The Completeness of Generalized Eigenfunctions of a Discrete Operator

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1 Joural of Mathematical Aalysis ad Applicatios 26, (200) doi:0.006/jmaa , available olie at o The Completeess of Geeralized Eigefuctios of a Discrete Operator Liapig Zhag Departmet of Mathematics, Shaxi Uiversity, Taiyua , People s Republic of Chia zhaglp@mail.sxu.edu.c Submitted by P. S. Muhly Received Jue, 2000 I this paper we preset some coditios for the completeess of geeralized eigefuctios of a discrete operator. These coditios ivole the boudedess of its eigeprojectios or the properties of its resolvet. 200 Academic Press Key Words: discrete operator; geeralized eigefuctios; completeess; basis; ifiitesimal geerator of C 0 semigroup.. INTRODUCTION We start by recallig some basic defiitios ad properties of discrete operators (see []). Let X be a complex separable Brach space with the orm ad A a liear operator with the domai DA i X. A is said to be discrete if there is a complex umber z i its resolvet set ρa for which the resolvet Rz A =zi A is compact, ad to be desely defied if DA =X, where I is the idetity operator ad S is the closure of the set S. IfA is discrete, the its spectrum σa is a deumerable set of poits with o fiite limit poit ad every λ i σa is a eigevalue of fiite multiplicity. Let with σa =λ λ 2 λ λ λ 2 λ () X/0 $35.00 Copyright 200 by Academic Press All rights of reproductio i ay form reserved.

2 242 liapig zhag ad lim λ =+.ByE λ, we deote the eigeprojectio associated with λ = 2. Defie The followig results hold. QA =xe λ x = 0= 2x X (2) Theorem A. Let A be a discrete operator. The (i) the space Q( A) either is ifiite dimesioal or cosists oly of zero, (ii) the space Q(A) is the set of all x i X for which R(z, A) x is a etire fuctio of z. Let SpA, the spectral spa of A, be the smallest closed maifold cotaiig all the maifolds E λ X = 2 with λ i σa. We have the followig theorem. Theorem B. If A is a discrete operator i a reflexive Baach space X, the SpA =QA, where A is the adjoit operator of A ad S is the aihilater of the set S. Referece [] emphasizes that the coditio for SpA =X give i Theorem B is QA =0 ad ot QA =0, ad poits out that it is possible that SpA X while QA =0. I this paper we study uder what coditios QA = 0 implies SpA =X. I Sectio 2, we obtai the mai results, Theorem ad Theorem 2, which aswer the above questio, by discussig the boudedess of the eigeprojectios E λ. I Sectio 3, we obtai Theorem 3 ad Theorem 4, which describe the structures of the sets QA ad SpA by meas of the properties of the resolvet Rz A. 2. RESULTS RELATED TO EIGENPROJECTIONS We eed some defiitios. Defiitio. Let X be a complex Baach space. A etire fuctio F C X is of order µf, orµ, if where l l Mr lim sup = µf r + l r Mr = max 0 θ 2π Freiθ

3 eigefuctios of a discrete operator 243 Defiitio 2. Let X be a complex Baach space. A fuctio F of the complex variable z, with values i X, is called a meromorphic fuctio of fiite order µ if it is expressible as Fz = Pz (3) qz where P C X ad q C C are all etire fuctios of order at most µ, ad µ = maxµpµq < + Theorem. Let X be a complex separable Baach space. A is a discrete operator defied i X, whose resolvet R(z, A) is a meromorphic fuctio of fiite order µ, ad there is a iteger N 0 such that (i) DA N+µ+ =X, where µ deotes the iteger part of µ; (ii) E λ =Oλ N, where λ is a eigevalue of A, E λ is the correspodig eigeprojectio, ad E λ deotes the operator orm of E λ ; ad (iii) D m λ CE λ m = 2K = 2 where C is a costat. The QA =0 implies SpA =X. Proof. Let λ λ 2 λ eigevalues of A, be a arragemet satisfyig (). By G /, we deote the pricipal part of the Lauret expasio of Rz A at the poit z = λ σa. The (see [2]) ( ) Rz AE λ = G ad = E λ + D λ 2 + D2 λ E λ D λ = D λ E λ = D λ D K λ = 0 D K λ K where K is the algebraic multiplicity of λ. Furthermore, we have ( ) R N+µ+ z AE λ = G N+µ+ From coditio (ii) of Theorem, we may assume E λ N M = 2

4 244 liapig zhag where M is a costat. Note coditio (iii) ad put M 2 = CM N +µ+!. Whe r> ad by the polyomial theorem, R N+µ+ zae λ ( ) = GN+µ+ z λ [ N +µ+! Eλ z λ µ+ z λ + D λ + + D K ] λ N z λ N+ z λ N+K CM [ N +µ+! + ] z λ µ+ z λ + + z λ K ( M = 2 / z λ ) K z λ µ+ /z λ M 2 z λ µ+ /r = M z λ µ+ where M =rm 2 /r. From the assumptio of Theorem, Rz A has the expressio (3), amely, Rz A =Pz/qz. Thus, the λ s i σa are zeros of the etire fuctio qz. By the relatio betwee the order of the etire fuctio qz ad the covergece expoet of the zeros of qz, we have < + λ µ+ Suppose that z ρa ad z <r. For a sufficietly large atural umber k ad k, it holds that λ > 2r ad λ z λ z > 2 λ >r> Hece, for ay y X, ( ) G N+µ+ y z λ My z λ µ+ 2µ+ My λ =k =k We ifer that, for ay y X z ρa z <r, the series ( ) G N+µ+ coverges strogly. From the arbitrariess of r, ( ) Gz = G N+µ+ =k µ+

5 eigefuctios of a discrete operator 245 is a operator valued meromorphic fuctio i the complex plae C. I view of the defiitio (2) of QA, we have R N+µ+ z Ay Gzy QA y X Cosequetly, the assumptio QA =0 yields that R N+µ+ z Ay = Gzy y X Sice Gzy SpA, we have DA N+µ+ SpA. By virtue of DA N+µ+ = X, we coclude that SpA = X. The proof is complete. Remark. The coditio (iii) i the above theorem does ot always hold. But it may be realized, for example, uder the coditio that there is a atural umber N such that λ is algebraically simple as >N. Corollary. assumptios that If coditio (iii) of Theorem is substituted with the ( ) D =Oλ, (2 ) K is uiformly bouded, the the coclusio of Theorem still holds. Proof. From the proof of Theorem, for z ρa, R N+µ+ zae λ ( ) = GN+µ+ z λ [ N +µ+! Eλ z λ µ+ z λ + N K D λ + + D λ z λ N+ z λ N+K By coditio (ii) of Theorem ad the assumptios of Corollary, we may assume E λ + D λ D K λ + + N N+ M N+K where M is a costat. Whe λ > 2z for z ρa (z fixed) ad k (k sufficietly large), it holds that ] λ z λ z > 2 λ k

6 246 liapig zhag Thus, for ay y X, ( ) G N+µ+ y z λ =k =k MN +µ+!y µ+ 2 µ+ MN +µ+!y It follows that, for y X z ρa, the ( G N+µ+ ) =k λ µ+ coverges strogly. The, similar to the proof of Theorem, we ca obtai the coclusio of Theorem. The proof is complete. I Theorem the coditio (ii) plays a key role. As N, it is possible that sup E λ =+ < As N = 0, the coditio (ii) chages ito sup E λ < + (4) < The coditio (i) of Theorem is automatically satisfied whe A is a ifiitesimal geerator of a liear operator semigroup. Referece [3] discusses the case that A is a ifiitesimal geerator ad satisfies (4). However, the proof of its mai theorem seems to have mistakes. If coditio (ii) of Theorem is substituted with m sup E λ < + (5) < we shall have a better result. This is the followig theorem. Theorem 2. Let A be a discrete operator i a complex separable Baach space X. Its eigevalues are λ with correspodig eigeprojectios E λ ad correspodig geeralized eigefuctios φ i i = 2K, where the positive iteger K is the algebraic multiplicity of the eigevalue λ. If A satisfies (5), the DA SpA QA (6) ad φ i i = 2K costitute a basis of the subspace Sp(A), or a basic sequece of X, where deotes the direct sum.

7 eigefuctios of a discrete operator 247 Proof. First, we prove that φ i i = 2K costitutes a basis of SpA. Suppose m E λ M m = 2 (7) where M is a costat. By S, deote the set of all liear combiatios of φ i i = 2K. The S is dese i SpA. For ay K l y = α li φ li S we have lim m l= i= ( m E λj )y = y (8) For ay x SpA ad ɛ>0, there is a y ɛ S such that x y ɛ < ɛ/2m. Cosider ( ( m x E λj )x x y ɛ+ y m ɛ E λj )x By (8), there exists a atural umber N such that y ɛ = m E λj y ɛ as m>n. From (7), for m>n, ( ( m y ɛ E λj )x = m E λj )y ɛ x My ɛ x < ɛ 2 Therefore, ( m x E λj )x < ɛ 2M + ɛ 2 ɛ for m>n. It follows that ( m E λj )x = x x SpA (9) Sice from (9), we have lim m x = ( m K m j E λj )x = α ji φ ji i= K j α ji φ ji x SpA (0) i=

8 248liapig zhag where coefficiets α ji are depedet o x. Suppose K l α li φ li = 0 () l= i= Let E λ = 2 act o the two sides of (). We have K α i φ i = 0 = 2 i= Because of the liear idepedece of φ i K i= α i = 0 i = 2K = 2. Thus, the expressio (0) of x is uique. It is proved that φ i i = 2K is a basis of the subspace SpA. Secod, we show (6). From (9), the sequece m E λj m= coverges strogly. Note that G ( ) x = E λ Rz Ax ( m = lim m E λ )Rz Ax x X where G / is the pricipal part of the Lauret expasio of Rz A at the poit λ σa. G / is also strog covergece. For ay x X, defie ( ) Gzx = G x ad wzx = Rz Ax Gzx By the defiitio (2) of QAwzx QA. For ay y DA, there is a x X such that y = Rz Ax = wzx + Gzx (2) where wzx QA ad Gzx SpA. For ay x QAx 2 SpA, suppose 0 = x + x 2 (3) By (0), x 2 = K i= α iφ i.lete λ = 2 act o the two sides of (3). We have K 0 = α i φ i i= = 2 So, α i = 0 i = 2K = 2 ad x 2 = 0. From (3), x = 0. It follows that the expressio (2) is uique. We the obtai (6). The proof of Theorem 2 is complete.

9 eigefuctios of a discrete operator 249 Remark 2. Iequality (5) is also a ecessary coditio for geeralized eigefuctios of the discrete operator A to be a basis of SpA. Its proof is omitted. Corollary 2. Let A be a discrete ad desely defied operator i a complex separable Baach space X. If A satisfies (5) ad QA =0, the the geeralized eigefuctios of A costitute a basis of X. 3. RESULTS RELATED TO RESOLVENT Now we discuss the problem from aother poit of view. Theorem 3. Let A be a discrete ad desely defied operator with spectrum σa =λ i a complex separable Baach space X, ad R(z, A) a meromorphic fuctio of fiite order µ>0. Assume that there is a set of m rays, arg λ = θ j j = 2m, such that (i) the agles betwee adjacet rays are less tha π/µ, (ii) for z sufficietly large, all the poits o the m rays belog to the resolvet set ρa of A ad Rz A is bouded for these z, ad (iii) o at least oe of the rays Rz A 0 as z. The SpA =X ad QA =0. Proof. Take the quotiet space X = X/SpA. X is a Baach space with the orm x = if u = if x y u x y SpA where x X. Defie the operator à i X Dà =xx DA Ãx =Ax for x DÃ. We have that à is liear ad Rz Ãx = Rz Ax for z ρa ad x DA. Sice à has o eigevalues, Rz Ãx is a etire fuctio of fiite order at most µ o the complex plae C ad possesses the correspodig properties of Rz Ax, (i), (ii), ad (iii) metioed above. Cosider f Rx Ãx for ay x X ad f X, where X deotes the cojugate space of X. By the Phragmet Lidelof theorem ad the Liouville theorem, f Rz Ãx 0 for ay z C. Thus, Rz Ãx 0 for ay z C, amely, x =0 x X, where 0 deotes the zero elemet of X. I view of 0 =SpA, we have x SpA for ay x X, that is, SpA =X. Note that, for ay y QARz Ay is a etire fuctio of z by Theorem A. By the same demostratio above we get Rz Ay 0 for ay z C. Soy = 0, ad QA =0. The proof is complete. Theorem 3 improves the correspodig result of [4]. We cosider the case that A is a ifiitesimal geerator of a liear operator C 0 semigroup T tt 0. Related defiitios ad techical terms may be foud i [5, 6].

10 250 liapig zhag Without loss of geerality, we ca rescale ad study the C 0 semigroup T t =e wt T t, where ω>ω 0, ω 0 = lim t + t lt t So, i the sequel, we shall assume that T t M ad σa z C z < 0 where z deotes the real part of z. This will i particular simplify the otatio. Theorem 4. Let A be a discrete operator ad a ifiitesimal geerator of a liear operator C 0 semigroup T tt 0 defied o a separable complex Baach space X. If there is a sequece of cotours C l l = 2 such that (i) C l is cotaied i the left plae z < 0, ad C l iτα l τ β l costitutes a closed cotour, where α l β l are two real umbers, (ii) if z C l, the z + as l, (iii) the poles of Rz Ax x X, are uiformly bouded away from the cotours C l, that is, there exists a strictly positive real umber ν so that { ν = mi z λj z C l l= 2 } j where λ j σa j = 2, meatime, there is a t 0 > 0 such that lim e zt 0 Rz Axdz = 0 x DA (4) l C l the SpA =RT t 0 ad QA =NT t 0, where RT t 0 is the rage of T t 0 ad NT t 0 is the ull space of T t 0. Proof. First, we prove QA = NT t 0. Let x NT t 0. The T t 0 x = 0. It follows that Rz Ax = + 0 e zt T txdt = t0 0 e zt T txdt (see [5, p. 25, (7.)]) is a etire fuctio of z. Hece x QA by Theorem A. Coversely, let x QA. The for some z 0 ρarz 0 Ax QA DA because QA is a ivariat subspace of Rz 0 A. Put y = Rz 0 Ax DA. We have that Rz Ay is a etire fuctio of z ad T t 0 y = i+ e zt 0 Rz Aydz 2πi i = 2πi lim l ( 2πi lim l βl + C l α l ) e zt 0 Rz Aydz C l e zt 0 Rz Aydz = 0

11 eigefuctios of a discrete operator 25 Here we use Theorem 3. from [6, p. 92] ad the coditio (4) of Theorem 4. This is y NT t 0. Sice z 0 I ANT t 0 NT t 0 x NT t 0. Secod, we show SpA =RT t 0. Because A λ E λ is a bouded liear operator o X, where E λ is the eigeprojectio associated with λ σa A λ E λ geerates the C 0 semigroup e λt T te λ o the subspace E λ X ad for x X, e λ t T te λ x= A λ t m E m! λ x=e λ x+! A λ te λ x m=0 + + K! A λ t K E λ x =E λ x+ K m= td λ m E m! λ x where K is the algebraic multiplicity of λ ad D λ =A λ E λ is the eigeilpotet associated with λ. Note that, for x X, e zt Rz Axdz 2πi C λ = eλ t ( t k ) 2πi C λ k! k=0 ( G m+ λ m + E K λ + z λ m=0 m= = eλ t ( 2πi C λ k=0 t k k! )( K Eλ + m= D m λ m+ { = e λ t E λ x +! td λ x + 2! td λ 2 x + + = T te λ x = E λ T tx ) xdz D m λ m+ ) xdz } K! td λ K x where C λ is ay couterclockwise circle =h with 0 <h<δad δ is such that all of σa except λ lie outside the circle =δ, G λ = z λ 2πi Rz Adz C λ

12 252 liapig zhag Therefore, it holds that e zt 0 Rz Axdz + 2πi C l 2πi N l βl α l e zt 0 Rz Axdz = E λj T t 0 x x DA (5) where the itegral is alog couterclockwise ad N l deotes the umber of poles eclosed by the cotour C l ad the imagiary axis. Let l go to i the formula (5). We have N l T t 0 x = lim E λj T t 0 x l x DA I the above formula we use the coditio (4). Put S =T t 0 xx DA. The S SpA. For ay y RT t 0, there exists a x X such that y = T t 0 x. Sice DA =X, for ay ɛ>0, there is a x 2 DA such that x x 2 < ɛ/m, where M is the boud of T t 0. Cosider y T t 0 x 2 =T t 0 x x 2 Mx x 2 <ɛ Thus, S RT t 0 ad RT t 0 SpA. Cosider the coverse. Because T t 0 A AT t 0 ad for ay x X, E λ x DA, ad T t 0 E λ x = E λ T t 0 x, we have E λ X is a ivariat subspace of T t 0.Letλ be a poit spectrum of A. The e λ t 0 is that of T t 0 ad E λ X = E e λt 0 X, where E e λt 0 is the eigeprojectio associated with the eigevalue e λ t 0 of T t 0. From the formulas (2.2) ad (2.3) i [5, p. 45], for x DA where B K λ λ A K x =e λ t 0 T t 0 K x B λ x = t0 For ay y E λ X λ A K y = 0 ad 0 e λ t 0 s T sxds e λ t 0 T t 0 K y = 0 (6) Equatio (6) meas y RT t 0. It follows E λ X RT t 0, that is, SpA RT t 0. The proof of Theorem 4 is complete. Corollary 3. Let A satisfy the coditios of Theorem 4. (i) If RT t 0 = X, the Sp(A)=X. (ii) If 0 is the residual spectrum of T t 0, the SpA X.

13 eigefuctios of a discrete operator 253 ACKNOWLEDGMENTS I express my thaks to the Shaxi Provicial Natural Sciece Foudatio ad the Shaxi Provicial Fuds for Retured Research Workers for their support. REFERENCES. N. Duford ad J. T. Schwartz, Operators. Part III. Spectral Operators, pp , Wiley Itersciece, New York, T. Kato, Perturbatio Theory for Liear Operators, pp , Spriger-Verlag, Berli, Heidelberg, New York, Geqi Xu ad Sheghua Wag, The completeess of geeralised eigefuctios of a class of ifiitesimal geerators of Riesz semigroups, Acta Math. Siica 39, No. 2 (996), P. Lag ad J. Locker, Deseess of the geeralized eigevectors of a H-S discrete operator, J. Fuct. Aal. 82 (989), A. Pazy, Semigroups of Liear Operators ad Applicatios to Partial Differetial Equatios, Spriger-Verlag, New York, Qua Zheg, Strogly Cotiuous Semigroups of Liear Operators, Huazhog Uiv. of Sciece ad Techology Press, Wuha, 994. [I Chiese]