) is the set of all A E :DK such that the operator K(A) is boundedly invertible, that is 0 E p(K(A)), and the spectrum of K is the set a(K) := C \ p(K). The point spectrum or set of eigenvalues ~p(K) is the set of all Ao e T~c such that 0 E ap(K(A0)), and each nonzero element x0 such that K(A0)x0 = 0 is called an eigenvector of K at the eigenvalue Ao. The set consisting of all eigenvectors at Ao and the zero vector is called the eigenspace of K at the eigenvalue A0. The essential spectrum aess(K) is the set of all A E :D~c such that K(A) is not a Fredholm operator. The finite sequence Xo, x l , . . . , xk in 7-/is called a Jordan chain of K at the eigenvalue A0 if x0 r 0 and the polynomial k

K(A)

- A0) xj j=0

has a zero of order greater than k at A0. In particular, x0 is an eigenvector of K at A0; the elements x l , . . . , xk are also called associated vectors of K and x0 at A0. The set of the

Adamjan, Langer, M611er

129

elements of all the Jordan chains o f / ( at A0, augmented by the zero element, is linear; it is called the root subspace of K at )`0. By 7-I1 := {x E 7/ : [Ixll = 1} we denote the unit sphere of 7/. The approximate point spectrum O-app(K) of the pencil K is the set of all )` E ~)g for which there exists a sequence (x~)~~ C 7/1, satisfying [IK()`)x=l[ --+ 0 if n -+ oo. For quadratic and linear operator polynomials we also need the notion of the extended spectrum. If, for n = 1 or 2, K()`) = ~ - - o )`~K~ with a nonvanishing operator K,~, the extended spectrum ~(K) is the set a(K), augmented by the point oc if and only if 0 E a(K~). Similarly, the extended point spectrum ~p(K) is the set ap(K) augmented by c~ if and only if 0 E ap(K~), and the extended essential spectrum ~e~s(K) and the extended approximate point spectrum "happ(K) are defined analogously. Equivalently, oo belongs e. g. to the extended point spectrum of K if and only if 0 E av(~'), where _~ is the polynomial pencil K(#) := t*"~K't;).l~ In the following it is assumed that the analytic operator function K is self-adjoint, this means that :DK ----7)~ and K()`*) =

)` e

Here and in the sequel for a complex number A the complex conjugate number is denoted by),*. 2.2. S p e c t r a l points of definite t y p e . The eigenvector x0 at the real eigenvalue A0 of K is said to be positive (negative, neutral, respectively) if (K'(Ao)Xo, x0) > 0 (< 0, = 0, respectively). The real eigenvalue A0 is said to be an eigenvalue of positive (negative, neutral, respectively) type if all its corresponding eigenvectors have this property. If an eigenvector x0 at Ao is positive or negative, then there do not exist any associated vectors to x0, that is, the equation K()`0)x + K'()`o)xo = 0 does not have a solution x. In particular, for an eigenvalue of positive or of negative type there do not exist nontrivial Jordan chains. The real point A is said to be a spectral point of positive type of K if it is a point of the approximate point spectrum eapp(K) and the following holds: (x,~) C 7/1, limooHK()`)xn H = 0 ~

limn~oo(K'()`)xn,xn) > O.

The set of all spectral points of positive type of K is denoted by r Spectral points of negative type of K are defined similarly; the set of all spectral points of negative type of K is denoted by a_ (K). Observe that e.g. an eigenvalue of positive type need not be a spectral point of positive type: It can e.g. be an accumulation point of spectral points of negative type. As an example we consider the following pencil L0(A) in the Hilbert space 7/1G 7/2 which will be of interest later:

0 The extended spectrum of Lo is Y(L0) = {0} U a ( - B o ) U {oo}.

o)

130


Here 0 is an eigenvalue of L0, the corresponding eigenspace being ?-/1, and the inner product (L~ ( 0 ) - , . ) is given by

Therefore, if Bo is a positive (negative, respectively) operator, then 0 is an eigenvalue of positive (negative, respectively) type of L0. If A E a(-Bo) \ {0}, then for a sequence of /

\

t , ~ ,] elements x,~ = \( xxem

[]m~H = 1, the relation [[L0(A)m~,,---> 0 means

IIA(B0 + A)xl, II -+ 0, IIx2, ll

0.

Hence ][xz,~[[ --+ 1 and (xl,n) is an approximate eigensequence of Bo at -A. Further, (Z~(A)x,, x , ) = 2A][xl,,I] 2 + (Boxl,,~,xl,,~) = A][xl,,[[ 2 + o(1) if n -+ oz. It follows that A C a+(L0) if A > 0 and A E a-(Lo) if A < 0. 2.3. S p e c t r a l p o i n t s o f finite n e g a t i v e or finite p o s i t i v e t y p e . Let V be a self-adjoint operator function which is meromorphic in C + O C - and the values of which are bounded linear operators in a Hilbert space 7/. By definition, the function V belongs to the class N~(7/) if for arbitrary m E N, xl, x 2 , . . . , xm C 7 / a n d A1, A 2 . . . , Am e C + A • v the matrix

(2.2)

f f v(Ad - v(Aj)*

t,t

has at most ~ negative eigenvalues and for at least one choice of m, Xl,X2,...,Xm and A1, A2..., A,, it has exactly ~ negative eigenvalues. The class No(?-/) is the set of all selfadjoint operator functions V which are analytic on C + U C - and such that .c~V(A) > 0 if .~A > 0. Let K be again a self-adjoint analytic operator function. Recall (cf. [LMMa 2, Theorem 5.5]) that if the closed interval A C N belongs to a + ( K ) U p(K), then the operator f u n c t i o n / ( is invertible on some set (2.3)

U~ := {A e C : ~A e Z~, 0 < I~A] _< , }

with 77 > 0 and the operators - K ( A ) -1, A E b/z~, admit there the representation (2.4)

-K(A)-' =//,

dF(t) A

+

Ro(),),

where the operator function R0 is analytic on A and F is a nondecreasing function on A whose values are bounded self-adjoint operators. Evidently, the integral in (2.4) is an operator function of the class No('//). Now we can define the set cr_/(K): The real point c~ belongs to a _ / ( K ) if there exists a closed interval A containing a as an interior point such that

,~ \ {o~} C o'+(K) U p(K) and such that the operator function __/~--1 admits in a set b/~ (see (2.3)) a representation (2.5)

- K ( A ) - ' = V(A) + Ro(A),

where V E N,(7/) for some ~ = ~(a) > 0 and the operator function 17 is analytic outside A including oz and vanishes at oz, and the operator function R0 is analytic on the interior


131

of A. It follows from [LMMa 2] that if a E a_,f(K), then a is an eigenvalue of K with the following properties: 1. All the Jordan chains of K at a are of finite length (< 2~(a) + 1). 2. If the signature of the inner product (K'(a) -, 9) on the eigenspace of K at c~ is denoted by (~+(a), ~0(a), ~_(a)), then ~0(a) + ~_(a) = ~(a). 2.4. S e l f - a d j o i n t o p e r a t o r s in K r e i n spaces a n d c o m p a c t p e r t u r b a t i o n s . The above definitions of the type of a spectral point of a self-adjoint analytic operator function are closely related to the corresponding definitions for a self-adjoint operator in a Krein space /C, which we now recall for the convenience of the reader. In the Krein space/C with indefinite inner product [., 9] the element x is said to be positive if [x, x] > 0, and the real eigenvalue of a self-adjoint operator D is said to be of positive type if all its eigenvectors are positive. Further, A0 E R is said to be a spectral point of positive type of D if it belongs to the approximate point spectrum of D and the following holds: []x~[[=l, n = l , 2 , . . . ,

lim ][(D-Ao)Xn[[=O ~

lim[x~,x~] > 0,

~---~OO

see [LMMa 1]. The set of all spectral points of positive type of D is denoted by a+(D). Eigenvectors, eigenvaiues and spectral points of negative type are defined similarly; o_ (D) denotes the set of all spectral points of negative type of D. Further, in [LMMa 1] a _ j ( D ) is defined as the set of all real points A0 such that for some open interval A which contains A0 the set A \ {A0) belongs to cr+(D) U p(A) and on the maximal spectral subspace of D corresponding to A the indefinite inner product has a finite and positive number of negative squares. The operator D is said to be definitizable if p(D) ~ ~ and there exists a real polynomial p which is not identically zero such that ~v(D)x, x] > 0 for all x E K:. If D is definitizable, a real number in the spectrum of D is called a critical point of D if it is a zero of all definitizing polynomials of D. A definitizabte operator D has a spectral function ED, see [L2]. By means of this spectral function a critical point of D is characterized as follows: For each open interval A containing a and such that ED(A) is defined, the range of the projection ED(A) contains positive as well as negative elements. The critical point a is said to be a regular critical point if for all such intervals A in some neighbourhood of a the projections ED(A) are uniformly bounded, otherwise a is called a singular critical point. Now let a be a critical point of finite rank of indefiniteness. This means that for some open interval A containing a on the range of ED(A) the indefinite inner product has a finite number of positive or a finite number of negative squares. In the first case for a there exists at least one non-negative eigenvector x0 (i. e. Ix0, x0] _> 0), and in the second case there exists at least one non-positive eigenvector x0. The point a is a regular critical point if and only if the indefinite inner product is nondegenerate on the algebraic eigenspace of D at a, cf. [L2, Theorem II.5.7]. Suppose now that the Krein space inner product is obtained from a Hilbert space 7-/with (positive definite) inner product (., 9) by means Of a Gram operator G, that means G is a bounded and boundedly invertible self-adjoint operator in 7-/and the indefinite inner product is defined by the relation Ix, y] := (Gx, y), x, y E 7-l. An operator D which is selfadjoint in this Krein space ]C is sometimes called G-self-adjoint in 7-/. Besides the self-adjoint

132


operator D in the Krein space /C we consider the self-adjoint operator function A(A) := AG - GD in the Hilbert space 7-/. Then the positive (negative) eigenvectors or spectral points of positive (negative) type of D coincide with the positive (negative) eigenvectors or spectral points of positive (negative) type of the operator function A and cr+,f (D) = cr:uj (A). For positive eigenvectors and eigenvalues of positive type this follows easily from the relation A'(A) = G. If A e cr_j(D), then we choose an open interval A as in the definition of a _ j ( D ) , and with the corresponding spectral projection ED(A) we decompose the function - A -1 as follows: -A(A) -1 = (D - ,~)-IG-1 = (D - ,~)-~ED(A)G -1 + (D - A ) - I ( I - E D ( A ) ) G - L The second operator function on the right hand side has an analytic continuation on A, and if we denote the first term on the right hand side for a moment by V(A) it follows that the elements of the matrix in (2.2) become

['V(Ai) - V(Aj)* Xi, Xj ) = [ED(A)(D 9

- -

~

- -

A~)-t(D

- -

Ai)-lG-lxi, G-lxi]

9

Since on the range of the projection ED(A) the indefinite inner product has a finite number of negative squares, the function V belongs to some class N~(7/) and hence a _ j ( D ) C a _ j ( A ). The converse inclusion can be proved similarIy. We also recall [LMMa 2, Theorem 6.1], which is the main result of that paper, and formulate a simple corollary. Both will be applied later. T h e o r e m 2.1. Let K and 1(1 be two self-adjoint analytic operator functions defined on some domain D of the complex plane such that the operators KI(A) - K(A), A 6 7), are compact. Consider an open interval I C o+(K) U (p(K) AN:) and suppose that at least one point of this interval is not a C-inner point of or(K1). Then the following inclusion holds:

c o+(K~) o p(K~) o ~-z(K1). In particular, the points of I cannot be accumulation points of the nonreaI spectrum of K1 or of the real points of cr ,i(K1 ). C o r o l l a r y 2.2. In the HiIbert space 7-l, let G, G1 be bounded and boundedIy invertible self-adjoint operators, A a bounded G-self-adjoint and A1 a bounded Gl-self-adjoint operator. Suppose that the differences G1 - G and A1 - A are compact. If A is an open interval of the real axis such that A C p( A ) U cr+(A ) U a_,f (A ) and A does not contain any C-inner point of c~(A~), then A C p(At) Ua+(A~) U cr_,f(A~). In particular, the points of A cannot be accumulation points of the nonreal spectrum orAl or of points of a_ f(A1). In order to prove this corollary we write A - A = G -~ (GA - AG) and consider the analytic operator function K(A) := AG - GA and, similarly, KI(A) := AGz - G~A~. They satisfy the assumptions of the operator functions K and K1 in Theorem 2.1. It remains to observe that the spectrum of the pencil K coincides with the spectrum of the operator A, and A is a spectral point of positive or negative type of K or belongs to ~r_4 (K), respectively, if and only if it is a spectral point of the same type for the G-self-adjoint operator A or it belongs to cr_j(A), respectively, and that the same holds for the pencil K1 and the operator A1.


133

3. MONIC

3.1. Spectrum and numerical operator pencil

QUADRATIC range.

(3.1)

OPERATOR

PENCILS

In this section we consider the self-adjoint bounded

L(A) = A2A + AB + C,

in the Hilbert space 7{ under the assumption that

inf{](Ax, x)] + ](Bx, x)[ + [Cx, x)i : x E 7{1} > O,

(3.2)

where again 7{1 is the unit sphere of 7{. Note that assumption (3.2) is satisfied if at least one of the operators A, B, C is uniformly definite. For x E 7{, x ~ 0, we define the function ~x as follows: ~ ( A ) := A2(Ax, x) + A(Bx, x) + (Cx, x),

(3.3)

A E C.

Evidently, restricting A to R and if (Ax, x) :/: O, then the graph of 9~ is a parabola, if (Ax, x) = 0, it is a straight line. The assumption (3.2) implies that the function ~ , x ~ 0, is not identically equal to zero. By A we denote the numerical range of the pencil L, that is, the set of all real or complex zeros of all the functions ~ : (3.4)

A:={AEC: Theorem

3.1.

~(A)=0

for s o m e x E T { 1 } .

Under the assumption (3.2) the inclusion ~(L) c X

holds. Proof. If Ao E G(L), then there exists a sequence (xn) C 7/1 such that L(A0)xn --+ 0 or L(Ao)*x,~ -+ 0. This implies ~x~ (A0) -+ 0 if n --+ oo. Passing to subsequences, we may assume that for n --+ oo (3.5)

c~ := (Axn, xn) -+ a, ~n := (Bxn, xn) -+ fl, ~ := (Cxn, x~) --+ 5.

The assumption (3.2) implies that [al+l~]+]51 r 0, hence the polynomial ~(A) := aA2+~A+5 d o e s n o t vanish identically, and ~(A0) = 0. Since ~x~ (A) --+ ~(A) for all A E C, the Theorem of Hurwitz implies that there exists a sequence (AN) such that ~ (An) = 0 and AN -+ Ao, n --+ oo. [] 3.2. W e a k l y h y p e r b o l i c p e n c i l s . We introduce the discriminant of the function ~ : (3.6)

d(x) := (Bx, x) 2 - 4(Cx, x)(Ax, x),

x E 7{.

Evidently, ~x has only real zeros if and only if d(x) >_ 0 for all x E 7{1. If d(x) > 0 for all x E 7{, that is if all the polynomials ~x, x ~ 0, have only real zeros, then the pencil L is called weakly hyperbolic, see [M, Section 31]. In order to formulate a statement about the location of the nonreal spectrum of L we define the set I0 :=

2(Ax, x) : x E n 1, d(x) < 0

and the numbers %+ : = s u p / o , 70 := inflo, 7o : : max{lz+l, 17ol},

134


(Cz, ~) } (Ax, x) : x C T-Iz, d(x) < O .

71:--sup

Obviously, 71 > 0, and it is easy to see that 7o < v/-~. T h e o r e m 3.2.

If the assumption (3.2) is satisfied, then the following statements

hold: 1. If the pencil L is weakly hyperbolic, that is d(x) >_ 0 for all x E 74, then or(L) is real. 2. If d(x) < 0 for some x E 7/, then the nonreal spectrum of L is contained in the strip {A 9 C : 7o 0 for all x 9 7/, then A C ~, and hence ~(L) C ~ by Theorem 3.1. 2. Let A0 9 a(L) be nonreal. With the notations from the proof of Theorem 3.1 it follows that the polynomial ~o(A) must be quadratic, i. e. a r 0, and 4a5 >/32 as well as NA0 -

2ol"

Hence d(x~) =/3~ - 4 a ~ Q < 0 for sufficiently large n, and therefore

~o < - -

- -

Z" _0}, that is, jp[1 is the set of all x E x E A d 1 by l + ( x ) the zero of ~ which ~o~(l+) _ 0. Evidently, if l + ( x ) = A_(x). The following sets

~l:=~nn~,

7-I1 such that the zeros of ~x are real. We denote for for which ~o~(l+) _> 0, and by A_(x) the zero of ~x for ~o~ is a quadratic polynomial with a double root, then play an important role in the sequel:

A+ := {~+(x) : x e : . 1 } ,

A_ := {A_(x) : x 9 M 1 } .

In the following theorem, ~ := R tO {oo} is considered with the topology which makes it homeomorphic to the unit circle in C. If the operator A is uniformly positive, the


135

statements of this theorem are well known, see [L2], [M, Theorem 31.5]. The proof below is adapted from the proof for this case. T h e o r e m 3.4. Suppose that the assumption (3.2) is fulfilled and that the pencil (3.1) is weakly hyperbolic. Then A+ and A_ are connected subsets o f ~ and A+ N A_ consists of at most two points. If the operator A is boundedly invertible then either A+ and A_ are

bounded, or one of these sets is bounded and the other consists of two unbounded intervals. Proof. If (Ax, x) = 0 for x E 7-/1, then c~ is considered as a second zero of the linear function F~ which is denoted by A_(x) (A+(x), respectively) if the finite zero of 9~z is A+(x) (A_(x), respectively). Then the first claim follows from the fact that the functions A+ and A_ with values in ~ depend continuously ~ E 7t 1 and that 7/I is connected. In order to prove that A+ fq A_ consists of at most two points, we introduce the set A0 := (A E ]R :.~(A) = ~ ( A ) --- 0 for some x E 7/1} and sht)w first that (3.7)

A+ n A_ C Ao.

Indeed, assume (3.7) is false. Then there exists a Ao E A+ N A_ which does not belong to Ao. Hence there are elements u, v E 7/I such that ~(Ao) = ~(Ao) = 0, G(Ao) > 0, ~'~(Ao) < 0. By the Hausdorff-Toeplitz theorem the numerical range of the operator L(Ao) + iL'(Ao) is convex, and it follows that there exists an element x E 7-/1 such that ~(Ao) ~(Ao) = 0, hence Ao E Ao, a contradiction. If A+ M A_ would contain more than two points, then it would contain an interval (a, b). By (3.7), (a, b) C Ao. Fix some point c E (a, b). Then there exists an x E 7/1 such that (3.s)

=

0.

=

Since ~az is not identically zero, ~a~(c) ~ 0, and (3.9)

=

-

c ) 2.

Now choose d E (a, e). In the same way we find an element y E 7_/1 such that ~yl! (d) ~ 0 and (3.10)

qoy(A) = l ~ ( d ) ( A - d) 2.

We repeat this procedure for a number dl in (a, d) and an element Yl E 7'/t1. Then two of the numbers T"(c), ~ ( d ) , ~1 (dl) have the same sign, and renaming, if necessary, implies that (3.9) and (3.10) hold and that ~ ( c ) and ~ ( d ) have the same sign. For t E ]R the polynomial ~Pz+tv can be written in the form (3.11)

~+ty(A) = ~ ( A ) + t2~y(A) + 2t~(L(A)y, x).

If (L(c)y, x) ~ 0, then, multiplying y by a suitable complex number of modulus 1, we may assume that ~(L(c)y, x) has the same sign as p~(d). Then the three functions 9~(A),

i36

Aclamjan, Langer, MSlier

~v(;~), ~(L(A)y,x) have the same sign or are zero, the first two for all A, the last one for E (c - c, c + z) for some z > 0. Then the relations pz(d) ~ 0 and ~y(c) ~ 0 imply (3.12)

~+tv(A) # 0 for all t > 0 and A e (c - e, c + ~).

If, however, (L(c)y, x) = 0, then multiplying y by a suitable complex number of modulus one, we may assume that ~(L'(c)y, x) -- 0. Then

~(Z(A)y,x)

= ~(~ - ~)~,

where 7/---- ~(L"(d)y, x). We also may assume that necessary. Hence (3.11) has the form

t! ~?~v(c) > 0 by substituting - y for y, if

and again (3.12) follows. Since the quadratic polynomial ~ has a double zero at c, p~ changes its sign in passing through the point c. Choose A1, A2 e (c - e, c + r such that

(3.13)

~'(~1) < 0, ~'(~2) > 0.

Since

~+,~(~) = ~'(~) + t 2~(A) , ' + 2t~(L'(~)y, x), the relation (3.13) implies that for sufficiently small t > 0

(3.14)

~'§

< 0, ~ ' + ~ ( ~ )

> 0.

It follows from (3.12) and (3.14) that the polynomial ~x+ty(A) has a nonzero extremum on (A~,A2). Note that the sign in (3.12) coincides with the sign of ~ , that the sign of ~x coincides with the sign of p~, and that the signs of ~ and ~x+ty" coincide for small t. Therefore this extremum is either a positive minimum or a negative maximum. In both cases~ ~+t~ has nonreal roots, which is impossible because of the assumption that d(x) > 0 for all x C 7i 1. Now let A be boundedly invertible. Then the spectrum of the pencil L is bounded. We consider the pencil M(#) := #2 L ( _ # - I ) . Then 0 ~ a(M), and ~ ~ 0 belongs to a(L) if and only if # = A-1 belongs to a(M). If one of the sets A+, A_ would be an infinite interval, the corresponding interval A -~ for M would have zero as a boundary point, which would imply (see, e.g., [LMT]) that zero belongs to the spectrum of M, a contradiction. [] R e m a r k 3.5. In the middle part of the proof of Theorem 3.4 we have actually shown that for a not weakly hyperbolic pencil which satisfies the assumption (3.2) the points of int~ (A+ a A_) are accumulation points of the set of nonreal points of the numerical range A of the pencil L.

Adamjan, Lander, M611er

137

3.3. S p e c t r a l p o i n t s o f d e f i n i t e t y p e . In this subsection the results of Subsection 3.2 are partly generalized to the case that the pencil is not weakly hyperbolic; comp. also [Ba]. L e m m a 3.6. Suppose that the assumption (3.2) is fulfilled. Then each of the two sets A+ \ A_, A_ \ A+ is either empty, a single point, or a union of nondegenerate intervals.

Proof. We consider only the first set, for the second set the proof is analogous. Here and below we may assume for convenience that co ~ A+ \ A_. This can always be achieved by a fractional linear transformation of/~. If L is weakly hyperbolic, then A+ \ A-7_ is an interval in ~ by Theorem 3.4, i.e. either empty, or a single point, or a nondegenerate interval. Now consider the case that L is not weakly hyperbolic and let ,~ E A+ \ A_. Choose x E A/[ 1 such that ~o~(~) = 0. Since L is not weakly hyperbolic, there is y E 7/1 such that d(y) < 0. Let 7 : [0,1] --+ 7/1 be continuous such that 7(0) = x, 7(1) = y. Since d i s continuous on 7t 1, to = min{t E [0, 1]: d(7(t)) = 0} is well-defined. The functions A+(7(t)), A-(T(t)), t E [0, to], are continuous, and A_(v(t0) ) E A_ shows that A+(v(t0)) = # A. Therefore the closed interval with endpoints ~ = ,~+(7(0)) and -~+(7(t0)) intersected by ]R \ A_ is a nondegenerate interval containing A. [] 3.7. Suppose that the assumption (3.2) is fulfilled. Then

Theorem (3.15)

A+ \ A_ C a+(L) U p(L),

A_ \ A+ C (r_ (L) U p(L).

The proof of this theorem is based on the following lemma which contains a property used already in [Kr, p.132], [MMa, p.540], [BEL, L e m m a 8]. L e m m a 3.8. Suppose that the assumption (3.2) is fulfilled. Then for each A E A+ \ A_ there exist constants 7 > O, 5 > O, such that for all x E 7/1 the inequality [ ~ (~) I < 6

implies ~'(~) >_ % Proof. Step 1. In this part we prove the statement for A E A+ \ ~ and all x E 5t~ 1 with ~o~(~) = 0, that is A = A+(x). Assume that the claim is false, that is, assume that there exists a sequence (x,~) C 7-I1 such that ~o~,(A) = 0 and lim,,_+~ ~o" (A) _< 0. Since .~ tg A_, we have ~o~(A) _> 0, and therefore lim~_+~ ~o" (A) = 0. If (Axe, x~) ~ 0 for all n, then (3.16)

~ . ( ~ ) = 2(Ax=,x~)A + (Bx~,X~) = (Ax~,x~)(A - A_(x~)),

and since inf= IA - A_(x~)[ > 0, we must have (Ax=,x~) --~ 0 if n --+ co. The second expression in (3.16) implies now (Bx~, x~) --+ O, but then also (Cx~, x~) -+ 0 if n --+ ~ since ~ = (A) = o. This contradicts (3.2). If (Axn, x~) = 0 for all n, then for n -+ ~ we obtain

(Bx ,

=

-+ o,

(Cx., x.) = - a ( B x . , x . ) --+ o,

which again contradicts (3.2). Step 2. Let ,~ E (inteA+)\ X-2-_. Then there exists an e > 0 such that [ A - e, A + e ] C A+ \ A_, and we find an x E 7/1 such that ~oz(,~ + e) = 0, ~o~(,~ + e) > 0. The second zero

138


A_(x) of ~ox lies in A_, hence on the interval (A - e, A + r the function ~o~ is negative and, in particular, ~o~(A) < 0. Similarly, we find a y E ~/1, such that ~o~(A-r = 0, ~,o~(A-e) > 0 and ~y (A) > 0. Now we consider again the numerical range W of the operator L(A) + iL' (A). From what was shown in Step 1 it follows that c := inf {d : id E W } is positive. Since W is convex, it admits a support line at ic. This line has a finite slope as c,o~(A) takes positive and negative values. That is, there is a real number 77 such that >

+ c,

x e n 1.

Choosing now 5 > 0 such that 1~?15< c/2 yields > e / 2 =: v,

x c

Step 3. Assume that A+ \ A_ consists of one point A. By the proof of L e m m a 3.6 we know that the problem is weakly hyperbolic, and the connectness of A+ implies A+ = {A}. Therefore ~o~(A) = 0 for all x E ~/1 and the statement follows from Step 1. Step 4. Assume that A+ \ A_ consists of more than one point and let A E A+ \ A_. There is a sequence (An) in A+ \ A_ such that An --+ A as n --+ oo. By L e m m a 8.6, A+ \ A~_ is a union of nondegenerate intervals, and we may therefore assume that As C (int~A+) \ A_. Now let x E 7t 1 such that Ic.ox(A)] < ~, where 5 and V are as in Step 2. Then ]~,o~(An)I < 5 for sufficiently large n, and therefore ~ox(n) ~A -> 7 for these n by Step 2. For n -+ oo this gives ~'(A) > 7.

[]

Proof of Theorem 3.7. Let A e (A+ \ A-~--)D or(L) and let (x~) be a sequence in -/_/1 such that L(A)xn --+ 0 i f n --+ oo. Then lim~o~(A) -~ 0, and from L e m m a 3.8 we infer that lim~ot,~(A) _ V, where V > 0 is independent of the sequence (xn). But this means that A C a+(L). The proof of the second inclusion in (3.15) is similar. C o r o l l a r y 3.9. Suppose that assumption (3.2) is fulfilled. Then points of A \ N and hence also points of the nonreal spectrum of L cannot accumulate in points of

(3.17)

(a-7\

u (A_ \ A+).

Proof. Let (An) be a sequence in A\]R Such that An --+ A E N as n -+ oo. T h a t is, there exists a sequence of elements Xn E ./tl such that ~ox~ has two nonreal zeros An, An with An --+ A as n -+ oo. Then ~ox,(k) = (gxn, xn)(A - kn)(A - A-~) --+ 0, ~o" (A) = 2(Ax~,xn)(A - RAn) -+ 0 as n -4 co. In view of Lemma 3.8, A • A+ \ A_. Similarly, A r A_ \ A+. The claim about the nonreal spectrum follows now immediately from Theorem 3.7. [] From [LMT, Theorem 2.9] we obtain the following result (comp. [GLaR, Theorem 10.15]).

C o r o l l a r y 3.10.

Suppose there is a point A0 E C such that inf [(L(Ao)X,X)l > O. xET-tl

Then every ]~-boundary point of (-~+ \ -~-_) and of (A_ \ A--~+) belongs to a(L). Evidently, the assumption of this corollary implies (3.2).


139

3.4. C o m p a c t p e r t u r b a t i o n s . The following result about compact perturbations is an immediate consequence of Theorem 2.1 and Theorem 3.7. T h e o r e m 3.11. Let L and L1 be seIf-adjoint bounded operator pencils in the Hilbert space 74 of the form (3.1): (3.18)

L(A) = A2A + AB + C,

nl (A) = A2A1 + AB1 + C1,

such that the following assumptions are satisfied." 1. The differences A1 - A, B1 - B, C1 - C are compact operators. 2. The pencil L satisfies the assumption (3.2). 3. p(L) A p(nl) n R r O. Then an open i~terval X which is contained in J+in) u (p(L) n ~) (in ~_ (L) U (p(L) n ~), respectively) and for which at least one of its points does not belong to intc or(L1), is contained in ~+(L~) u ~-z(L1) U p(L~) (in ~_(L1) u ~+,~(L~) V p(L~), respectively). Zn particular, the points of I cannot be accumulation points of the nonreal spectrum of L1. Proof. If both or(L) and a(L1) are bounded, then this is an immediate consequence of Theorem 2.1. Otherwise, choose a real number a E p(L) n p(Lz). Then the substitution # = (A - a) -~ transforms L(A) and L~ (A) into self-adjoint quadratic operator pencils M(#) := #2L(#-~ + a) and M I ( # ) : = #2L2(#-t + a) with bounded spectra. Straightforward relations between the spectral properties of L and M and L1 and M1 complete the proof. [] Corollary 3.12. that intca(L1) = 0. Then

Let the pencils L and L1 be as in Theorem 3.11, and suppose

int~ (A+ \ A_) C ~+(L1) U ~-j(L1) U p(L1),

int~ (A_ \ A+) c r

U ~+z(L1) U p(L1).

In particular, the points of the set (3.17) cannot be accumulation points of the nonreal spectrum of LI. In the following corollary, if A+ and A_ are both unbounded, then co is considered to be a common boundary point of them. Corollary 3.13. Let the pencils L and Lz satisfy the same assumptions as in Theorem 3.11, and suppose that the pencil L is weakly hyperbolic, that is, d(x) > 0 for all x E 74. Then, with the possible exception of the common boundary points of A+ and A_, all the real points of a(L1) belong to a+(Li) Ua_ (LI) Ua-,I(L1 ) Ua+,I(L1), the nonreal spectrum of Lz is discrete and its only possible accumulation points are the (at most two) common boundary points of A+ and A_. If, additionally, the operator A is boundedly invertible, then the nonreal spectrum of Lz is bounded; irA is uniformly positive or uniformly negative, then the nonreal spectrum of Lz can accumulate in at most one point; if, additionally, A+ and A_ have a positive distance, then the nonreal spectrum of L1 is finite. Proof. We know from Theorem 3.2 that r C R since L is weakly hyperbolic. By Assumption 3 of Theorem 3.11, p(L) is connected. The compactness of the operators LI(A) - L(A) implies that LI(A) is a Fredholm operator for A C p(L). But p(L) N p(L~) ~ 0 implies that p(L) n a(L1) is a discrete set. Therefore the assumptions of Theorem 3.11 are satisfied, and all the results follow from that theorem and Theorem 2.1. []

140


4. A QUADRATIC OPERATOR PENCIL WITH NONINVERTIBLE LEADING COEFFICIENT

4.1. L i n e a r l z a t i o n a n d t r a n s f o r m a t i o n consider a self-adjoint quadratic pencil (4.1)

to a pencil with bounded

s p e c t r u m . We

L(A) = A2A + AB + C,

where the coefficients are in the Hilbert space (4.2)

7 / = 7-ll @ 7/2,

given by the block operator matrices

A=

(4.3)

, B = \B;2 B22]' C= [,C, 2 i + C i 2 .

Since the leading operator A is not invertible, the spectrum of L can be unbounded and even the whole complex plane. In order to exclude the latter situation we always suppose that there exists a real point a such that L(a) is invertible. For simplicity we also suppose that a is positive and such that B + a is uniformly positive. In a first step this pencil is iinearized. To this end the component 7/1, where the operator at Az is invertible, is added to 7/, that is, we consider the Hilbert space 7i := 7/|

7/1

and in it the operators

0);

here Q denotes the embedding of 7/1 into 7-I according to (4.2), whence projection in 7 / o n t o 7/1, i. e. QQ* = A. The linear pencil (4.4)

QQ* is the orthogonal

N(A) := AS - T

is then equivalent to a canonical extension of L(A). tn fact, it is easy to check that the relation

:

(4.5) .

\ -~Q*

,)

:

_

0)

holds. Since

L(a) is boundedly invertible, also the operator N(a) has this property, and

we write N(A) =

N(a) + ( A - a)S.

In the space 7 i we introduce the bounded operator (4.6)

A :=

f -L(a)-I(aA + B) L(a)-IQ ) -1V(a)-IS = \ _Q.L(a)_IC _aQ.L(a)_l Q .

Then, with #(A) = (), - a) -1 for x, f E 7-t and A ~

N(;~)~ = f r

a, the following equivalences hold:

( S + t,N(a)):~ = (;~ - a ) - ~ f

(4.7) 4==> (A - # ) x = - ( A -

a)-lN(a)-lf.


141

If we equip the space 7t with the inner product

Ix, v] := (N(a).,, y),

~, y e n ,

it becomes a Krein space, which will be denoted by K:. It is easy to see that A is a bounded self-adjoint operator in K:. The relation (4.5) implies that the spectra (and also the point spectra and the approximate point spectra) of the pencils L(A) and N(A) coincide. Consider now A E a+(L) and let (x,~) be a sequence of elements ofT-L1, x,~ =

Y,~ E ~ @ ~ 1 , such that [IN(A)x~]I ~

0 if n -4 e~. Then

-AQ*x,~ - y,~J

Hence L(A)x~ -+ 0 and AQ*xn + yn -40. The latter implies limllz~[i > 0 since otherwise a subsequence of (xn), and hence also of (y~) and of (xn) would tend to zero, which is impossible since [Ix~[[ = 1. Further, lim(L'(A)x~, x~) > 0 since A E a+(L). Then Xn

= \k

-Qx.

]'

-~Q*x.

Xn

+o(1)=(L'(;,)x,,,x,~)+o(1)

if n -4 c~. This shows that A E a + ( N ) . The other inclusion a+(N) C ,:r+(L) can be shown similarly, and it is now also easy to see that the eigenvalues of positive type of L and N coincide, and also that a_ (L) = ~_ (N). In order to prove the relation cr_,/(L) = c~_,I(N ) we observe that (4.5) implies

Next we use for the function -L(A) -1 in a neighborhood of a point a E e_.f(L) of index of negativity ~; the decomposition (2.5): - L ( A ) -1 = V(A) + n0(A), where V E N~(~) and the operator function V is analytic outside the closed interval including eo and vanishes at 0% and the operator function Ro is analytic on A. The function V, as a function of the class N~(7/), which is holomorphic at ee and vanishes there, admits an operator representation g(.~) = r * ( D - ~ ) - l r with some self-adjoint operator D in a Try-space II~ and a bounded linear operator from 7-/ into II~, see [KL 2]. This representation can be chosen minimal, that is, such that a~ = c.l.s. { ( D - a ) - l r x l x e n , a s p(D)}.

142


Then we obtain for N(A) -1 a representation

( r*(D- A)-IF

- F * D ( D - A)-IFQ ) - N ( A ) - I = \ - Q * F * D ( D - A)-IF Q*F*D2(D -

(4.9)

a)-lrQ/+""

where -.- stands for a function which is holomorphic on A. If we denote the first function on the right hand side of (4.9) by N1 we obtain with xi = ( xYii )

E ?-l, Ai E p(Di), i =

1, 2 , . . . , m: ( N I (.'~i) - N I ( )~j)*

)

and it is now easy to see that N1 E N~(7-/), where ~ is the index of the space H~ and hence also the index of the class N,(7-/) to which V belongs. Further, the relation (4.7) implies that the point A belongs to the spectrum (point spectrum, approximate point spectrum, respectively) of the pencil N if and only if the point# = (A - a) -1 belongs to the spectrum (point spectrum, approximate point spectrum, respectively) of the operator A. If, e.g., A E a+(N) and (x~) is a bounded sequence such that lim][xn]] > 0 and limN(A)x~ = 0, then it follows from the above calculations that x,~ =

-.~Q*x,~

+ o(1), limllx~ll > 0 and that lim(L'(A)x,.z~) > 0. Now we obtain

=

((0

- 0I )

(aQ*

~) ( - A Q ' x , ) '

( - A Q * x , ~ ) ) +~

\ k

= ((Z(a) - L(A))xn, Xn) - (a - A)2(Axm x~) + o(1) = (a - A)(2A(Ax~,x~) + ( B x , , x ~ ) ) + o(1) :

-

+

o(1);

here we have used that (L(A)xn, x~) = o(1). Now it follows easily that AEa+(N),Aa,p=(A-a) -lea-(A) and, similary, that A < a is an eigenvalue of positive type of N if and. only if # = (A - a) -1 is an eigenvalue of positive type of A and also that A E a - , l ( N ) , A < a, if and only if # = (A - a) -1 C a_,f(A) etc. Thus we have proved the following theorem. T h e o r e m 4.1. Let L be the pencil (4.1) with coefficients given by (4.3). Suppose that there exists a number a > 0 such that the operator L(a) is invertibIe and put #(A) = (A- a)-l, #(ce) = O. Then the following relations hold between the extended spectrum of the pencil L and the spectrum of the operator A: or(A) : #(5(L)), ap(A) = #(Sp(L)), a~ss(A) = #(5~s(L)).


143

Further, if A < a, then A 9 a•

~==>#(A) 9 cr•

A 9 a~,i(L) 4==> I-*(A) 9 a~,i(A),

and if A > a, then A 9 a•

#(A) 9

~

A 9 cr=~,i(L) r

a:F(A),

#(A) e a •

4.2. A s p e c i a l p e n c i l a n d its l i n e a r i z a t i o n . In this subsection we consider the pencil Lo given by (2.1), i.e., we suppose that B12 = 0, B22 = 0, C n = 0, C12 = 0, C~2 = 0, and that B n = /3o with a bounded self-adjoint operator Bo. Then we write So and To instead of S and T, and with respect to the decomposition 7i = 7-ll @ 7/2 @ 7/1 the pencil No(A) := ASo - To becomes ....

AoO No(A) = k - A 1 The factorization

(a I

No(A)=

0 I 0

C~ 0 0

I o A21 0 0 I 0 0 -I

AI

o)

0 0 I 0

implies that a(No) = ~ ( - B o ) U {0}. T h e positive number a as chosen at the beginning of Subsection 4.1 does not belong to or(No). Then

(a o ~

-aI

(4.1o)

No(a) = k - a 1

I 0

is a self-adjoint and invertible operator. Further,

No(A) = No(a) + (A - a)So = No(a)(A - a) ((A - a) -1 + No(a)-Z So) . In the space art we introduce the bounded operator Ao := - N o ( a ) - l S o . It follows from

(4.6) that (4.11)

Ao

0

,

o -(Bo + ~x) -~ ] and with # = (A - a) -z for x, f E 7 / a n d A ~ Ao, (4.7) yields the equivalences (4.12)

No(A)x = f 4==> (Ao - #)m = - ( A - a ) - l Y o ( a ) - I f . If we equip the space 7i with the inner product

(4.13)

[m,Y]o := ( N o ( a ) m , y ) ,

m , y e 7/,

it becomes a Krein space, which will be denoted by/Co. Evidently, Ao is a bounded selfadjoint operator in K:o, and Theorem 4.1 implies that the extended spectrum of the pencil 1 N o coincides with the set f(a(Ao)), where f is the function f(t) = ~ + a. On the other hand, from (4.11) we obtain cr(Ao) = { - a -1} U {0} U cr ( - ( B o + a I ) - l ) 9

144


It is easy to see that outside 0, - a -1 the point spectrum (discrete spectrum, or the continuous spectrum) of Ao and of -(B0 + aI) -1 coincide. The operator Ao is definitizable with the definitizing polynomial p(t) - t(t + a-l). Indeed,

p(Ao) =

(--ao01I 0 (aB~ (i 0 0 -(Bo + aI) -1 ]

0 (aBO+Oa2I)-I I a-lI 0 So(aBo + a2I) - l j

0 -aBo(aBo + a2I)-2,]... and

No(a)p(Ao) =

(i 00~ (aBo + a2I) -1

>_O.

The possible critical points of Ao are - a -1 and 0. They are eigenvalues of Ao, and their multiplicities coincide with the dimensions of the spaces 7tl and 7/2, respectively. We describe these eigenspaces in more detail. The root subspace/:o(A0), which is also the geometric eigenspace of Ao at 0, is

~o(Ao) = {(o ~ o)T ~ ~ 7/~} Hence, by (4.13) and (4.10) it is a uniformly positive subspace of/Co. The (geometric) eigenspace/:~ of Ao corresponding to - a -1 is

~oo_l(Ao) = {(xl o o)T: xl ~ 7/1}, and associated vectors exist if and only if kerBo r {0}. Indeed, consider the equation (Ao + a - l x ) x =

a-~I

0

0

Bo(aBo+ a2I) -1

~2

=

, :1 e 7/1 \ {0}.

Yl .

It has a solution if and only if kerBo 7~ {0} and fl E kerBo, and then this solution is, modulo an eigenvector, (0 0 a2fl) T. It follows that : Xl C 7-/1, Yl E kerBo , and the indefinite inner product on this subspace Jg-a-1 (Ao) becomes (4.14)

[(Xl 0 y l ) T , ( X i

0

yl)T]o=a(BoXl,Xl)--2a~(yl,Xl)--[lYl[] 2 = ((aBo + a2I)z~, z~) - Ilazl + yl]l 2,

If kerBo = {0}, then the expression on the right hand side simplifies to a(Bozl, xl). Finally, the orthogonal companion ~2 in/Co of the sum of the subspaces /2o(Ao), s (Ao) is also invariant under Ao; this subspace is

~_- {(~ o ~o~)~: ~1 ~ ( ~ o / }


145

with the inner product

[(xx o 13oxl)T,(xx o 13oxl)T]o=-a(13oxx, )-j)13o jr 2 The above considerations imply the following theorem. In order to formulate it we introduce the nonnegative operators 13+ := 13oEo((0, c~)) and B o := -13oEo((-~z, 0)) in ~1, where Eo denotes the spectral function of the operator 13o. T h e o r e m 4.2. Let a > 0 be such that 13o + a is a uniformly positive operator. Then: (i) a(Ao)= { - a -1,0} U { - ( A + a) -1 I A e a(Bo)}. (ii) 0 e c~p(Ao) n ~r+(Ao). " (iii) Cr=F(Ao) \ {--a -z, 0} = {-(A + a) -1 [ A e a(• ~) \ {0}}. (iv) - a -z is an eigenvaIue of positive type (of negative type) of Ao if Bo is a positive (negative, respectively) operator. (v) If Bo is a strictly positive(strictly negative, respectively) operator, then - a -1 E a+(Ao) (e a- (Ao), respectively). (vi) - a -1 is a critical point of Ao if and only if13o is not strictly positive or strictly negative. C o r o l l a r y 4.3. Under the assumptions of Theorem 4.2 the point - a -1 is the only possible critical point of the operator Ao. To explain the claims of Theorem 4.2, we introduce the numbers fii := rain a(Bo) and fls := maxq(130). The spectrum of A0 contains the point 0 as an isolated eigenvalue of positive type. Moreover,

~(Ao) \ {0, -~-~) c [ - ( ~ + ~)-~, - ( ~ + ~)-~J. and ( - a - I , -(fls + a) -1] C ~-(A0) U p(Ao),

[-(fli + a) -1, - a - i ) C a+(Ao) U p(Ao).

Here e.g. an interval (c, d l with d < c is the empty set. More detailed, it is convenient to distinguish three cases: i) fli < 0 < fls, then ( - o o , - ( f l i + a) -1) C p(Ao), [-(fli + a) -1, - a -1) C cr+(Ao) U p(Ao),

( - a -1, -(fls + a) -~] C a-(Ao) U p(Ao), (-(fls + a) -z, O) C p(Ao), o c ,~+(Ao), (0, ~ ) c p(Ao), ii) fli < fls < O, then

(-cr

+ a) -1)

C

p(Ao),

[- (/~ + a) -~, -(~, + a) -~] c ,,+(Ao) u p(Ao), (-(fl, + a) -1, - a -1) C p(Ao), - a -1 e a-(Ao),

( - a -1, O) C p(Ao), 0 e ~r+(Ao), (0, cx)) C p(Ao), and the relations for the case 0 < fl~ < fls are similar.

146


Let 7-/~ := ~

and define the subspaces

}, Then the next theorem follows easily from the previous considerations. T h e o r e m 4.4. Let a > 0 be such that Bo + a is a uniformly positive operator. Then: (i) /5+ ( E_, respectively) is a positive (negative, respectively) invariant subspace of Ao in ]Co and a(Ao ] 1:_) C [-a -1, 0), a(Ao I L+) C {0} U ( - c o , - a - i ] . (ii) The set s (Ao) + s + f~- is dense in leo, and it coincides with leo if and only if 0 is not an accumulation point of a(Bo). (iii) - a -~ is a singular critical point of Ao if and only ifO is an accumulation point of a(Bo). Proof. It only remains to prove (iii). To this end we consider the resolvent of A0:

0

0

- ((Bo + a• -1 +

-1

If A is an open interval of the real axis the closure of which does not contain 0 and - a -1, then from the inversion formula we find that the spectral projection Eo(A) of the operator A0 admits the representation

Eo(ZX) =

0

0

E0(-A -l-a)

and the claim follows since the integral in the upper right corner becomes singular if 0 is an accumulation point of or(B0) and A approaches - a -1 from the same side as the spectrum of B0 does. 4.3. C o m p a c t P e r t u r b a t i o n s . Now we return to the pencils L and N from (4.1), (4.3), and (4.4) and the corresponding self-adjoint operator A = - N ( a ) - I S from (4.6). We suppose that the following assumptions are satisfied: (a) B n -- B0 + B~I with a compact operator B~I; (b) The operators B12, B22, Cn, C12, C~2 are all compact. Then also the difference N ( A ) - N0(A) is compact, and we can choose a > 0 such that B + a is uniformly positive and that, additionally, L(a) and N ( a ) are invertible.


147

Recall that the operator A from (4.6) is self-adjoint in the Krein space K: which is the space 7/equipped with the inner product (4.15)

Ix, y] =: (N(a)x, y),

x, y e n .

The extended spectrum of the pencil N coincides with the set f(a(A)): ~ ( N ) = f ( a ( A ) ) , where f is the function f(t) = a + t -1. The difference A - A0 and also the difference of the corresponding Gram operators N(a) - No(a) are compact operators. Therefore Corollary 2.2 can be applied, and from Theorem 4.2 we get the following result. T h e o r e m 4.5. Suppose that for the pencil L from (4.1), (4.3) the conditions (a) and (b) are satisfied. Further, let a > 0 be such that Bo + a is a uniformly positive operator and that L(a) is boundedly invertible. Then for the spectrum a(A) of the self-adjoint operator A in (4.6) the following statements hold: i) The essential spectrum Oess(A) is the set {;, e R : - ( a - :,)-1 e a0ss(B0)} augmented by 0 if dimT-/2 = oo and by - a -1 if dimT/z = cr ii) ( - e c , - a =1) c p(A) U a+(A) U a_,f(A), ( - a - l , 0 ) c p(A) 0 a_(A) U a + j ( A ) , and (0, oo) C p(A) U a+(A) U a _ j ( A ) . iii) 0 e a+(A) U a _ j ( A ) . iv) If Bo is uniformly positive, t h e n - a - Z E a+(A) U a _ : ( A ) , if Bo is uniformly negative, then - a -1 e a_(A) U a+j(A). v) The only possible accumulation point of the nonreal spectrum of A is - a -1. If Bo is uniformly definite, then the nonreal spectrum of A is finite. The corresponding theorem for the pencil L is now a consequence of Theorem 4.5 and Theorem 4.1. T h e o r e m 4.6. Suppose that for the pencil L from (4.1), (4.3) the conditions (a) and (b) are satisfied. Further, suppose that there exists an a > 0 such that L(a) is boundedly invertible. Then the following statements hold: i) The extended essential spectrum of L is aess(B0), augmented by oo if dimT-/2 = eo, and by 0 if dim~l = 0o. ii) (-oo, O) C p(L) Ua_(n) Ua+j(L), (O, oc) C p(L) Ua+(L) l..Ja_,f(i). iii) cr e Y+(L) U Y_,l(L). iv) If Bo is uniformly positive, then 0 E a+(L) U a_j(L), if Bo is uniformly negative, then 0 e a_(L) U a+j(L). v) The only possible accumulation point of the nonreal spectrum of L is O. If Bo is uniformly definite, then the nonreaI spectrum of L is finite.

5. AN EXAMPLE

FROM

MAGNETOHYDRODYNAMICS

5.1. The problem. The following example is taken from Section 7.10 of the monograph [Li] of A. Lifschitz. It leads to a quadratic operator pencil with noninvertible leading operator of the form (4.1), (4.3). Suppose that an equilibrium state of a non-ideal incompressible plasma, ~yhich is surrounded by a perfectly conducting wall, is described by the density p0, the total pressure

148


p~, the magnetic field Bo and the resistivity 70. in the single-fluid approximation the perturbed quantities which describe small oscillations of the plasma near the equilibrium state satisfy the equations

p(~, t) = po(~) + ~pl(~, t), v(x, t) --~ CVl(X , t), p*(~, t) = p~(~) + ~p~(~, t), B ( x , t ) = Bo(x) + cBl(m,t), ~(~, t) = ~o(~) + ~ ( ~ , t), where v(m, t) is the velocity vector. We consider the simple case of a resistive gravitating plasma layer

{x = (x,y,z) : O < x < d, O