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On Simplicity of the Maximal Eigenvalue
Journal of Mathematical Analysis and Applications 259, 338᎐345 Ž2001. doi:10.1006rjmaa.2001.7508, available online at http:rrwww.idealibrary.com on
On Simplicity of the Maximal Eigenvalue Bojan Kuzma1 Uni¨ ersity of Ljubljana, Institute of Mathematics, Physics and Mechanics, Department of Mathematics, Jadranska 19, 1000 Ljubljana, Slo¨ enia E-mail: [email protected] Submitted by Joyce R. McLaughlin Received March 5, 2001
It is shown that a maximal eigenvalue of a rank-one perturbed, compact, self-adjoint operator is automatically simple, if the norm of perturbation is large enough. 䊚 2001 Academic Press Key Words: maximal eigenvalue; simple eigenvalue; eigenvector; norm; rank-one perturbation.
0. INTRODUCTION There are several physical problems that depend heavily on finding an eigenvector of some linear, self-adjoint operator and where, moreover, the main interest is associated with the smallest eigenvalue. A sample example is the wave equation, where the first eigenvector of certain linear differential operator represents the ‘‘shape’’ of a basic vibration. The inverses of these are self-adjoint, compact operators on some Hilbert space. Of course, the minimal eigenvalue then becomes a maximal one, while the corresponding eigenvector remains the same. In many applications, it is important to know how the solution of a problem depends on perturbations of the data. Comparatively, a lot is known about the behaviour of eigenvalues of perturbed operators Žcf. w1x., but much less is known about the corresponding eigenvectors. Part of the problem in investigating the latter is no doubt the fact that they lack a ‘‘smoothness property’’ found in the former. To simplify the matter, we 1 This work was supported by grants from the Ministry of Science and Technology of Slovenia.
338 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
SIMPLICITY OF THE EIGENVALUE
339
restrict ourselves to the simplest possible perturbation, i.e., that of rank one. In w2x, extending the results from w3x, the perturbation of the form AŽ t . [ A q t ² y , z : z was considered Ž A being a compact, self-adjoint operator on some Hilbert space.. It was shown that if max Ž t . exists and the corresponding eigen¨ ector x max Ž t . is ne¨ er orthogonal to z, then its normᎏsubject to a normalization ² x max Ž t ., z : ' 1ᎏis strictly decreasing, provided that z is not an eigenvector of A. Here we are able to give a complete description of the self-adjoint, rank-one perturbation to show that in fact the maximal eigenvalue must exist whenever t is large enough and that it is automatically simple for such t’s; moreover, the corresponding eigenvector is never orthogonal to z. This ultimate result is described in detail in Section 2, where two examples are given to amplify the picture. Although the conclusions are valid in general, our main interest lies within the framework of the infinite-dimensional complex Hilbert space H . In contrast to the finite-dimensional case, where the only obstacle in applying the results from w2x is the fact that z could be orthogonal to KerŽ A y max Ž t .. for some t, here we are facing a possibility that max Ž t . does not even exist. It does not exist not only for some t, but also for any t from some interval, which is illustrated in Example 1. In a way, however, the situation within dim H s ⬁ is more simple as we shall show in the first Lemma.
1. MAIN RESULTS We begin by stating a Lemma, alluded to in the Introduction, and then go straight to the main Theorem. LEMMA 1.1. If H is an infinite-dimensional Hilbert space and A: H ª H a compact, self-adjoint operator, then whene¨ er it exists, its maximal eigen¨ alue is nonnegati¨ e. If it does not exist, then A is a negati¨ e definite operator. Proof. This follows easily once we recall that 0 g Ž A., that H is a direct sum of eigensubspaces of A corresponding to different eigenvalues, and that, moreover, all of them Žexcept possibly Ker A. are finite dimensional. THEOREM 1.1. Suppose I s w t 0 , ⬁. is a half-closed inter¨ al and A: H ª H a compact, self-adjoint operator. Choose a ¨ ector z g H and form a family of compact, self-adjoint operators AŽ t . [ A q t² y , z : z
Žt gI..
Ž 1.
340
BOJAN KUZMA
If the maximal eigen¨ alue, max Ž t 0 . exists and if z is not orthogonal to Yt 0 [ KerŽ AŽ t 0 . y max Ž t 0 .., then max exists for all t G t 0 and z is ne¨ er orthogonal to Yt [ KerŽ AŽ t . y max Ž t ... Moreo¨ er, if the corresponding maximal eigen¨ ector x max Ž t . satisfies ² x max Ž t . , z : ' 1; Ž t ) t 0 . ,
Ž 2.
then there exists a limit ¨ ector x [ lim t ot 0 x max Ž t .. This ¨ ector is the maximal eigen¨ ector of AŽ t 0 . and satisfies Ž2. as well. Proof. Without loss of generality, we may assume that t 0 s 0; otherwise, we would consider the shifted family A˜Ž t . [ AŽ t q t 0 .. We proceed by decomposing H s Y0 [ Ker A [ [n ² yn : according to the normalized eigenvectors yn , corresponding to nonzero eigenvalues n of A s AŽ0., which are counted with their multiplicities. ŽIf, accidentally, 0 [ max Ž0. s 0, then we omit the middle term.. The first part of the Theorem will be proven once it is shown that for each fixed t ) 0 there exists an eigenpair Ž , x . of AŽ t . with ) 0 and ² x, z : s 1. To see this, we first formulate an expression equivalent to the existence of eigenpair Ž , x . with the above properties. With regard to our decomposition of H , we can write z s y0 q u0 q
Ý n yn
and
n
x s yt q u t q
Ý ␣ n yn ; Ž u 0 , u t g Ker A, y 0 , yt g Y0 . . n
Observing at this point the eigenvector-eigenvalue equation
x s A Ž t . x s Ax q t ² x, z : z s Ax q tz s Ž 0 yt q ty 0 . q tu 0 q
Ž 3.
Ý Ž n ␣ n q t n . yn n
and considering only the orthogonal projections to the eigensubspaces ² yn :, we get the Žpossibly infinite . system of equations with solutions
␣n [
tn
y n
Ž n ) 0. ,
yt s 0 yt q ty 0 ,
and
u t s tu 0 . Ž 4 .
ŽBy assumption, ) 0 ) n , and consequently the denominator in the above set of solutions is different from 0; also, 0 s 0 implies u t , u 0 s 0, by initial agreement.. Furthermore, if dim H s ⬁, then, by the previous lemma, ) 0 G 0, and if dim H - ⬁, then 0 s ) 0 implies Ker A s 0. Thus it is clear that either u t s t u 0 or s 0 Žand thus u t s 0 s u 0 ., in ty 0 which case we define t u 0 [ 0. Equation Ž4. also gives yt s y 0 .
341
SIMPLICITY OF THE EIGENVALUE
Consequently, we have 1 s ² x, z : s ² yt , y 0 : q ² u t , u 0 : q
t < n < 2
Ý y n
s
t 5 y0 5
t 5 u0 5
2
y 0
q
< n <
2
q tÝ n
n
2
y n
.
Ž 5.
Inversely, if ) 0 solves Eq. Ž5., then clearly x [ yt 0 y 0 q t u 0 q t n Ý n y n yn is an eigenvector of AŽ t . corresponding to , provided that 0 / x g H Ži.e., 0 - 5 x 5 - ⬁.. However, by assumption, z H u Y0 , thus y 0 / 0 2 and obviously 5 x 5 2 G Ž yt 0 . 5 y 0 5 2 ) 0. On the other hand, we also have 2
5 x52 s
s
F F
t2
Ž y 0 .
5 y0 5 2 q 2
t2
t2
5 u0 5 2 q 2
t2
5 u0 5 2 q 2
5 u0 5 2 q 2
5 y0 5 2 q 2
Ý
t2
5 y0 5 2 q 2
t2
2
Ý n
n
t2
5 u0 5 2 q 2
t2
Ý n
t 2 < n < 2
Ž y n .
2
Ž 6.
t 2 < n < 2
Ž q 0 y n .
2
t 2 < n < 2
2
5 z 5 2 - ⬁;
here [ y 0 ) 0. Therefore, Ž , x . is an eigenpair to AŽ t . with ) 0 , whenever ) 0 is a solution of Ž5.. The right side of Eq. Ž5. is obviously a continuous function of since < <2 the series Ý n yn n converges uniformly for g Ž 0 q , ⬁. for any ) 0. 5 52 In fact, by arguments similar to those above, it is less than z . Since Ž . ªo 0 ⬁ and Ž . ªp⬁ 0, it follows by continuity that this equation has a solution ) 0 . Consequently, for every t ) 0 there exists an eigenvalue ) 0 of AŽ t ., and, by the previous Lemma, for any such t there exists a maximal eigen¨ alue max Ž t . ) 0 s max Ž0.. Further, from the eigenvector-eigenvalue equation Ž3. we can quickly conclude that zH u Yt , for otherwise max Ž t . x max Ž t . s Ax max Ž t . q t ² x max Ž t ., z : z s Ax max Ž t ., and thus max Ž t . would equal one of the eigenvalues of A, which is impossible. Finally, suppose t o 0. By assumption, z H u Y0 so y 0 / 0. But then it follows from Ž5. that ma x t y 0 is bounded as t o 0. From Ž6., and as Žt .
342
BOJAN KUZMA
max Ž t . ) 0 ) n , we have t2
5 x max Ž t . 5 2 s
Ž max Ž t . y 0 . qÝ n
F
t2
2max Ž t .
5 u0 5 2
t 2 < n < 2
Ž max Ž t . y n . t2
Ž max Ž t . y 0 .
2
5 y0 5 2 q 2
t 2 < n < 2
K
q
5 y0 5 2 q 2
Ý ns1
Ž max Ž t . y n .
2
t2
2max Ž t .
q
Ý n)K
5 u0 5 2 t 2 < n < 2
Ž max Ž t . y 0 .
2
,
where the last term is, for any t ) 0 close to zero, as small as we please if K is sufficiently large. But with t o 0, the third term also converges to zero since max Ž t . y n ) 0 y n ) 0; the same happens to the second term if 0 ) 0, and if 0 s 0 the second term is zero by initial assumption. Therefore, d Ž x max Ž t ., Y0 . ªt o 0 0. Considering again the identity Ž5., we deduce easily that lim t o 0 ma x t y 0 s 1r5 y 0 5 2 ; therefore x max Ž t . actually y converges to lim sup t o 0 ma x t y 0 y 0 s 5 y 005 2 . Žt .
Žt .
The previous Theorem says, roughly speaking, that if max exists for some t s t 0 and if z H u Yt 0 , then this is true for any t G t 0 . But what if it is discerned that max Ž t . does exist for any t G t 0 , and that z H u Yt for t ) t 0? Does it follow that actually z H u Yt 0? When dim H - ⬁ or 0 / 0 the answer is yes. If this was not the case, then we could decompose, in accordance with the above proof, z s u 0 q Ý n n yn . As we may well suppose ² x max Ž t ., z : ' 1; Ž t ) t 0 [ 0., we can use Ž4., with y 0 s 0, to come to yt s 0 yt ; [ max Ž t .. Recalling that max is a strictly increasing function, it follows that yt s 0 Ži.e., x max Ž t . H Y0 . for all t ) 0. In this case, however, x max Ž t . s
t
u0 q
tn
Ý y n
yn s t ⭈ n
ž
1
u0 q
n
Ý y n
/
yn , n
and since s max Ž t . ) 0 ) n Žand as n converge to 0 / 0 or else the sum is finite. the norm of the sum on the right side is bounded for t g ⺢q. Consequently, lim t o 0 x max Ž t . s 0, contradicting Ž2.. The next Corollary is in fact an extension of w2, Theorem 1.1x, where it was assumed that the function f from Eq. Ž7. is at least continuous.
SIMPLICITY OF THE EIGENVALUE
343
COROLLARY 1.1. Suppose f is a strictly increasing Ž possibly noncontinuous . function in the inter¨ al I s w t 0 , ⬁., and A and z as in Theorem Ž1.1.. Form a family of compact, self-adjoint operators, AŽ t . [ A q f Ž t . ² y , z : z
Žt gI..
Ž 7.
If z is not an eigen¨ ector of A, if max Ž t 0 . exists, and if z is not orthogonal to Yt 0 , then, for t ) t 0 the maximal eigen¨ alue, max Ž t . exists and is simple. Further, for each t G t 0 , we can define x max Ž t . in such a way that ² x max Ž t ., z : ' 1 and that the norm of x max is strictly decreasing on I. Proof. Without loss of generality, we assume that f ' Id I ; then virtually all were proven in the previous Theorem, with the sole except for the simplicity of max and claims concerning 5 x max 5. But the latter was settled in w2x once simplicity is established, which follows easily from the proof of the previous Theorem; namely, it was shown there that max Ž t . ) max Ž t 0 .. Thus, if x max Ž t ., ˆ x max Ž t . were two linearly independent eigenvectors, then some nonzero linear combination would be orthogonal to z. This, however, would imply the existence of ˜ x max Ž t . for which
max Ž t . ˜ x max Ž t . s A t 0 ˜ x max Ž t . q Ž t y t 0 . ² ˜ x max Ž t 0 . , z : s At0 ˜ x max Ž t . ,
Ž At
0
[ AŽ t0 . . .
Consequently, ˜ x max Ž t . would be an eigenvector of A t 0 , and thus max Ž t . F max Ž t 0 ., a contradiction. Finally, suppose that the operator A does not have a maximal eigenvalue. Then A must be injective, having infinitely many negative eigenvalues converging to 0. We can decompose H s [n ² yn : as we did in the proof of the Theorem. Following the lines, we come to the same conclusion that ) 0 is an eigenvalue of AŽ t . iff solves Eq. Ž5. for some t Žin this situation, of course, 5 y 0 5 s 0 s 5 u 0 5.. But the sum in Ž5. is convergent for, say, s 1; it is in fact less than Ý n < n < 2 s 5 z 5 2 . It follows that there exists such t s t 0 ) 0 for which Ž5. holds with [ 1. Therefore, max Ž t 0 . of AŽ t 0 . does exist and is larger than or equal to s 1. Moreover, the corresponding eigenvector x max Ž t 0 . is, by similar argument as in the proof of the Theorem, not orthogonal to z. As a consequence of the Theorem, we see that in fact this must be true for every t G t 0 .
344
BOJAN KUZMA
´ ´ 2. RESUME We begin this short section with an example on nonexistence of max . EXAMPLE 1. Let e1 , e2 , . . . denote the orthonormal basis of l 2 and define A: e n ¬
¡y10 e , 1 ¢y n e ,
~
1
ns1
n
nG2
.
It is immediate that A is a negative definite, compact operator on l 2 . Next, set z [ Ý n n12 e n and form AŽ t . [ A q t ² y , z : z. Now, if Ž Ž t ., x Ž t .. is an eigenpair, then either ² x Ž t ., z : s 0, in which case Ž t . s Ž0. - 0, or else we may well assume that the scalar product equals 1. But then, by decomposing x Ž t . s Ý n ␣ n e n , we can easily deduce from the eigenvectoreigenvalue equation Ax Ž t . q tz s x Ž t . that
¡
␣n
t
q 10 s~ t
ns1
,
¢ Ž n q 1. n
. ,
nG2
Thus, for G 0 and t g ⺢q small enough, we have 1 s ² x Ž t . , z: s t ⭈ Ft⭈
ž
1 10
q
⬁
Ý ns2
ž
1
q 10
1 n3
/
q
⬁
1
Ý 3 ns2 Ž n q 1 . n
/
, t ⭈ 0.3020569 z 1,
a contradiction. Consequently, AŽ t . is negative definite for all sufficiently small t. Since dim l 2 s ⬁, it follows from Lemma Ž1.1. that max Ž t . does not exist for any of such t. In this example, however, there exists the smallest such number t 0 [ Ž 101 q Ý⬁ns 2 n13 .y1, with a property that max Ž t . exists for t G t 0 . In the next Example we show that the abovementioned t 0 need not exist at all. EXAMPLE 2. With the same notations as above, let A: e n ¬ y
1 n
e and z [ 2 n
⬁
Ý ns1
1 n
en .
345
SIMPLICITY OF THE EIGENVALUE
Again A is a negative definite operator. Arguing as above, we see that the existence of max Ž t . implies ² x max Ž t ., z : s 1. This time, the equation is 1 s ² x max Ž t . , z : s t Ý n
Ž 1rn .
2
q 1rn
2
st
⬁
1
Ý
1 q n2
ns1
.
Since the sum on the right is a strictly decreasing, continuous function of , going to ⬁ as o 0, it follows that whenever t s t 1 solves the above equation, ) 0. Consequently, some t 2 - t 1 must also solve this equation for some 0 - ⬘ - . Thus, there does not exist the smallest such t s t 0 , for which AŽ t 0 . has a maximal eigenvalue. The following picture therefore emerges. If A is a self-adjoint, compact operator, perturbed by an arbitrary self-adjoint, rank-one projection Žcf. Eq. Ž1.., then there exists t for which AŽ t . has a maximal eigen¨ alue and the corresponding maximal eigenvector is not orthogonal to z. If t 0 is an infimum of all such parameters t, then for t ) t 0 , max Ž t . is always simple and z is ne¨ er orthogonal to x max Ž t .. However, max Ž t 0 . could or could not exist, depending on the convergence of the sum in Ž5. at s 0; if dim H - ⬁, it does exist and then z H u Yt 0 . Moreover, for each t G t 0 Žresp. t ) t 0 , if max Ž t 0 . does not exist., the maximal eigenvector x max Ž t . satisfying ² x max Ž t ., z : s 1 can be found, depending continuously on t Žcf. w2x and the proof of the Theorem.. If, in addition, z is not an eigen¨ ector of A, then its norm is strictly decreasing. Finally, if the perturbation is of the form Ž7., all of the foregoing remains valid with two exceptions: Maximal eigenvalue need no longer exist, and x max is continuous Žperhaps only. at the points of continuity of f.
REFERENCES 1. T. Kato, ‘‘Perturbation Theory for Linear Operators,’’ Springer-Verlag, Berlin, 1980. 2. B. Kuzma, Eigenvectors of perturbed operators, J. Math. Anal. Appl. 233 Ž1999., 623᎐633. 3. J. de Pillis and M. Neumann, The effect of the perturbation of Hermitian matrices on their eigenvectors, Siam J. Alg. Disc. Meth. 6Ž2. Ž1985., 201᎐209.
On Simplicity of the Maximal Eigenvalue Bojan Kuzma1 Uni¨ ersity of Ljubljana, Institute of Mathematics, Physics and Mechanics, Department of Mathematics, Jadranska 19, 1000 Ljubljana, Slo¨ enia E-mail: [email protected] Submitted by Joyce R. McLaughlin Received March 5, 2001
It is shown that a maximal eigenvalue of a rank-one perturbed, compact, self-adjoint operator is automatically simple, if the norm of perturbation is large enough. 䊚 2001 Academic Press Key Words: maximal eigenvalue; simple eigenvalue; eigenvector; norm; rank-one perturbation.
0. INTRODUCTION There are several physical problems that depend heavily on finding an eigenvector of some linear, self-adjoint operator and where, moreover, the main interest is associated with the smallest eigenvalue. A sample example is the wave equation, where the first eigenvector of certain linear differential operator represents the ‘‘shape’’ of a basic vibration. The inverses of these are self-adjoint, compact operators on some Hilbert space. Of course, the minimal eigenvalue then becomes a maximal one, while the corresponding eigenvector remains the same. In many applications, it is important to know how the solution of a problem depends on perturbations of the data. Comparatively, a lot is known about the behaviour of eigenvalues of perturbed operators Žcf. w1x., but much less is known about the corresponding eigenvectors. Part of the problem in investigating the latter is no doubt the fact that they lack a ‘‘smoothness property’’ found in the former. To simplify the matter, we 1 This work was supported by grants from the Ministry of Science and Technology of Slovenia.
338 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
SIMPLICITY OF THE EIGENVALUE
339
restrict ourselves to the simplest possible perturbation, i.e., that of rank one. In w2x, extending the results from w3x, the perturbation of the form AŽ t . [ A q t ² y , z : z was considered Ž A being a compact, self-adjoint operator on some Hilbert space.. It was shown that if max Ž t . exists and the corresponding eigen¨ ector x max Ž t . is ne¨ er orthogonal to z, then its normᎏsubject to a normalization ² x max Ž t ., z : ' 1ᎏis strictly decreasing, provided that z is not an eigenvector of A. Here we are able to give a complete description of the self-adjoint, rank-one perturbation to show that in fact the maximal eigenvalue must exist whenever t is large enough and that it is automatically simple for such t’s; moreover, the corresponding eigenvector is never orthogonal to z. This ultimate result is described in detail in Section 2, where two examples are given to amplify the picture. Although the conclusions are valid in general, our main interest lies within the framework of the infinite-dimensional complex Hilbert space H . In contrast to the finite-dimensional case, where the only obstacle in applying the results from w2x is the fact that z could be orthogonal to KerŽ A y max Ž t .. for some t, here we are facing a possibility that max Ž t . does not even exist. It does not exist not only for some t, but also for any t from some interval, which is illustrated in Example 1. In a way, however, the situation within dim H s ⬁ is more simple as we shall show in the first Lemma.
1. MAIN RESULTS We begin by stating a Lemma, alluded to in the Introduction, and then go straight to the main Theorem. LEMMA 1.1. If H is an infinite-dimensional Hilbert space and A: H ª H a compact, self-adjoint operator, then whene¨ er it exists, its maximal eigen¨ alue is nonnegati¨ e. If it does not exist, then A is a negati¨ e definite operator. Proof. This follows easily once we recall that 0 g Ž A., that H is a direct sum of eigensubspaces of A corresponding to different eigenvalues, and that, moreover, all of them Žexcept possibly Ker A. are finite dimensional. THEOREM 1.1. Suppose I s w t 0 , ⬁. is a half-closed inter¨ al and A: H ª H a compact, self-adjoint operator. Choose a ¨ ector z g H and form a family of compact, self-adjoint operators AŽ t . [ A q t² y , z : z
Žt gI..
Ž 1.
340
BOJAN KUZMA
If the maximal eigen¨ alue, max Ž t 0 . exists and if z is not orthogonal to Yt 0 [ KerŽ AŽ t 0 . y max Ž t 0 .., then max exists for all t G t 0 and z is ne¨ er orthogonal to Yt [ KerŽ AŽ t . y max Ž t ... Moreo¨ er, if the corresponding maximal eigen¨ ector x max Ž t . satisfies ² x max Ž t . , z : ' 1; Ž t ) t 0 . ,
Ž 2.
then there exists a limit ¨ ector x [ lim t ot 0 x max Ž t .. This ¨ ector is the maximal eigen¨ ector of AŽ t 0 . and satisfies Ž2. as well. Proof. Without loss of generality, we may assume that t 0 s 0; otherwise, we would consider the shifted family A˜Ž t . [ AŽ t q t 0 .. We proceed by decomposing H s Y0 [ Ker A [ [n ² yn : according to the normalized eigenvectors yn , corresponding to nonzero eigenvalues n of A s AŽ0., which are counted with their multiplicities. ŽIf, accidentally, 0 [ max Ž0. s 0, then we omit the middle term.. The first part of the Theorem will be proven once it is shown that for each fixed t ) 0 there exists an eigenpair Ž , x . of AŽ t . with ) 0 and ² x, z : s 1. To see this, we first formulate an expression equivalent to the existence of eigenpair Ž , x . with the above properties. With regard to our decomposition of H , we can write z s y0 q u0 q
Ý n yn
and
n
x s yt q u t q
Ý ␣ n yn ; Ž u 0 , u t g Ker A, y 0 , yt g Y0 . . n
Observing at this point the eigenvector-eigenvalue equation
x s A Ž t . x s Ax q t ² x, z : z s Ax q tz s Ž 0 yt q ty 0 . q tu 0 q
Ž 3.
Ý Ž n ␣ n q t n . yn n
and considering only the orthogonal projections to the eigensubspaces ² yn :, we get the Žpossibly infinite . system of equations with solutions
␣n [
tn
y n
Ž n ) 0. ,
yt s 0 yt q ty 0 ,
and
u t s tu 0 . Ž 4 .
ŽBy assumption, ) 0 ) n , and consequently the denominator in the above set of solutions is different from 0; also, 0 s 0 implies u t , u 0 s 0, by initial agreement.. Furthermore, if dim H s ⬁, then, by the previous lemma, ) 0 G 0, and if dim H - ⬁, then 0 s ) 0 implies Ker A s 0. Thus it is clear that either u t s t u 0 or s 0 Žand thus u t s 0 s u 0 ., in ty 0 which case we define t u 0 [ 0. Equation Ž4. also gives yt s y 0 .
341
SIMPLICITY OF THE EIGENVALUE
Consequently, we have 1 s ² x, z : s ² yt , y 0 : q ² u t , u 0 : q
t < n < 2
Ý y n
s
t 5 y0 5
t 5 u0 5
2
y 0
q
< n <
2
q tÝ n
n
2
y n
.
Ž 5.
Inversely, if ) 0 solves Eq. Ž5., then clearly x [ yt 0 y 0 q t u 0 q t n Ý n y n yn is an eigenvector of AŽ t . corresponding to , provided that 0 / x g H Ži.e., 0 - 5 x 5 - ⬁.. However, by assumption, z H u Y0 , thus y 0 / 0 2 and obviously 5 x 5 2 G Ž yt 0 . 5 y 0 5 2 ) 0. On the other hand, we also have 2
5 x52 s
s
F F
t2
Ž y 0 .
5 y0 5 2 q 2
t2
t2
5 u0 5 2 q 2
t2
5 u0 5 2 q 2
5 u0 5 2 q 2
5 y0 5 2 q 2
Ý
t2
5 y0 5 2 q 2
t2
2
Ý n
n
t2
5 u0 5 2 q 2
t2
Ý n
t 2 < n < 2
Ž y n .
2
Ž 6.
t 2 < n < 2
Ž q 0 y n .
2
t 2 < n < 2
2
5 z 5 2 - ⬁;
here [ y 0 ) 0. Therefore, Ž , x . is an eigenpair to AŽ t . with ) 0 , whenever ) 0 is a solution of Ž5.. The right side of Eq. Ž5. is obviously a continuous function of since < <2 the series Ý n yn n converges uniformly for g Ž 0 q , ⬁. for any ) 0. 5 52 In fact, by arguments similar to those above, it is less than z . Since Ž . ªo 0 ⬁ and Ž . ªp⬁ 0, it follows by continuity that this equation has a solution ) 0 . Consequently, for every t ) 0 there exists an eigenvalue ) 0 of AŽ t ., and, by the previous Lemma, for any such t there exists a maximal eigen¨ alue max Ž t . ) 0 s max Ž0.. Further, from the eigenvector-eigenvalue equation Ž3. we can quickly conclude that zH u Yt , for otherwise max Ž t . x max Ž t . s Ax max Ž t . q t ² x max Ž t ., z : z s Ax max Ž t ., and thus max Ž t . would equal one of the eigenvalues of A, which is impossible. Finally, suppose t o 0. By assumption, z H u Y0 so y 0 / 0. But then it follows from Ž5. that ma x t y 0 is bounded as t o 0. From Ž6., and as Žt .
342
BOJAN KUZMA
max Ž t . ) 0 ) n , we have t2
5 x max Ž t . 5 2 s
Ž max Ž t . y 0 . qÝ n
F
t2
2max Ž t .
5 u0 5 2
t 2 < n < 2
Ž max Ž t . y n . t2
Ž max Ž t . y 0 .
2
5 y0 5 2 q 2
t 2 < n < 2
K
q
5 y0 5 2 q 2
Ý ns1
Ž max Ž t . y n .
2
t2
2max Ž t .
q
Ý n)K
5 u0 5 2 t 2 < n < 2
Ž max Ž t . y 0 .
2
,
where the last term is, for any t ) 0 close to zero, as small as we please if K is sufficiently large. But with t o 0, the third term also converges to zero since max Ž t . y n ) 0 y n ) 0; the same happens to the second term if 0 ) 0, and if 0 s 0 the second term is zero by initial assumption. Therefore, d Ž x max Ž t ., Y0 . ªt o 0 0. Considering again the identity Ž5., we deduce easily that lim t o 0 ma x t y 0 s 1r5 y 0 5 2 ; therefore x max Ž t . actually y converges to lim sup t o 0 ma x t y 0 y 0 s 5 y 005 2 . Žt .
Žt .
The previous Theorem says, roughly speaking, that if max exists for some t s t 0 and if z H u Yt 0 , then this is true for any t G t 0 . But what if it is discerned that max Ž t . does exist for any t G t 0 , and that z H u Yt for t ) t 0? Does it follow that actually z H u Yt 0? When dim H - ⬁ or 0 / 0 the answer is yes. If this was not the case, then we could decompose, in accordance with the above proof, z s u 0 q Ý n n yn . As we may well suppose ² x max Ž t ., z : ' 1; Ž t ) t 0 [ 0., we can use Ž4., with y 0 s 0, to come to yt s 0 yt ; [ max Ž t .. Recalling that max is a strictly increasing function, it follows that yt s 0 Ži.e., x max Ž t . H Y0 . for all t ) 0. In this case, however, x max Ž t . s
t
u0 q
tn
Ý y n
yn s t ⭈ n
ž
1
u0 q
n
Ý y n
/
yn , n
and since s max Ž t . ) 0 ) n Žand as n converge to 0 / 0 or else the sum is finite. the norm of the sum on the right side is bounded for t g ⺢q. Consequently, lim t o 0 x max Ž t . s 0, contradicting Ž2.. The next Corollary is in fact an extension of w2, Theorem 1.1x, where it was assumed that the function f from Eq. Ž7. is at least continuous.
SIMPLICITY OF THE EIGENVALUE
343
COROLLARY 1.1. Suppose f is a strictly increasing Ž possibly noncontinuous . function in the inter¨ al I s w t 0 , ⬁., and A and z as in Theorem Ž1.1.. Form a family of compact, self-adjoint operators, AŽ t . [ A q f Ž t . ² y , z : z
Žt gI..
Ž 7.
If z is not an eigen¨ ector of A, if max Ž t 0 . exists, and if z is not orthogonal to Yt 0 , then, for t ) t 0 the maximal eigen¨ alue, max Ž t . exists and is simple. Further, for each t G t 0 , we can define x max Ž t . in such a way that ² x max Ž t ., z : ' 1 and that the norm of x max is strictly decreasing on I. Proof. Without loss of generality, we assume that f ' Id I ; then virtually all were proven in the previous Theorem, with the sole except for the simplicity of max and claims concerning 5 x max 5. But the latter was settled in w2x once simplicity is established, which follows easily from the proof of the previous Theorem; namely, it was shown there that max Ž t . ) max Ž t 0 .. Thus, if x max Ž t ., ˆ x max Ž t . were two linearly independent eigenvectors, then some nonzero linear combination would be orthogonal to z. This, however, would imply the existence of ˜ x max Ž t . for which
max Ž t . ˜ x max Ž t . s A t 0 ˜ x max Ž t . q Ž t y t 0 . ² ˜ x max Ž t 0 . , z : s At0 ˜ x max Ž t . ,
Ž At
0
[ AŽ t0 . . .
Consequently, ˜ x max Ž t . would be an eigenvector of A t 0 , and thus max Ž t . F max Ž t 0 ., a contradiction. Finally, suppose that the operator A does not have a maximal eigenvalue. Then A must be injective, having infinitely many negative eigenvalues converging to 0. We can decompose H s [n ² yn : as we did in the proof of the Theorem. Following the lines, we come to the same conclusion that ) 0 is an eigenvalue of AŽ t . iff solves Eq. Ž5. for some t Žin this situation, of course, 5 y 0 5 s 0 s 5 u 0 5.. But the sum in Ž5. is convergent for, say, s 1; it is in fact less than Ý n < n < 2 s 5 z 5 2 . It follows that there exists such t s t 0 ) 0 for which Ž5. holds with [ 1. Therefore, max Ž t 0 . of AŽ t 0 . does exist and is larger than or equal to s 1. Moreover, the corresponding eigenvector x max Ž t 0 . is, by similar argument as in the proof of the Theorem, not orthogonal to z. As a consequence of the Theorem, we see that in fact this must be true for every t G t 0 .
344
BOJAN KUZMA
´ ´ 2. RESUME We begin this short section with an example on nonexistence of max . EXAMPLE 1. Let e1 , e2 , . . . denote the orthonormal basis of l 2 and define A: e n ¬
¡y10 e , 1 ¢y n e ,
~
1
ns1
n
nG2
.
It is immediate that A is a negative definite, compact operator on l 2 . Next, set z [ Ý n n12 e n and form AŽ t . [ A q t ² y , z : z. Now, if Ž Ž t ., x Ž t .. is an eigenpair, then either ² x Ž t ., z : s 0, in which case Ž t . s Ž0. - 0, or else we may well assume that the scalar product equals 1. But then, by decomposing x Ž t . s Ý n ␣ n e n , we can easily deduce from the eigenvectoreigenvalue equation Ax Ž t . q tz s x Ž t . that
¡
␣n
t
q 10 s~ t
ns1
,
¢ Ž n q 1. n
. ,
nG2
Thus, for G 0 and t g ⺢q small enough, we have 1 s ² x Ž t . , z: s t ⭈ Ft⭈
ž
1 10
q
⬁
Ý ns2
ž
1
q 10
1 n3
/
q
⬁
1
Ý 3 ns2 Ž n q 1 . n
/
, t ⭈ 0.3020569 z 1,
a contradiction. Consequently, AŽ t . is negative definite for all sufficiently small t. Since dim l 2 s ⬁, it follows from Lemma Ž1.1. that max Ž t . does not exist for any of such t. In this example, however, there exists the smallest such number t 0 [ Ž 101 q Ý⬁ns 2 n13 .y1, with a property that max Ž t . exists for t G t 0 . In the next Example we show that the abovementioned t 0 need not exist at all. EXAMPLE 2. With the same notations as above, let A: e n ¬ y
1 n
e and z [ 2 n
⬁
Ý ns1
1 n
en .
345
SIMPLICITY OF THE EIGENVALUE
Again A is a negative definite operator. Arguing as above, we see that the existence of max Ž t . implies ² x max Ž t ., z : s 1. This time, the equation is 1 s ² x max Ž t . , z : s t Ý n
Ž 1rn .
2
q 1rn
2
st
⬁
1
Ý
1 q n2
ns1
.
Since the sum on the right is a strictly decreasing, continuous function of , going to ⬁ as o 0, it follows that whenever t s t 1 solves the above equation, ) 0. Consequently, some t 2 - t 1 must also solve this equation for some 0 - ⬘ - . Thus, there does not exist the smallest such t s t 0 , for which AŽ t 0 . has a maximal eigenvalue. The following picture therefore emerges. If A is a self-adjoint, compact operator, perturbed by an arbitrary self-adjoint, rank-one projection Žcf. Eq. Ž1.., then there exists t for which AŽ t . has a maximal eigen¨ alue and the corresponding maximal eigenvector is not orthogonal to z. If t 0 is an infimum of all such parameters t, then for t ) t 0 , max Ž t . is always simple and z is ne¨ er orthogonal to x max Ž t .. However, max Ž t 0 . could or could not exist, depending on the convergence of the sum in Ž5. at s 0; if dim H - ⬁, it does exist and then z H u Yt 0 . Moreover, for each t G t 0 Žresp. t ) t 0 , if max Ž t 0 . does not exist., the maximal eigenvector x max Ž t . satisfying ² x max Ž t ., z : s 1 can be found, depending continuously on t Žcf. w2x and the proof of the Theorem.. If, in addition, z is not an eigen¨ ector of A, then its norm is strictly decreasing. Finally, if the perturbation is of the form Ž7., all of the foregoing remains valid with two exceptions: Maximal eigenvalue need no longer exist, and x max is continuous Žperhaps only. at the points of continuity of f.
REFERENCES 1. T. Kato, ‘‘Perturbation Theory for Linear Operators,’’ Springer-Verlag, Berlin, 1980. 2. B. Kuzma, Eigenvectors of perturbed operators, J. Math. Anal. Appl. 233 Ž1999., 623᎐633. 3. J. de Pillis and M. Neumann, The effect of the perturbation of Hermitian matrices on their eigenvectors, Siam J. Alg. Disc. Meth. 6Ž2. Ž1985., 201᎐209.