SPECTRUM OF THE DIRECT SUM OF OPERATORS

Transcription

1 Electroic Joural of Differetial Equatios, Vol. 202 (202), No. 20, pp. 8. ISSN: URL: or ftp ejde.math.txstate.edu SPECTRUM OF THE DIRECT SUM OF OPERATORS ELIF OTKUN ÇEVIK, ZAMEDDIN I. ISMAILOV Abstract. We study the coectio betwee spectral properties of direct the sum of operators i the direct sum of Hilbert spaces ad its coordiate operators.. Itroductio It is kow that ifiite direct sum of Hilbert spaces H, ad ifiite direct sum of operators A i H, are defied as H = H = u = (u ) : u H,, u 2 H = u 2 H < +, ad A = A, D(A) = u = (u ) H : u D(A ),, Au = (A u ) H, A : D(A) H H (see [3]). The geeral theory of liear closed operators i Hilbert spaces ad its applicatios to physical problems has bee ivestigated by may mathematicias (see for example [3]). However, may physical problems of today arisig i the modelig of processes of multi-particle quatum mechaics, quatum field theory ad i the physics of rigid bodies support to study a theory of liear direct sum of operators i the direct sum of Hilbert spaces (see [5, 9, 0, 2, 3] ad refereces i it). I this paper, a coectio betwee spectrum, resolvet sets, discreteess of the spectrum (sec. 2) ad asymptotical behavior of the eigevalues (sec. 3) of direct sum of operators defied i the direct sum of Hilbert spaces ad suitable properties of coordiate operators has bee established. The obtaied results has bee supported with applicatios. These ad related problems i the case cotiuous direct sum of the Hilbert space operators have bee ivestigated i works ([, 2, 4, ]).But i these works has ot bee cosidered a coectio betwee parts of the spectrum of direct sum 2000 Mathematics Subject Classificatio. 47A0. Key words ad phrases. Direct sum of Hilbert spaces; spectrum; resolvet set; compact operators; discrete spectrum; eigevalues. c 202 Texas State Uiversity - Sa Marcos. Submitted November 30, 20. Published November 27, 202.

2 2 E. OTKUN ÇEVIK, Z. I. ISMAILOV EJDE-202/20 operator ad suitable parts of the spectrum their coordiate operators.i this paper give sharp formulaes i the this sese. 2. O the spectrum of direct sum of operators I this sectio, the relatioship betwee the spectrum ad resolvet sets of the direct sum of operators ad its coordiate operators will be ivestigated. First of all it will be ivestigated the cotiuity ad compactess properties of the operator A = A i H = H i case whe A L(H ) for each. It is easy to see that the followig propositios are true i geeral. Theorem 2.. Let A = A, H = H ad for ay, A L(H ). I order for A L(H) the ecessary ad sufficiet coditio is sup A < +. I additio, i this case whe A L(H) it is true A = sup A (see []). Theorem 2.2. Let A C (H ) for each. I this case A = A C (H) if ad oly if lim A = 0. Furthermore, the followig mai result ca be proved. Theorem 2.3. For the parts of spectrum ad resolvet sets of the operator A = A i Hilbert space H = H the followig statemets are true σ p (A) = σ p (A ), σ c (A) = ( σ p (A )) c ( σ r (A )) c ( σ c (A )) λ ρ(a ) : sup R λ (A ) =, σ r (A) = ( σ p (A ) ) c ( σ r (A ) ), ρ(a) = λ ρ(a ) : sup R λ (A ) <. Proof. The validity of first claim of give relatios is clear. Moreover, it is easy to prove the fourth equality usig the Theorem 2.. Now we prove the secod relatio o the cotiuous spectrum. Let λ σ c (A). I this case by the defiitio of cotiuous spectrum A λe is a oe-to-oe operator, R(A λe) H ad R(A λe) is dese i H. Cosequetly, for ay a operator A λe is a oe-to-oe operator i H, there exists m N such that R(A m λe m ) H m ad for ay liear maifold R(A λe ) is dese i H or λ ρ(a m ) for each m but sup R λ (A m ) : m =. This meas that λ ( [σ c (A ) ρ(a )]) ( σ c (A )) λ ρ(a ) : sup R λ (A ) = O the cotrary, ow suppose that for the poit λ C the above relatio is satisfied. Cosequetly, either for ay, or λ σ c (A ) ρ(a ), λ ρ(a ) : sup R λ (A ) =,

3 EJDE-202/20 SPECTRUM OF THE DIRECT SUM OF OPERATORS 3 ad there exist m N such that λ σ c (A m ). That is, for ay, A is a oe-to-oe operator, R(A λe ) = H ad R(A m λe m ) H m. Ad from this it implies that the operator A = A is a oe-to-oe operator, R(A λe) H ad R(A λe) = H. Hece λ σ c (A m ). O the other had the simple calculatios show that [ (σ c (A ) ρ(a ))] [ σ c (A )] = [ σ p (A )] c [ σ r (A )] c [ σ c (A )]. By a similarly techique, we ca proved the validity of the third equality of the theorem. Example 2.4. Cosider the multi-poit differetial operator for first order, A u = u (t), H = L 2 ( ), = (a, b ), < a < b < a + < < + ; A : D(A ) H H, D(A ) = u W 2 ( ) : u (a ) = u (b ), ; A = A, H = H. For ay operator A ad A are ormal, σ(a ) = σ p (A ) = 2kπi b a : k Z ad eigevectors accordig to the eigevalue λ k,, k Z are i the form u k (t) = c k exp(λ k (t a )), t, c k C 0 (see [8]). I this case u k 2 H = c k 2 exp(λ k (t a )) 2 dt = c k 2 (b a ). The coefficiets c k may be chose such that the last series to be coverget. This meas that λ k σ p (A). From this ad Theorem 2. it is obtaied that σ p (A) = σ p (A ). Defiitio 2.5 ([6]). Let T be a liear closed ad desely defied operator i ay Hilbert space H. If ρ(t ) ad for λ ρ(t ) the resolvet operator R λ (T ) C (H), the operator T : D(T ) H H is called a operator with discrete spectrum. Note that if the operator A = A is a operator with discrete spectrum i H = H, the for every the operator A is also i H. The followig propositio is proved by usig the Theorem 2.2. Theorem 2.6. If A = A, A is a operator with discrete spectrum i H,, ρ(a ) ad lim R λ (A ) = 0, the A is a operator with discrete spectrum i H. Proof. I this case for each λ ρ(a ) we have R λ (A ) C (H ),. Now we defie the operator K := R λ (A ) i H. I this case for every u = (u ) D(A), we have K(A λe)(u ) = R λ (A )( (A λe ))(u ) = R λ (A )((A λe )u ) = (R λ (A )(A λe )u ) = (u )

4 4 E. OTKUN ÇEVIK, Z. I. ISMAILOV EJDE-202/20 ad (A λe)k(u ) = (A λe)( R λ (A ))(u ) = (A λe)(r λ (A )u ) = ( (A λe ))(R λ (A )u ) = ((A λe )R λ (A )u ) = (u ) These relatios show that R λ (A) = R λ (A ). followig operators K m : H H, m i the form Furthermore, we defie the K m u := R λ (A )u, R λ (A 2 )u 2,..., R λ (A m )u m, 0, 0, 0,..., u = (u ) H. Now the covergece i operator orm of the operators K m to the operator K will be ivestigated. For the u = (u ) H we have K m u Ku 2 H = ( ( = =m+ =m+ sup m+ sup m+ R λ (A )u 2 H R λ (A ) 2 u 2 H ) 2 R λ (A ) u 2 H R λ (A ) ) 2 u 2 H From this, K m K sup m+ R λ (A ) for m. This meas that sequece of the operators (K m ) coverges i operator orm to the operator K. The by the importat theorem of the theory of compact operators K C (H) ([3]), because for ay m, K m C (H). Example 2.7. Cosider that the family of the operators i the form A := d dt + S, S = S > 0, S C (H), A : D(A ) L 2 L 2, = (a, b ), sup(a, b ) <, D(A ) = u W2 (H, ) : u (b ) = W u (a ), A W = W A, where L 2 = L 2 (H, ),, H is ay Hilbert space ad W is a uitary operator i H, (for this see [8]). For ay a operator A is ormal with discrete spectrum ad ρ(a ). For the λ ρ(a ) ad sufficietly large a simple calculatio shows that R λ (A )f (t) = e (S λe)(t a)( E We (S λe)(b a)) W e (S λe)(b s) f (s)ds t + e (S λe)(t s) f (s)ds, f L 2,. a

5 EJDE-202/20 SPECTRUM OF THE DIRECT SUM OF OPERATORS 5 O the other had the followig estimates hold: t e (S λe)(t s) f (s)ds 2 L 2 a ( t ) 2dt e (S λe)(t s) f (s) H ds a ( t ) e (S λe)(t s) 2 ds dt f (s) 2 Hds a ( t ) = e (S λre)(t s) 2 ds dt f 2 L 2 a ( t ) = e 2λr(t s) e S(t s) 2 ds dt f 2 L 2 a ( t ) () = )(t s) ds dt f 2 L 2 = a e 2(λr λ 4(λ r λ [2(λ () r λ () )2 )(a b) + e 2(λr λ() )(b a ) ] f 2 L, 2 (E W e (S λe)(b a) ) W = (We (S λe)(b a) ) m m=0 e (S λre)(b a) m m=0 = ( e (S λre)(b a) ) = ( e (λr λ() )(b a ) ), e (S λe)(b s) f (s)ds 2 )[e2(λr λ () 2(λ r λ () )(b a ) ] f 2 L 2 Hece from (2.2) ad (2.3), we have e (S λe)(t a)( E We (S λe)(b a)) W e (S λe)(b s) f (s)ds 2 L 2 e 2λr(t a) e S(t a) 2 dt (E We (S λe)(b a) ) W 2 e (S λe)(b s) f (s)ds 2 L 2 ( 4λ r (λ r λ () ) e 2λr(b a) )( ) e (λr λ() )(b a ) ) (e 2(λr λ() )(b a ) f 2 L, 2 (2.) (2.2) (2.3) (2.4) where λ r is the real part of λ ad λ () is the first eigevalue of the operator S,.

6 6 E. OTKUN ÇEVIK, Z. I. ISMAILOV EJDE-202/20 From estimates (2.) ad (2.4) the followig result is obtaied. Propositio 2.8. If λ ρ(a ), sup(b a ) < ad λ () (S ) as, the R λ (A ) 0 as. Cosequetly, the operator A = A is a operator with discrete spectrum i L 2 = L Asymptotical behavior of the eigevalues I this sectio, asymptotical behavior for the eigevalues of the operator A = A i H = H is ivestigated, i a special case. Theorem 3.. Assume that the operators A i H ad A i H, are operators with discrete spectrum ad for i, j, i j, σ(a i ) σ(a j ) =. If λ m (A ) c m α, 0 < c, α <, m, α <, ad there exists q N such that α q = if α > 0, the λ (A) γ α, 0 < γ, α = α q < as. c Proof. First of all ote that by the Theorem 2.3 σ p (A) = σ p (A ). Here it is deoted by N(T ; λ) :=, λ 0, that is, a umber of eigevalues of the some λ(t ) λ liear closed operator T i ay Hilbert space with modules of these eigevalues less tha or equal to λ, λ 0. This fuctio takes values i the set of o-egative iteger umbers ad i case of ubouded operator T it is odecreasig ad teds to as λ. Sice for every i, j, i j, σ(a i ) σ(a j ) =, the N(A; λ) = N(A ; λ). I this case it is clear that N(A; λ) λ α c α λ α α = c α λ α α αα = c α The last series is uiformly coverget i (, ) o λ. The lim c α ( λ ) α α αq αα = cq. λ ( λ ) α α αα, λ 0. Therefore N(A; λ) cλ α, 0 < c = c /αq q, α < as λ. We have the followig asymptotic behavior of eigevalues of the operator A i H λ (A) γ α, 0 < γ, α < as. Remark 3.2. If i the above theorem the coefficiets α, satisfy the coditio if α > 0, the for every 0 < α < if α, N(A; λ) = o(λ α ) as λ. Remark 3.3. If the every fiitely may sets of the family σ(a ), i complex plae itersect i the fiitely may poits, the it ca be proved that claim of Theorem 3. is also valid i this case. Example 3.4. Let H = H, H = l 2 (N),, A : D(A ) H H, A (u m ) := (c m u m ), u = (u m ) D(A ), c m C, c m c km, k,, k, m, c m k m α, 0 < k <, α < as m, α is coverget ad there exists q N such that α q = if α. I this case, for ay, A is a liear ormal operator ad σ(a ) = σ p (A ) = m= c m. k

7 EJDE-202/20 SPECTRUM OF THE DIRECT SUM OF OPERATORS 7 Now we obtai the resolvet operator of A. Let λ ρ(a ). The from the relatio (A λe )(u m ) = (v m ),, (v m ) H, i.e c m u m λu m = v m, m. It is established that u m = vm c, m ; i.e., R v m λ λ(a )(v m ) = ( m c ), m λ. O the other had sice c m k m α, α as m, the for ay v = (v m ) H we have R λ (A )(v m ) 2 v m H = c m λ 2 c m λ 2 v m 2 = m= m= c m λ 2 v 2 H. m= Cosequetly, for ay, ( R λ (A ) c m λ 2) /2 m= m= (3.) Moreover, it is kow that a resolvet operator R λ (A ), is compact if ad oly if c m λ 0 ([7]). Sice λ c m,, m ad coditios o c m, the m the last coditio is satisfied. Hece for ay, R λ (A ) C (H ). O the other had sice the series α is coverget, the from the iequality (3.) for the R λ (A ), it is easy to see that lim R λ(a ) = 0, λ ρ(a ). Hece by the Theorem 2.6 for the λ ρ(a ) it is established that R λ (A) C (H). The by the Theorem 2.3 it is true that σ p (A) = σ p (A ). Furthermore, the validity of the relatio σ(a i ) σ(a j ) =, i, j, i j is clear. Therefore by the Theorem 3. λ (A) γ α, 0 < γ, α < as. k Ackowledgmets. The authors are grateful to Research Asst. R. Öztürk Mert (Departmet of Mathematics, Hitit Uiversity) for the iterestig ad helpful commets. Refereces [] E. A. Azoff; Spectrum ad direct itegral, Tras. Amer. Math. Soc., 97 (974), [2] T. R. Chow; A spectral theory for the direct sum itegral of operators, Math. A., 88 (970), [3] N. Duford, J. T. Schwartz; Liear Operators I, II, Secod ed., Itersciece, New York, 958; 963. [4] L. A. Fialkow; A ote o direct sums of quasiilpotet operators, Proc. Amer. Math. Soc., 46 (975), [5] F. R. Gatmakher, M. G. Krei; Oscillatig Matrices ad Kerels ad Small Oscillatios of Mechaical Systems, Gostekhteorizdat, Moscow, 950, (i Russia). [6] V. I. Gorbachuk, M. L. Gorbachuk; Boudary value problems for operator-differetial equatios, First ed., Kluwer Academic Publisher, Dordrecht, 99. [7] J. K. Huter, B. Nachtergaele; Applied Aalysis, Uiversity of Califoria, Davis, [8] Z. I. Ismailov; Compact iverses of first - order ormal differetial operators, J. Math. Aal. Appl., USA, 2006, 320 (), p [9] Z. I. Ismailov; Multipoit Normal Differetial Operators for First Order, Opusc. Math., 29, 4, (2009), [0] A. N. Kochubei; Symmetric Operators ad Noclassical Spectral Problems, Mat. Zametki, 25, 3 (979), [] M. A. Naimark, S. V. Fomi; Cotiuous direct sums of Hilbert spaces ad some of their applicatios, Uspehi Mat. Nauk., 0, 2(64) 955, -42. (i Russia).

8 8 E. OTKUN ÇEVIK, Z. I. ISMAILOV EJDE-202/20 [2] S. Timosheko; Theory of Elastic Stability, secod ed., McGraw-Hill, New York, 96. [3] A. Zettl; Sturm-Liouville Theory, First ed., Amer. Math. Soc., Math. Survey ad Moographs vol. 2, USA, Elif Otku Çevik Istitute of Natural Scieces, Karadeiz Techical Uiversity, 6080, Trabzo, Turkey address: e otkucevik@hotmail.com Zameddi I. Ismailov Departmet of Mathematics, Faculty of Scieces, Karadeiz Techical Uiversity, 6080, Trabzo, Turkey address: zameddi@yahoo.com