Spectral Properties of a Multiplication Operator

Math. Nachr. 178 (1996), 135 - 156

Spectral Properties of a Multiplication Operator

By VOLKER HARDT and EKKEHARD WAGENFUHRER of Regensburg

Dedicated to Prof. R. MENNICKEN on occasion of his 60th birthday

(Received January 16, 1995) (Revised Version August 10, 1995)

Abstract . In this note we study the spectral properties of a multiplication operator in the space Lp(X)” which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities” .

1. Introduction

Let ( X , C , p ) be a measure space and A = (&ij)Tjz1 be an m by m matrix the elements &j of which are complex-valued measurable functions on X . Further, 1 5 p < 00 being fixed, let L p ( X ) denote the space of all p-integrable complex-valued functions on X modulo the functions which are zero almost everywhere with respect to p. Then the operator A = T ( a ) defined by

( A f ) ( 4 := a<,, f($) (. E x , f E D ( 4 c Lp(Wrn)

with domain D ( A ) = { f E Lp(X)rn : A( . ) f( a ) E Lp(X)rn}

is a linear operator in L p ( X ) m and is called a multiplication operator. I t is well-known that D(A) is dense in Lp(X)rn and A is a closed operator. The informations on the spectrum of such a multiplication operator are needed in several fields of application, one of which is the theory of semigroups (see [7], [14], [20]). As an example, we consider an m by m system of evolution equations

- w ( t , 2 ) d = P(&)w(t ,s) ( t l 0 , z E R ” , w(0, * ) = 210, at 1991 Mathematics Subject Classification. Primary 34B05; Secondary 47B99. Keywords and phrases. Differential equations, boundary value problems in ODE’S, multiplication

operator.

136 Math. Nachr. 178 (1996)

where P ( & ) is an m by m matrix of partial differential operators with constant coefficients and vo E L2(Rk)", and we are interested in solutions v ( t , z ) such that v( t , . ) E Lz(R'))" for all t 2 0. Then Fourier transformation with respect to z leads to a system

in which now A is a multiplication operator in L ~ ( n t ' ) ~ and $0 is the Fourier- transformed of VO. The operators A in such examples are unbounded, in general. According to the Hille-Yoshida Theorem the main question in the theory of semigroups are the localization of the spectrum of A and the behaviour of the resolvent ( X I - A)-' as X tends to + 00.

In this paper we are interested in questions of somewhat different type because our investigations will be needed for the spectral analysis of operators

.;l(t) = A u ( ~ ) , ~ ( 0 ) = 6 0 ,

in which A , B, C are ordinary or partial differential operators and D is a bounded multiplication operator in Lz(R), where R is an interval in R or a domain in Rd, d 2 2, respectively. Operators of this form arise from problems in magnetohydrodynamics, astrophysics and fluid mechanics (see [l], [lo], [18], for example), and the physicists are interested to know the essential spectrum. Such operators L have been investigated in [2], [17] and [13] in the case of ordinary differential operators, and a research of the partial differential operator case has begun at present, see [9].

If B is a linear operator in some Banach space H , e(B) denotes the resolvent set of B which is the set of all points X E G such that B - X I is invertible and ( B - X I ) - 1 is bounded. By o(B) := G \e(B) we denote the spectrum of B, N ( B ) denotes its kernel and R(B) its range. The point spectrum op(B) is the set of all eigenvalues of B and Lx(B) denotes the algebraic eigenspace of B for X f op(B). If, additional, B is a densely defined closed operator we define

nu1 ( B ) := dim N ( B ) , def(B) := codimR(T) ,

these being finite numbers or 00, and, if both are finite,

ind ( B ) := nu1 (B) - def (I?)

(see [IS]). Recall that B is called a Fredholm operator if nu1 ( B ) < 00 and def ( B ) < 00.

If B is a Fredholm operator, then it follows from def ( B ) < 00 that R(B) is closed (see [ll]). The essential spectrum aess(B) is the set of all points X E 42 such that B - X I is not a Fredholm operator. It is well-known that aess(B) is a closed subset of a(B) (see [ll], [16], for example). We remark that this definition of the essential spectrum is following T. KATO [16] and is the same as in [2] but different from the definition in [17], [9] and [13]. In those papers the essential spectrum is the set of all points of the spectrum o(B) which are not isolated normal eigenvalues of finite algebraic multiplicity, where an eigenvalue X E op(B) is called normal if H = HO + Cx(B) and B - X I maps Ho bijectively onto itself. The latter "essential spectrum" will be denoted

Hardt/Wagenfuhrer, Spectral Properties of a Multiplication Operator 137

by und(B) in the present paper. Obviously, aess(B) C and(B) in any case, and for a multiplication operator both sets coincide as we see in Section 3.

Section 2 is concerned with a general discussion of the spectrum of an unbounded multiplication operator A in Lp(X)m, where X is a measurable space with the finite subset property. The first problem is the characterization of the spectrum by means of the spectra of the matrix A(,), 2 E X. We have adopted a result of HOLDERRIETH [14] from 1993 in our Proposition 2.3. In the special case that A is a bounded multiplication operator and p = 2, this result can also be derived from a paper of DUNFORD [6] from 1966, in which more general multiplication operators in a space H", H a Hilbert space, are discussed. In the remaining part of Section 2 we describe the point spectrum a,(A) and discuss some properties of isolated spectral points.

In Section 3 we discuss the special case that A is a multiplication operator in the Hilbert space L2(CL)m, where now R is an open subset of Rd (d 2 1) equipped with the Lebesgue measure (or, more general, a Lebesgue measurable subset of IRd with p ( 0 ) > 0). For this case we obtain the equality a ( A ) = aess(A) which has been needed in [2, Theorem 4.51 for the spectral analysis of an operator L mentioned above. This spectral property is based on the implication

p ( M ) > 0 * dim L2(M) = 00

which is valid for the Lebesgue measure. We further give a necessary condition for so-called approximate eigenvalues of A and show that each spectral point of A is an approximate eigenvalue, if A is bounded.

In Section 4 we apply the preceding spectral analysis to certain boundary eigenvalue problems in ordinary differential equations which are rational in the eigenvalue parameter A. Such problems can be reduced to the spectral analysis of operators L as above in which now B and C are multiplication operators given by rectangular matrices. We are able to generalize some results of the paper [17] in which B , C and D are only scalar functions. Further generalizations of results in [17] concerning selfadjoint problems can be found in [13]. Additionally, we give an example which shows that the sets aess( l ) and Vnd(L) do not coincide, in general. In this example the operators B and C are finite-dimensional operators instead of multiplication operators, because we do not have an analogous example with multiplication operators to date.

The present paper is based on a part of the doctoral thesis of the first author [13].

2. Basic properties

In this section (X, C, p) is a measure space with the finite subset property, i. e., for all M E C with 0 < p ( M ) 5 00 there exists a subset N 2 M , N E C with 0 < p ( N ) < 00. We use the following notations

meas ( X ) := { f : X --f C measurable with respect to p } , M ( m x m, S) := { m by m matrices with elements in S} ,

138 Math. Nachr. 178 (1996)

where S is an appropriate set of func,tions or numbers. Now for 1 5 p < 00, let L p ( X ) m be equipped with the norm

where 11 .I\ denotes a fixed norm in C”. The following result is well-known - see [14], for example.

Proposition 2.1. Let 2 E M ( m ~ m , meas(X)) and A = T ( A ) be the correspond-

1. A is bounded if and only if A E M (m x m,L,(X)).

2. If

ing multiplication operator in L p ( X ) m . Then

E M ( m x m, L , (X) ) , then the norm of the operator A equals

(A( = esssup{ \lA(z)\\ : 2 E X}, where the norm llBll of a matrix B E M ( m x m, C) is given b y

IlBll = SUP {IIBYII : Y E C r n , llYll = I} *

In the following we give a first characterization of the resolvent set of a multiplication operator. Without loss of generality we consider the point X = 0.

Proposition 2.2. Let A E M ( m x m, meas (X)) and A = T ( A ) . W e consider the following statements a) 0 E @(A) , p) A(z) is invertible for p-almost all x E X and

esssup{IlA(z)-’II : x E X } < co,

y) essinf { ldet (a(,)) I : x E X } > 0 . Then, in any case

a) - a) * 71,

a) * P ) - 7 ) .

and if /3) is valid then A-l = T (a-’). Moreover, if a E Len

This proposition can be concluded from the proof of Proposition 1 in [14], a more detailed proof for the case A E M ( m x rn, L,(X)) can be found in [13], for example. The next proposition literally correspondens to Proposition 1 in [14].

Proposition 2.3. Let A = T ( A ) with A E M ( m x m, meas(X)) and suppose @(A) # 0. Then

= n u .(Aw). M E I :

p ( X \ M ) = O 2 E M


We note some direct consequences of the preceding results. At first, if @(A) # 0 then

Further, owing to Proposition 2.2, a) =$ r ) , for any X E C

(2.2)

If a E M ( m x m, L,(X)), then also the converse implication in (2.2) holds.

X E e(A) a essinf (Idet (a(,) - X I ) I : 2 EX} > 0 .

Next we turn to the characterization of the point spectrum of A which is defined by

c p ( A ) = { A E C : A - X I not injective}

= { A E C : 3f E Lp(X)m\{O} with ( A - XI)f = O } .

Lemma 2.4. Let a E M ( m x m, meas (X)) and set

I?' := {X E X : deta(X) = 0) (E C) .

Suppose p ( r 0 ) > 0. Then there exist Q1, Q2 E M ( m x m,meas(rO)) with Q1(z), Q2(z) unitary fur all z E I?' such that f o r x E r0

where Rl(z), &(x) are m by m - 1 and m - 1 by m matrices, respectively.

P r o o f . The well-known Schur factorization theorem says that for every B E M(m x m, C ) there exists a unitary matrix Q such that Q*BQ is upper triangular. According to AZOFF [3], such a Q can be chosen Borel measurable as dependent on the elements of B. By similar arguments as in the paper [3], we can conclude that each non-invertible m by m-matrix B can be transformed into an upper triangular matrix Q*BQ with zero in the upper left corner, where Q is also Borel measurable with respect to B. Then the first assertion of the lemma is clear when we choose Q l ( x ) that Q which corresponds to B := A ( x ) for 2 E I?'. The second assertion is proved in an analogous way. 0

Theorem 2.5. Let X E C, A E M ( m x m, meas (X)). Let FA be defined by

FA := {X E X : det (a(,) - X I ) = 0 )

140 Math. Nachr. 178 (1996)

and A = T ( A ) . Then

X f uTp(A) if and only if p ( r X ) > 0 .

P r o o f . At first, let X E up(A) and f E Lp(X)" be a corresponding eigenvector. Then there exists a set M E C with p ( M ) > 0, f(z) # 0 and (A(z) - XI)f(z) = 0 for all 2 E M . Then M C_ I'X and, consequently, p ( r X ) > 0.

On the other side, suppose p ( r X ) > 0. We choose Ql(z) as in the preceding lemma with respect to a(,) - X I instead of A(z), further we choose a subset M with M E C, 0 < p ( M ) < m. If el denotes the first unit vector in C", then the function .f defined by

is in Lp(X)\{O} and satisfies ( A - XI)f = 0.

We are able to generalize the choice of f.

0

Corollary 2.6. Suppose p ( I " ) > 0 and 4 E L P ( r X ) arbitrary. Then the function

is in L p ( X ) and satisfies ( A - X I ) f = 0.

Example 2.7. 1. When we apply Proposition 2.2 to the scalar case m = 1 then we obtain the

following well-known results ([21], for example): If ii E L,(X) and a = T(ii), then

.(a) = { A E C : essinf {I&(a) - A ( : z E X } = 0 } ,

and for X E @(a)

2. Next, let m E IN be arbitrary and X = [0, 11, for simplicity. If

A E q m x m,C([O,11)),

i. e., all entries of A are continuous in the closed interval1 [0,1], then the spectrum of A = T ( A ) is characterized by

u(A) = U .(A(z)). XEIO,11

This is an easy consequence of Proposition 2.3. The same characterization of u(A) remains valid if A is regarded as an operator in C( [0, 11)". The latter result is gen- eralized in [20, p. 901 to the case that X is a locally compact space, Co(X) denotes

Hardt/Wagenfiihrer, Spectral Properties of a Multiplication Operator 141

the space of continuous functions vanishing a t infinity and A E M ( m x m, C(X)) is bounded. Then the spectrum of A = T ( A ) as an operator in C O ( X ) ~ is

u(A) = U ,(A(,)). +EX

3. Finally, suppose that A is a matrix-valued step function on (0, l), i.e., all entries of A are step functions and LI, . . . , ~1 denotes a decomposition of (0 , l ) with A j := AILj E M ( m x m,C) ( j = 1 , . . . , 1 ) and p ( ~ j ) > 0 ( j = 1, . . . , I ) . Then

u(A) = { X E C : 3 j E (1,. . . , 1 } det ( A j - X I ) = O}.

Obviously, in this case, all points of the spectrum a(A) are eigenvalues and if X E a(A) there exists j E (1, . . . , 1 ) such that X E u(AIL,).

At the end of this section we prove that each isolated spectral point of a multiplication operator is an eigenvalue.

Theorem 2.8. Let A E M ( m x m,meas(X)) , A = T ( A ) be the corresponding multiplication operator in L p ( X ) , further let A E C and I” be defined as in Theorem 2.5. If X E a(A) is an isolated spectral point of A, then X E ap(A) and X is in the

resolvent set of the multiplication operator generated b y A restricted to the measurable set x\r

Proof . On one hand we conclude from Proposition 2.3

On the other hand there exists a 6 > 0 such that every A’ E C with 0 < JX - X’I < 6 is in @(A). From (2.1) we conclude that for every A’ E C with 0 < IX - X‘I < 6 there is a subset MA! E C with p(X\Mx!) = 0 and a constant E X ’ > 0 such that

By Lindelof’s Covering Theorem the set {A’ E (I: : 0 < IX - X’I < 6) is covered by a m m

countable union U V(Au). If we set M = n MA, then p ( X \ M ) = 0 and

142 Math. Nachr. 178 (1996)

- - Now suppose p ( r X ) = 0. Then the measure of X\M, M := M\rX, is also equal to 0, and X is not in U o(a(x)) by the definition of rX. Moreover, X is no accumulation

point of U o(A(z)), because for each A' with 0 < IX - X'l < 6 owing to (2.4) X+

X€G /

Consequently, X $! U n ( A ( x ) ) which is a contradiction with (2.3). X€G

For the proof of the second assertion we extend the right-hand side of (2.4) by chang- ing M into M\rX and note that X E @(A(,)) by the definition of rX. Conse-

quently, the whole open disk {A' E C : 1X - X'l < 6) is contained in n ,(A(,)).

As p((X\l?A)\(M\l?X)) 5 p(X\M) = 0, the characterization (2.1) of e(A) implies 0

n x E i w \ r

xEM\rX

that X is in the resolvent set of T ( A ) , with a restricted to X \ r X .

3. A further characterization of the spectrum

In this section we restrict ourselves to the case X = R with R a nonvoid open subset of Rd, d 2 1, p is the Lebesgue measure, and further we choose p = 2. That is, each A = T(2) with A E M ( m x m),meas(SZ)) is an operator in the Hilbert space L2(f l )m, the inner product in which is given by

P

n

and the corresponding norm is

where 11 . 112 is the Euclidean norm in 43". We note that the norm of a constant matrix B as defined in Proposition 2.1 satisfies

IlBll = T(B*B)i

in the present case, where T(B*B) is the largest eigenvalue of the matrix B*B. The following technical lemma will be used in the proof of Theorem 3.3.

Lemma 3.1. Let X be a measurable subset of 0, A E M ( m x m, meas ( X ) ) . Then there exists a measurable function v : X + 43" such that for all x E X

Il4.>112 = 1, Il~c4V(x)112 = llA(411.


P r o o f . I f B E M(mxm,C)isafixedmatrixandvECm\{O}isaneigenvector of B*B corresponding to the eigenvalue r(B*B), then

IJBWllg = w*B*Bv = r(B*B) l lwl~;

which means llBvll2 = IlBll IIwll2. We can find such a v with 1)w(12 = 1 as the first column of a unitary matrix with Q*B*BQ is upper triangular with r(B*B) in the upper left corner. Using the same arguments as in Lemma 2.4, v can be choosen Bore1 measurable as dependent on B. 0

The next proposition is a consequence of Corollary 2.6 combined with a specific property of sets with positive Lebesgue measure. Let N(A) denote the nulspace (kernel) of a linear operator A in L2(52)m.

Proposition 3.2. Suppose A = T ( A ) , A E M ( m x m,meas(R)) and X E up(A). Then dim N(A - X I ) = m, and therefore

Proof . If X E up(A) then, owing to Theorem 2.5 and Corollary 2.6, there exists a Lebesgue measurable subset rX of 52 with p ( r X > > 0 such that N(A - X I ) contains a subspace which is isomorphic with L2 (I'

Further, suppose X E a(A)\u,(A) and A-XI is aF'redholm operator. Then R(A-XI) is closed and 0 # R(A - X I ) i = N(A* - XI) has finite dimension. This is impossible because A* is also a multiplication operator corresponding to the matrix A* (see [8,

and La (r has infinite dimension.

Satz 2.21, for example). 0

We remark that for a multiplication operator A the different definitions of the essential spectrum are equivalent, i.e. und(A) = aess(A).

According to a definition given in [21], for example, a point X E (I: is called an approximate eigenvalue of A if and only if there exists a sequence ( f n ) n G ~ in ~ 5 2 ( 5 2 ) ~ with l f n l = 1 for all n E N such that the sequence of the fn weakly converges to 0 and (A - X I ) f n + 0 in the strong sense.

Theorem 3.3. Suppose A E M ( m x m, meas ( 5 2 ) ) and X E C. For all n 6 IN let

and let A = T ( A ) be the multiplication operator in Lz(52)" corresponding to a. A. If p ( r ; ) > 0 for all TZ E IN, then A E u(A), and X i s an approximate eigenvalue of

P r o o f . X must be a spectral point of A because otherwise p ( r A ) = 0 for all n sufficiently large - see (2.2). Next we distinguish two cases. If, in the first case, lim p ( r ; ) > 0 then p ( r X ) > 0 and therefore X is an eigenvalue of A according to

n-+m

144 Math. Nachr. 178 (1996)

Theorem 2.5 and N ( A - X I ) has infinite dimension. Then we can choose an orthonor- ma1 sequence (fn)F=p=o in N ( A - X I ) . In the second case we have lim p ( r 2 ) = 0.

Then there exists a subsequence (r&)kEN such that c (k E IN) and p(r'Ak\rik+l) > 0 for all k E IN. We choose measurable subsets M k of r&\rik+l such that 0 < p ( h f k ) < 00 for all k. We write

n- 00

A,(,) := A(z) - XI (z E n),

for brevity. Then by the choice of the h f k

Then f k E L 2 ( R ) m with I f k 1 2 = 1 for all k E lN. Moreover, the fk are orthonormal functions because the sets M k are pairwise disjoint, and, consequently, f k --f 0 weakly. Further we have

the latter because of (3.1). These estimates complete the proof of the theorem. 0

Corollary 3.4. If A is bounded then each spectral point is an approximative eigenvalue of A .

HardtlWagenfuhrer, Spectral Properties of a Multiplication Operator 145

4. Applications

Let Ao, Al , A2 E M ( m x m, L,(O, 1)) with m E IN be given, let the same symbols denote the corresponding operators in Lz(0,l)" and suppose 0 E e(A2). First we consider the second order differential operator L1 in the Hilbert space Lz(0, 1)" which is defined by

and the domain of which is chosen

LlY := A2y " + Aly ' + Aoy

(4.1) Drn := {y E (H2(0,1))" : y(0) = y(1) = o} It can be shown by using perturbation methods (see [ll] and [l3]) that the spectrum of L1 is discrete and for all A E e(L1) the operators (L1 - XI)-' are compact. Next let k be a further positive integer and let u, q1, 42 be matrices of the size k by k, rn by k, k by m, respectively, with elements in Lw(O,l). We consider the differential equation

(4.2)

(4-3)

L1Y - ( X I + Ql(U - X I ) - l q z ) Y = f

L1y - (XI + Ql(U - XI)-lqz)y = 0

with an f E Lz(0, I)", or the corresponding homogeneous equation

both with the boundary conditions y(0) = y(1) = 0 (We use the symbol I without making a distinction between the different matrix sizes.). We note that the class of problems (4.2) contains all differential equations of the type

(4.4) L1y - ( X I + W(X))y = 0

in which W : fE - M ( m x na, L,(O, 1)) is rational and W(co) = 0, i. e.,

r

W(X) = c X-"w,, T E IN, w, E M(rn x m, L,(O, 1)) . n= 1

According to a result of the theory of nodes ([4], p. 53), there exists I E IN such that W(X) = ql(u - XI)-lqZ with k = Zm and ql, q2, u as above.

The expressions on the left-hand sides of (4.2) - (4.3), namely

T(X)y := Liy - ( X I + qi(u - X I ) - l q z ) y

make sense for all X E e(u). For any X E e(u) , T(X) is a linear operator in Lz(0, l)rn the domain of which is D" as defined in (4.1), and the dependence on the eigenvalue parameter is nonlinear. As usual, the resolvent set, the spectrum and the point- spectrum of T are respectively defined by

e(T) := { A E e(u) : T(X)-l exists and is a continuous operator in L2(0, l)rn} , 4T) := e(u)\e(T), n p ( ~ ) := { A E e(u) : 3 y E D ~ \ { o } T ( X ) Y = 0).

146 Math. Nachr. 178 (1996)

Each X E up(T) is called an eigenvalue of T and y # 0 with T(X)y = 0 is an eigenvector. A Jordan chain (yo,. . . , y1) corresponding to XO E np(T) is an ( I + 1)-tuple (yo,. . . , yl) with y j E V m and yo # 0 such that

T(Xo)Yo = 0 , T(Ao)Y1 + + T (X0)yo = 0 , ... , ( 1 ) T (Xo)yo = 0 , T(Xo)yl + 5 T (X0)yl-l + . +

where T (A), etc., denote the formal derivatives of T(X) with respect to A, namely

T (A) = - I - q1(u - XI) -%& , ( j ) T (A) = - j ! * 4 1 ( ~ - X I ) - ( j + ' ) q 2 ( j > 1).

For any X E e(u) the problem (4.2) can be transformed into a problem which contains X linearly by setting

Then (4.2) is equivalent to

or

(3 ( L - X l ) y =

where L is the operator in the Hilbert-space Lz(0, l )m@L2(0 , l)k with domain D(L) = Dm @ L2(0, l)k which is represented by the matrix

The following two lemmas are generalizations of Proposition 1.2 in [17]:

Lemma 4.1. 1. e (T) = @(L) n e(u).

3. For any XO E u,(T) the Jordan chains of T and those of L at XO are correlated as follows: a system (yo,. . . ,yl) in D(L) is a Jordan chain of L at XO if and only if the y j 's have the form

2. rp (T) = c p ( L ) n e(u).


and (yo, . . . , y1) is a Jordan chain of T at XO.

Proof . For X E e(u) we make use of the following relation between T(X) and L - A 1 which we obtain by an easy calculation:

(4.5)

where

For X E e(u) both Wl(X) and W2(X) are linear continuous operators in L2(0, l )m @ L2(0,1)& having continuous inverses. From this reaon the assertions 1) and 2) are obvious. The relations between the Jordan chains follow by straightforward calculations using (4.5). 0

We note that T(X) is defined only for X E e(u), whereas L - X I makes sense for all X E (I:. There is the following correspondence between d,L) and CT(U).

Theorem 4.2. 1. U(U) = cess(L) C and(L). 2. Suppose that A is a component of e(u) for which there exists a XO E A rl e(L1)

with (4.6) 1 E e((u - ) - l q a ( - h - )-'q1).

Then ond(L) fl A = 8.

and ( L ) . 3. If e(u) is connected and there exists a XO E ~ ( u ) il e(L1) with (4.6), then U(U) =

4. W e denote by A0 the union of all components A of e(u) for which

P r o o f . First we prove ~(u) C cess(L). We have to modify the arguments of the proof of Proposition 1.2 in [171. First suppose X to be an isolated spectral point of u, which implies X E cp(u) because of Theorem 2.8. Owing to Theorem 2.5 and Lemma 2.4, there exists a measurable set M C (0 , l ) with p ( M ) > 0, u1 E M ( k - 1 x Ic, L,(M)), V E M(Ic x Ic,meas(M)) unitary such that

We conclude that for f1 E Dm, fi E L2(0, l)k the following equality

148 Math. Nachr. 178 (1996)

holds in M . Now by Egorov's Theorem, successively applied to the elements of Vq2, there is a measurable set MI C M with p(M1) > 0 such that Vq2 is continuous in Ml and, consequently, for any fi E D"(5 C[O, 11") and f 2 E Lz(0, l)k the last component of

(4.7) .If2

V q z f 1 f ( ) is a continuous function in MI. Therefore the space spanned by the expressions (4.7) has infinite codimension in L ~ ( M I ) ~ . The latter remains true if the operator V * is applied to (4.7), because V * is a topological isomorphism. The final result is that the range of L - X I has infinite codimension in Lz(0,l)" @ Lz(0, l)k which implies A E aess(L).

Next we have to prove that each accumulation point of a(.) is contained in aess( l ) . For X E e ( L l ) the operator L - X I can be factored in the Frobenius-Schur sense:

with

Since TI and T2 are bounded and bounded invertible operators L - X I are Fkedholm for X E e(L1) if and only if

) 0 ( ( L 1 0 (u - X I ) - q2(L1 - X I ) - l q 1

is a Fredholm operator. Evidently, in the case X E e(LI) , the latter operator is Fredholm if and only if (u - X I ) - qz(L1 - X I ) - l q l is Fredholm. Now we use the fact that a compact perturbation of a Fredholm operator is again a Fredholm operator (see [16], [ll], for example) and that (L1 - XI)-l is compact for X E e(L1). We conclude that L - X I is a Fredholm operator if and only if (u - X I ) is a Fkedholm operator, provided X E e(L1). Therefore, by Proposition 3.2 we have

e(L1) n Uess(L) = e(L1) n aess(u) = e(L1) n a(.). (4.8)

As a(L1) is discrete and aess(L) is closed, it follows that each accumulation point of a(.) is contained in ues8(L).

Conversely, let X E aess(L). Obviously, if X E e (L1) then X E a(.) because of (4.8). Therefore we must consider the case X F. Oess(L) n a(&). Since L1 has a compact resolvent, we conclude that X is an isolated normal eigenvalue of L1. With the Riesz projection


E sufficiently small, we introduce the operator

L: := Ll( I -PA)+bPX, S # X .

Then X E e(Lt ) , L t is a finite-dimensional perturbation of L1. We define

L: Q1 LA := ( q2 ) -

Since LA is a finite-dimensional perturbation of L and L - X I is not Redholm, also LA - X I has this property. By an analogous formula as (4.8) it follows X E a(u).

Finally, aess(L) c Cnd(L) in any case. The proof of part 2) works in a similar way as in [17]. Here we have to consider

the function X H (u - XI)-lqz(L1 - XI)-'ql which is finitely meromorphic on A, it values being compact operators. Since there exists a XO E (A r l e(L1)) such that ( I - (u - XoI)-lq2(L1 - XoI) - lq l ) - l exists and is a continuous operator, also R(X) := (I - ( t ~ - XI)-lq2(L1 - X I ) - ' q 1 ) - ' is finitely meromorphic on A. Hence one concludes that also (L - X I ) - ' is finitely meromorphic on A by using an appropriate representation of (L - X I ) - ' involving R(X). This representation of (L - XI)-l is deduced from the factorization of L - X I mentioned above: the details can be found in (131, p. 68. That paper also contains some presuppositions on L1 under which a XO with the condition (4.6) or, which is stronger, with ( ( u - X O I ) - ' ~ ~ ( L I -XoI)-lqlI < 1 exists.

Part 3) is a direct consequence of 1) and 2). It remains to prove part 4). The inclusions a(.) C Und(L) C a(.) U A0 are direct

consequences of 1) and 2). It remains to be shown that A0 C Und(L). At first let XO E A0 n e(L1). Then, as XO E e(u) and 1 E a,((. - XoI)-'q2(L1 - XoI)-lql), by definition, the factorization of L - X I implies XO E ap(L), but Xo is no isolated spectral point of L, of course. Consequently

AOne(L1) C "nd(L)*

Now A0 C and(L) follows from the facts that and(L) is closed and a(L1) is discrete.

We conclude from that theorem that aess(L) and and(L) do not coincide if and only

In the following Example 4.3 we construct the operator L such that if A0 # 0.

Ao = { z E G : IzI < l} and a(u) = { z E G : IzI = 1 ) .

In this example q1 and q2 are finite dimension operators instead of multiplication operators but it is easy to see that the Lemma 4.1 and the Theorem 4.2 are also true for such an operator L. We set A1 := C\(Ao U a(u)) = e(u)\Ao.

Example 4.3. Let m = 1 and the operator L1 be defined by

L1 := - y f f (y E D = {. E Hz(0,l) : y(0) = y(1) = O}). We thank S. NABOKO to communicate Prof. R. MENNICKEN an abstract version of this example.

150 Math. Nachr. 178 (1996)

The following statements for L1 are well-known ((211, for example) 1. L1 is a selfadjoint operator, 2. a(L1) = {k2n2 : k E IN} and therefore L1 1. r21. 3. { fi sin ( k m ) : k E IN} is a orthonormal basis of the space Lz(0,l) and consists

We define the multiplication operator u by of eigenvectors of L1.

(.f)(.) := eir(2z-1) f(.) (. E (0, I), f E L2(0,1)) .

4 U ) = { eia(2z - 1) : 5 E [O,l]} = { z E c : J Z J = l},

F'rorn the Example 2.7 2) it follows

and @(u) = { z E C : I z I < l } U { z E C : I z ( > l } =: A U A , .

It remains to define the operators q1 and q 2 . For this we put $(z) := fi sin (nz), which belongs to L2(0, l), and define q1 : L2(0,1) --t Lz(0, l ) by

q1.f := (f,4)lc, (f E J52(0,1))

Q 2 f = ( f ,$)d (f E L2(0,1)).

with a function 4 E Lz(0,l) which will be chosen later. Now we set q2 := qT, i.e.,

It is obvious that q1 and q2 are one-dimensional operators, i.e., rank(q1) = 1 = rank(q2). Let 77 E IR with 171 > 1, then

Therefore I ( u - iq l ) - 'qz (L~ - iqI)- lql( < 1 for 77 large enough and we conclude A, C Al. Now we are going to construct 4 E Lz(0,l) such that A C Ac,, i. e.,

1 E a,((. - AI)-lqZ(L1 - A1)-'ql) (A E A n e(L1) = A ) ,

which is equivalent to

(4.9) N ( 1 - (U - AI)-'qz(Li - XI)-lql) # (0) (A E A)

Let f E L2(0,1) and A E A, then


The last equality follows from the fact that .II, is the eigenvector of L1 to the eigenvalue lr2. Therefore (4.9) is equivalent with

(4.10)

and, in this case, v(z) := (eirr(2z-1) - X)-'c$(z) is an eigenvector of

I - (u - XI)-lq2(L1 - XI)--lq1

and

) ( - (L1 - ;I )-lw

is an eigenvector of L to the eigenvalue X because of the factorization of L - XI. Consequently, we have to choose 4 E Lz(0,l) in such way that the function $ defined on the unit circle by $(eirr(22-1)) := 4(z) (2 E [0,1]) satisfies

(4.11) dz = l r 2 - X . 2lra z(z - A)

( z l = l

For convenience, $ ought to be red-valued on {z E C : 1zI = 1) with a continuation of & z ) ~ to the whole unit disc which is holomorphic apart form a pole at zero. An appropriate choice is

$(z) = d + ( z + z - l ) - (22+z-Z) +co (IZI = 1)

where Co is an arbitrary real constant with CO > 27r2+2. For this function, the relation (4.11) is verified by the Residue Theorem, and the corresponding b, is obviously in Lz(0,l). Therefore (4.9) is satisfied where A is the open unit disc. In this case we conclude from Theorem 4.2

oess(L) = a(u) = {Z €(I: : I z ~ = I }

whereas ond(L) = a(.) U A = {Z E C : 1x1 5 I}.

The latter example shows that the different definitions of the essential spectrum are not equivalent. We remark that in the original version of the example, as it was communicated by S. NABOKO, L1 was an arbitrary selfadjoint operator in a Hilbert space H with L1 2 2 and the element II, € H was chosen arbitrary. Then it follows from L1 2 2 that ((L1 - is holomorphic in X on {z E G : 1.1 < 2} and the example works with certain infinite sums in the definition of $(z).

In the rest of the paper we return to the case that 91, 92 are multiplication operators defined by rn by k and k by m matrices with elements in L,(O, I), respectively. We have shown in Theorem 4.2 that in any case a(.) C a(L). But in general it is not true that ap(u) C op(L). In fact, Lemma 1.3 in (171 which will be quoted as Corollary 4.6

152 Math. Nachr. 178 (1996)

below, gives a sufficient condition such that a X is in up(.) but not in up(L) €or the case m = k = 1. The generalization reads as follows:

Proposition 4.4, Let m, Ic E IN be arbitrary, let X be an isolated spectral point of u and denote

Suppose that for all y the following implication

(4.12) y E D ( L ) , ((L-XI)y)(x) = 0 a.e. inr‘

as valid. Then X $! a,(L) .

may suppose that I’x has the property

(4.13) p ( r ’ n ([ - E , [ + E ) ) > 0 , for all < E r’ and for all E > 0 ,

which is obtained by removing a set of memure zero from J? ’. Now we have to show the following: if y E D(L), ( L - XI)y = 0, then y = 0, which means y(x) = 0 a.e. in

( 0 , l ) . If we set y = ( ii ) then (4.12) implies yl(z) = 0 a.e. in r’, and from the

property (4.13) of I?’ and the continuity of y1 and of yi we conclude yl(z) = yi(x) = 0 for all 2 E T’. Now suppose FA c [0,1], f;’ # [0,1] and let I1 be a component of [0, l]\F’, Then at least one of the boundary points (1 of 11 belongs to FA and therefore yl([1) = yi([1) = 0. Then we make use of Theorem 2.8 which says that (u - X I ) - 1 as restricted to the interval 1 1 is in M(k x k,Lm(I1)), and ( L - AI)y(z) = 0 in 11

implies

which after multiplying with Agl from the left is a system of differential equations with coefficients in Lw(I1). Therefore the uniqueness theorem yields y1 = 0 in I1 and, consequently yl(z) = 0 for all z E [0, 1). I t remains to be shown yz(x) = 0 a. e. in (0, l)\I”. This is clear because ( L - XI)y(x) = 0 implies q2yl + (u - M ) y 2 = 0. 0

In the following theorem we present a sufficient condition for the implication (4.12). We restrict ourselves to the case m 5 k which is important in view of the applicability to general rational systems (4.4). Let the symbols R and N denote the range and the null space, respectively.

I’ ’ := ( I E (0 , l ) : det (u([) - X I ) = 0} .

a y(x) = 0 a.e. inr’

P r o o f . The proof works in a similar way as that of Lemma 1.3 in [17]. First we

~~y~ - (XI + ql(u - ~1)-’q2)y1 = O in I1

Theorem 4.5. Let m 5 k, let X be an isolated eigenvalue ofu and let I?’ be defined as in Proposition 4.4. Suppose that for x E F A a. e. the following conditions are satisfied

rankqz(x) = m ,

(4.14) R(&)) n R(+) - XI) = (01 )

N(ql(x)) n N ( u ( ~ ) - XI) = (01. Then X 4 up(L) .


P r o o f . We show the implication (4.12). The equality ( L - X1)y = 0 is equivalent to

(4.15) (L1 - X J ) Y 1 +41Y2 = 0 , 42y1 + ( u - X I ) y 2 = 0

when we omit the argument x E I”, for brevity. The second equation and ~ ( q z ( x ) ) n R(u(z) - X I ) = ( 0 ) implies

42y1 = (u - XI)m = 0 ,

which yields y ~ ( x ) = 0 a. e. because the matrix 42(2) is injective a. e.. We conclude by similar arguments as in the preceding Proposition that y i(x) = y g ( x ) = 0 a. e. in FA, and the first equation in (4.15) yields qly2 = 0 a.e. in FA. We finally obtain y2 = 0

0

We obtain Lemma 1.3 in [17] as a direct consequence of the latter Theorem. We

a.e. because y2(x ) E N ( q l ( x ) ) n N ( u ( x ) - X I ) = ( 0 ) a.e..

note:

Corollary 4.6. Let m = k = 1, let X be an isolated eigenvalue of u and denote rX := {( E ( 0 , l ) : u(() = A}. Suppose 41 = 1 and q2(() # 0 a. e. in FA. Then

In the case of that corollary, the scalar differential equation T ( X ) y = 0 has the form

f O P ( L ) .

L1y - (A + 5) y = 0 .

Next we are interested how the conditions (4.14) in Theorem 4.5 read in the case of a more general scalar equation of the form (4.4), namely

L l Y ( Z ) - (A + W(Z, X))Y(.) = 0

Wx,X) = 4 1 ( x ) ( u ( x ) - XI)- lq2( . )

in which W ( z , A) is rational with respect to X and W ( x , 00) = 0. In the representation

with 41 E M ( l x k,L,(O, l)) , 42 E M ( k x 1, L,(O, 1)), u E M ( k x k, L,(O, 1)) the dimension k as well as the other data are not unique, but it will turn out that the conditions (4.14) restrict the choices.

Theorem 4.7. Let m = 1, k 2 1 be arbitray and 41, 42, u be given as in (4.2). Further let

W(x ,X) := q1(x ) (u (2 ) - XI) -1q2(x ) (x E ( O , l ) , X E a). Suppose that XO is an isolated eigenvalue of the multiplication operator u and let x E r A O be fixed. Then the conditions (4.14) hold for X = XO i f and only if the algebraic multiplicity of XO as an eigenvalue of u(x) coincides with the pole order of W ( x , . ) in XO.

154 Math. Nachr. 178 (1996)

P r o o f . As q2 (z) is a column in the present case, we note

rank q2(a) = 1 if and only if q2(2) # 0 , R(q2(x)) nR(u(z) - A o I ) = 0 if and only if q2(z) $! R(u(z ) - Aol).

Since N ( u ( z ) - A o I ) # (0) is our assumption, the condition

N(ql(z)) n N ( + ) - x o ~ ) = (0)

implies q1(x) # 0, and further, dimN(u(x)-AXgl) = 1. Let N(u(z) -A~gl ) be spanned by c E C'\{O}. Then

N(ql(x)) nN(4z) - ~ ~ 1 ) = (0) - c $! N ( q d 4 ) - ql(+ # 0 .

We conclude that the conditions (4.14) are equivalent with the following ones (the implication (4.16) + (4.14) is obvious):

q2(z) 4 R ( u ( 4 - x o q , (4.16) dimN(u(z) - A O I ) = 1 ,

q l ( z ) c # o for all c E N ( u ( s ) - A O I ) \ { o ) .

Now we choose an invertible k by k matrix C in such way that for the x given C-'u(s)C = J is the Jordan canonical form of u(x ) . Then for that x and for all A € C

with ij1 := ql(z)C, 42 := C-'qz(x). The conditions (4.16) are transferred to the same condition for J , 41, 42 in the place of u(z), q1(z), qz(z), respectively. The middle line in (4.16) means that there is only one Jordan block corresponding to A0 the extension of which being the algebraic multiplicity of the eigenvalue A0 say K . For simplicity, let this Jordan block be the leading one. Then @2 q! R(J - ,401) means that the component no. K. of 42 does not vanish, and the third line of (4.16) means that the first component of 41 is not zero. Finally we have to use (4.17) which yields the partial fraction representation of W(x, .). Then it is clear that a term c ~ o , ~ ( X - A o ) - " with an aO,& # 0 occurs in this representation if and only if the conditions (4.16) are fulfilled.

0

(4.17) (.,A) = Q1(J-A1)-l42

As we have seen in the scalar case, the conditions (4.14) are equivalent to the fact that each eigenvalue of ~ ( x ) occurs as a pole of W(z,A) in an appropriate manner. But in the nonscalar case the latter is not sufficient. We conclude the paper with an example in which the middle condition of (4.14) is violated although both eigenvalues of u(x) occur as poles in W(x, A).

Example 4.8. Let m = k = 2, Lly = -y",

Hardt/Wagenfiihrer, Spectral Properties of a Multiplication Operator

Here X = 0 is an isolated eigenvalue of u, and if we set

155

then y E V ( L ) is an eigenfunction of L corresponding to the eigenvalue X = 0.

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Universitit Regensburg N W F I - Malhematik 0-93040 Regensburg Germany

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Spectral Properties of a Multiplication Operator