Sh. Al-sharif - R. Khalil

Red. Sem. Mat. Uiv. Pol. Torio - Vol. 62, 2 (24) Sh. Al-sharif - R. Khalil C -SEMIGROUP AND OPERATOR IDEALS Abstract. Let T (t), t <, be a oe parameter c -semigroup of bouded liear operators o a Baach space X with ifiitesimal geerator A ad R(, A) be the resolvet operator of A. The Hille-Yosida Theorem for c -semigroups asserts that the resolvet operator of the ifiitesimal geerator A satisfies R(, A) ω M for some costats M > ad R (the set of real umbers), > ω. The object of this paper is to ivestigate whe a c -semigroup ca be i some operator ideals ad to show that the previous iequality may ot hold for certai ideal orms, icludig the p-summig, uclear ad the Schatte orms. Itroductio Let X be a Baach space. A oe parameter family T (t), t <, of bouded liear operators from X ito X is called a oe parameter semigroup of bouded liear operators o X if : (i) T () I (the idetity operator) ad (ii) T (s+t) T (s)t (t) for all s, t i [, ). A semigroup, T (t), is called strogly cotiuous if lim (t)x x +T for every x X. A strogly cotiuous semigroup of bouded liear operators is called a c -semigroup. If T (t) is a c -semigroup, the there exist two costats M > ad ω (, ), such that T (t) Me ωt for all t [, ). The liear operator A defied by : T (t)x x T (t)x x D(A) {x X : lim t + t exists} ad Ax lim t + t for x D(A), is called the ifiitesimal geerator of the semigroup T (t) ad D(A) is the domai of A. The resolvet set of A is deoted by ρ(a) ad for ρ(a), the operator R(, A) ( A) 1 is the resolvet operator of A. It is kow, for c -semigroups, that D( A) is dese i X, A is a closed operator ad the resolvet operator R(, A) is a bouded operator for all ρ(a). We refer to [6] ad [12] for excellet moographs o semigroups. The Hille-Yosida Theorem (see[12]) asserts that, the resolvet operator R(, A) of the ifiitesimal geerator A of a c -semigroup T (t) satisfies the iequality : (1) R(, A) M ω for ω ad M as above ad for > ω. The orm of the resolvet operator i (1) is the usual operator orm. However, there are may importat orms o differet classes of bouded liear operators o X. It is atural to ask : Does iequality (1) hold true for orms other tha the operator orm? I this paper we address two questios : (i) If T (t) J(X) L(X) for some ideal J(X) with ideal orm. J, ca we 165 t

166 Sh. Al-sharif - R. Khalil prove a similar type of iequality of the form: (2) R(, A) J M ω for ω ad M as above ad for > ω with ρ(a)? (ii) Whe ca a c -semigroup be i some ideal J(X) L(X)? Problem (ii) was studied by Pazy [1] for compact operators K (X) L(X) o ay Baach space X. For the ideal of Hilbert-Schmidt operators o a Hilbert space H, C 2 (H ), the problem was discussed by Pazy [11]. Khalil ad Deeb [8] studied the problem for the ideal of Schatte Classes C p (H ) L(H ). Throughout this paper, the dual of a Baach X is deoted by X, ad B 1 (X) is the ope uit ball of X. For x X ad x X the value of x at x is deoted by < x, x > ad x y deotes the operator (x y)(x) < x, x > y. The set of real umbers will be deoted by R, ad the set of positive itegers by N. 1. Basic properties ad examples of operator ideals Let X be a Baach space ad L(X) be the space of all bouded liear operators from X ito X. For T L(X), T deotes the operator orm of T. DEFINITION 1. Let J(X) be a subset of L(X). The set J(X) is called a ideal of operators i L(X) if : (i) J(X) is a subspace of L(X). (ii) PT Q J(X) for all T J(X) ad P, Q L(X). (iii) J(X) cotais all fiite rak operators i L(X). A fuctio. J : J(X) [, ) is called a ideal orm o J(X) if the followigs are satisfied : (i). J is a orm o J(X). (ii) x y J x y for all oe rak operators x y. (iii) PT Q J P Q T J for all P, Q i L(X) ad T J(X). We refer to [7] ad [13] for a excellet treatmet of operator ideals. Now we preset the basic examples of ideals of operators i L(X) ad the associated ideal orms. Let p, q [1, ). (a) A operator T L(X) is called (p, q)-summig if there exists > such that (3) ( ) 1 p T (x i ) p sup x 1 ( x i, x q ) 1 q for every fiite set {x 1, x 2,...x } X. Let p,q (X) deote the set of all (p, q)- summig operators i L(X). For T p,q (X), the (p, q)-summig orm of T is T (p,q) if{ : (3) holds}. It is kow that, p,q (X) is a ideal of operators i L(X) ad. (p,q) is a ideal orm o p,q (X).

c -Semigroup, Ideals 167 (b) A operator T L(X) is called strogly (p, q)-summig if there exists > such that : (4) sup T xi, x ( ) 1 p i x i p for all fiite sets {x 1, x 2,...x } X, where the supremum is take over all fiite sets {x1,...x } X for which < x i, x > q 1 for all x B 1 (X). Let D p,q (X) deote the set of all strogly (p, q)-summig operators i L(X). For T D p,q (X), the strogly (p, q)-summig orm of T is T D(p,q) if{ : (4) holds}. It is kow that, D p,q (X) is a ideal of operators i L(X) ad T D(p,q) is a ideal orm o D p,q (X). (c) Let H be a Hilbert space ad T be a compact operator i L(H ). The T has a represetatio T i e i f i, where (e i ), ( f i ) are sequeces of orthoormal vectors i H ad lim i. The Schatte class of idex p o H, deoted by C p (H ), is i the set of compact operators T i e i f i i L(X) for which i p <. For T C p (H ), set T p sup( < T θ i, δ i > p ) 1 p, where the supremum is take over all orthoormal sequeces (θ i ), (δ i ) i H. It is kow that, C p (H ) is a ideal of operators i L(H ) ad. p is a ideal orm o C p (H ). (d) A operator T L(X) is called Cohe ( p, q)-uclear operator if there exists > such that : (5) sup T xi, x i sup x 1 ( x i, x q ) 1 q for all fiite sets { x 1, x 2,...x } X, where the supremum o the left had side of the iequality (5) is take over all (x i ) i X for which < x i, x > p 1 for all x B 1 (X). Let N p,q (X) deote the set of all Cohe (p, q)-uclear operators i L(X). For T N p,q (X), set T N(p,q) if { : (5) holds}. It is kow,[2], that N p,q (X) is a ideal of operators i L(X) ad. N(p,q) is a ideal orm o N p,q (X). (e) A operator T L(X) is called itegral operator if T admits a factorizatio: T X X P Q L j (, µ) L 1 (, µ), where µ is a fiite regular Borel measure o some compact Hausdorff space, J : L (, µ) L 1 (, µ) is the caoical iclusio of L (, µ) ito L 1 (, µ), Q L(L 1 (, µ), X ), P L(X, L (, µ)) ad I is the caoical embeddig of X ito X. Let I (X) deote the set of all itegral operators i L(X). For T I (X), set i X

168 Sh. Al-sharif - R. Khalil T it µ ( ). It is kow that I (X) is a ideal of operators i L(X) ad. it is a ideal orm o I (X). (f) A operator T L(X) is called uclear operator if T has a uclear represetatio : T xi y i with xi y i <. The set of uclear operators o X is deoted by N(X) ad for T N(X), set T if x i y i, where the ifimum is take over all uclear represetatios of T. It is kow that, N(X) is a ideal of operators i L(X) ad T is a ideal orm o N(X). Further, if X is reflexive, the N(X) I (X). Examples (a), (b), (c), (e) ad ( f ) are discussed i details i [7] ad [13]. 2. Operator ideals ad the s.d. Property Let J(X) be a ideal of operators i L(X) with ideal orm. J. DEFINITION 2. The ideal J(X) is said to have the strog domiatio property if for every sequece (T ) of operators i J(X) for which lim T x T x for all x X ad T J ξ for some ξ > ad all 1, 2, 3..., the T J(X) ad T J ξ. We will write s.d property for the strog domiatio property. If X is ifiite dimesioal, the the ideal of compact operators K (X) L(X) does ot have the s.d. property sice I, the idetity operator, is ot compact. Now, we preset some examples of operator ideals that have the s.d. property. LEMMA 1. The followig ideals of operators have the s.d. property : p,q (X), C p (H ), p 2, D p,q (X), N p,q (X) ad I (X). Proof. That p,q (X) satisfies the s.d. property follows from the defiitio. The case of C p (H ) follows from C p (H ) p,2 (H ), p 2, [9]. The proof for D p,q (X) ad N p,q (X) follows from the defiitio of such ideals. So we prove oly the case of I (X). Let (T ) be a sequece i I (X) such that sup T i ξ for some ξ > ad lim T x T x for all x X. By Propositio 17.5.2 i [7] we have : tr(t S) T it S for every fiite rak operator S i L(X). If S m xi x i, the m tr(t S) T x i, x i T it S ξ S.

c -Semigroup, Ideals 169 Sice lim T x T x for all x X we get : m tr(t S) T xi, x i ξ S. Aother applicatio of Propositio 17.5.2 i [7] implies T is a itegral operator ad T i ξ. COROLLARY 1. If Y is reflexive, the the ideal of uclear operators N(X) satisfies the s.d. property. Proof. Sice Y is reflexive, the by Theorem 17.6.4 i [7] we get N(X) I (X). Hece, N(X) satisfies the s.d. property. 3. c -semigroups ad operator ideals Let J(X) be a ideal of operators i L(X) with ideal orm. J such that (J(X),. J ) is a Baach space. We call J(X) a Baach ideal. Let T (t), t <, be a oe parameter c -semigroup of operators i L(X) with ifiitesimal geerator A ad resolvet operator R(, A). The mai result of this sectio is : Iequality (2) ca t hold true for ay ideal orm. J differet from the operator orm, for which J(X) satisfies the s.d. property. LEMMA 2. Let T (t) be a c -semigroup o a Baach space X ad J(X) be a ideal i L(X). If T (t ) J(X) for some t >, the T (t) J(X) for all t > t. Proof. For t > t, T (t) T (t t + t ) T (t )T (t t ). Sice T (t t ) is a bouded liear operator for all t > t ad T (t ) J(X), the T (t) J(X) for all t > t. LEMMA 3. Let T (t) be a c -semigroup o a Baach space X with ifiitesimal geerator A ad J(X) be a ideal i L(X) that has the s.d. property. If T (t) J(X) for all t > ad T (t) J < ξ i (, ɛ) for some ɛ >, the for < a < ɛ, the a operator G a : X X, G a x e s T (s)x ds, for ay x X, belogs to J(X) ad G a J < ξ 1 e a. Proof. For all N, sice T (t) J(X) for all t > ad T (t) J < ξ i (, ɛ) for some ɛ >, the operators G a defied by G a x is i J(X) ad G a J k1 e t k T (t k ) J k1 k1 e t k T (t k ) J e t k T (t k )x < ξ k1 e t k < ξ 1 e a,

17 Sh. Al-sharif - R. Khalil where, (k 1)a < t k < ka. Sice T (s) is strogly cotiuous, the operators G a coverge strogly to the operator G a. By the s.d. property the operator G a J(X) ad G a J < ξ 1 e a. Now we prove oe of the mai results of this paper. THEOREM 1. Let T (t) be a c -semigroup o a Baach space X with ifiitesimal geerator A ad J(X) be a Baach ideal i L(X) that has the s.d. property. The followig are equivalet : β ω for some β > (i) R(, A) J(X) for all ρ(a) ad R(, A) J ad > ω. (ii) T (t) J(X) for all t > ad T (t) J ξ i (, ε) for some ε >. Proof. (ii) (i). For N, R, > ω, defie : R (, A)x e s T (s)x ds 1 e s T ( 1 )T (s 1 )x ds 1 T ( 1 ) e s T (s 1 )x ds. Sice J(X) is a ideal i L(X), T ( 1 ) J(X) for all N, ad the operator P, defied by P(x) e s T (s 1 ) xds, is a bouded liear operator i L(X) for 1 > ω ad all N, the operators R (, A) J(X) for all ad all R, > ω >. Further more, sice T (t) is a c -semigroup, the T (s 1 ) Me ω(s 1 ), [12]. So usig (ii) for N, 1 < ɛ, we get: R (, A) J T ( 1 J )P 1 T ( 1 J ) P ξ e ω 1 e s Me ωs ds Sice T (t) J(X), by Lemma 3, the operator R (, A) R(, A), (R (, A) R(, A)) x 1 e s T (s)x ds Mξ ω e.

c -Semigroup, Ideals 171 is i J(X) ad R (, A) R(, A) J ξ (1 e / ) for N, 1 < ɛ. Sice R (, A) J(X) for all, lim (1 e / ) space, it follows that R(, A) J(X) for > ω > ad R(, A) J lim R (, A) The resolvet idetity, ad (J(X),. J ) is a Baach J lim R (, A) J Mξ lim ω e Mξ ω. R(, A) R(µ, A) (µ )R(, A)R(µ, A), implies R(µ, A) J(X) for all µ ρ(a). Coversely (i) (ii). Sice T (t) L(X) for all t > ad R(, A) J(X) for all ρ(a), it follows that R(, A)T (t) J(X). But R(, A) ω M for > ω, ad lim R(, A)T (t)x T (t)x, for all x X, sice T (t) is a c -semigroup,[12]. Hece R(, A)T (t) J R(, A) J T (t) Cosequetly, there exists γ > ad ε > such that R(, A)T (t) J γ β ω β T (t). ω for all t (, ε) ad > ω. The s.d. property of J(X) gives T (t) J(X) for all t (, ε). Lemma 2 the implies that, T (t) J(X) for all t >. Further, sice { ω : > ω } is a bouded set, it follows that for t (, ε), T (t) J γ β sup ω. As a corollary we get : THEOREM 2. Let X be a ifiite dimesioal Baach space ad T (t) be a c - semigroup i L(X). The Hille-Yosida iequality for the resolvet operator R(, A) of the ifiitesimal geerator A of T (t) is ot true if the operator orm, R(, A), of R(, A) is replaced by a ideal orm R(, A) J for ay Baach ideal J(X), properly cotaied i L(X), that satisfies the s.d. property. Proof. Sice (ii) i Theorem 1 implies that I J(X), the result follows. I the ext result, the boudedess coditio o T (t) i Theorem 1 is replaced by a itegrability coditio.

172 Sh. Al-sharif - R. Khalil THEOREM 3. Let T (t) be a c -semigroup o a Baach space X with ifiitesimal geerator A ad J(X) be a Baach ideal i L(X) that has the s.d. property. If T (t) J(X) for all t >, T L 1 ((, t ), J(X)) for some t > ad the itegral t e s T (s) J ds is bouded i, the R(, A) J(X) for all ρ(a) ad R(, A) J is bouded for large. Proof. Sice T L 1 ((, t ), J(X)), it follows that for < t < t, T L 1 ((, t), J(X)) ad < t < t, defie: lim t t T (s) J ds. For ρ(a), R, > ω ad R t (, A)x e s T (s)x ds T (t) e s T (s t)x ds. t t Sice T (t) J(X) ad the operator P, P(x) e s T (s t) xds, is a bouded operator i L(X) for > ω, the operator R t (, A) J(X) ad R t (, A) R(, A) J t e s T (s) ds t J t T (s) J ds, otig that sup e s 1. Cosequetly, lim R t (, A) R(, A) J. Further, sice J(X) is a Baach space, it follows that R(, A) J(X) for all R, s (,t) t > ω ad R(, A) J (+1)t e s T (s)ds e s T (s) ds J t (+1)t T (t ) e s T (s t ) ds t J J

c -Semigroup, Ideals 173 T (t ) t T (t ) e t t e s T (s) J ds e t e s T (s) ds t J e s T (s) J ds ( T (t ) e t ) 1 e t T (t ) t e s T (s) J ds, otig that for large, e t t T (t ) < 1. But e s T (s) J ds is bouded by assumptio. Thus R(, A) J is bouded for large. The resolvet idetity, R(, A) R(µ, A) (µ )R(, A)R(µ, A), implies that R(µ, A) J(X) for all µ ρ(a). 4. Whe is a semigroup i some operator ideal? Throughout this sectio, J(X) is a ideal i L(X) with ideal orm. J (J(X),. J ) is a Baach space. for which THEOREM 4. Let T (t) be a c differetiable semigroup o a Baach space X with ifiitesimal geerator A. If there exists ρ(a) such that R(, A) J(X), the T (t) J(X) for all t >. t Proof. With o loss of geerality, assume. Defie : B(t)x B L(X). Usig Theorem 2.4 [12] we have : t AB(t)x A T (s)x ds T (t)x x (T (t) I)x for all x X. Hece AB(t) ( A)B(t) I T (t). This implies B(t) R(, A)(I T (t)). T (s)x ds. The

174 Sh. Al-sharif - R. Khalil Sice R(, A) J(X) ad I T (t) L(X), the operator B(t) J(X) for all t >. But T (t) is strogly cotiuous. So, T (t)x d dt t ( T (s)x ds lim B(t + 1 ) )x B(t)x i X. Put D (t)x (B(t + 1 ) )x B(t)x ( R(, A) (I T (t + 1 ) ) ) x R(, A) (I T (t)) x R(, A) (T (t)x T (t + 1 ) )x. ( ) Sice T (t) is differetiable, so lim T (t) T (t + 1 ) T (t) ad Thus, T (t) J(X). T (t)x lim D (t)x R(, A)T (t). 5. Further results I this sectio we preset some results o some classes of semigroups of operators o Hilbert spaces. THEOREM 5. Let T (t) be a c -semigroup o a Hilbert space H. If T (t) C p (H ) for some p >, the T (t) C p (H ) for all p >. Proof. If p > p, the T (t) C p (H ) sice C p (H ) C p (H ), [7]. If p < p, the choose N such that p < p. Sice T ( t ) C p (H ), the T ( t ) C p, [4]. But Hece, T (t) C p (H ) C p (H ). T (t) T ( t ) T ( t ). REMARK 1. For self adjoit operators, Theorem 5 was proved i [8]. COROLLARY 2. Let T (t) be a c -semigroup of compact ormal operators o a Hilbert space H ad ( ) be the eigevalues of the ifiitesimal geerator of T (t). The followig are equivalet : (i) T (t) is compact ad e t <. (ii) T (t) C 2 (H ). 1

c -Semigroup, Ideals 175 Proof. (i) (ii). Let T (t) be a compact ormal operator i L(H ). By the spectral mappig theorem for semigroups,[1], we have T (t) e t e (t) e (t), where 1 (e (t)) is a sequece of orthoormal vectors i H. Sice 1 e t <, we have T (t) C 1 (H ) C 2 (H ). Coversely (ii) (i). Let T (t) C 2 (H ) (so T (t) is compact). Theorem 5 implies T (t) C 1 (H ). Cosequetly, e t <. 1 REMARK 2. For self adjoit operators, Corollary 2 is to be foud i [1]. THEOREM 6. Let T (t) be a c -semigroup of ormal operators o a Hilbert space H with ifiitesimal geerator A. The for ρ(a) ad Re > ω, R(, A) is a ormal operator. Proof. Sice T (t) is ormal for all t, the T (t)t (t) T (t)t (t) for all t. For s, t ad s, t Q (the set of ratioal umbers) there exist positive itegers m,, k, ad r such that s m ad t r k. Thus T (t)t (s) T ( k r )T ( m ( ) T ( 1 ) k ( r ) T ( 1 ) mr r ). Sice T (t) is ormal for all t, we have : T (t)t (s) ( T ( 1 ) mr ( r ) T ( 1 ) k r ) T (s)t (t). The desity of Q i R gives T (t)t (s) T (s)t (t) for all s, t. Now, for ρ(a), Re > ω we have : R(, A)(R(, A)) R(, A)R(, A ) e t T (t)dt e t T (t)dt e t e s T (t)t (s)dtds Thus, R(, A) is a ormal operator. R(, A )R(, A) R(, A) R(, A).

176 Sh. Al-sharif - R. Khalil REMARK 3. R(, A) eed ot be a uitary operator for ρ(a) eve if the c -semigroup, T (t), is a semigroup of uitary operators. The fuctio T : [, ) L ( L 2 [, 1] ) defied by : T (t) f (s) (t, s) f (s), where (s, t) cos st + i si st, t > defies a c -semigroup of uitary operators i L(L 2 [, 1]). But its ifiitesimal geerator A, A f (s) is f (s), is ot a uitary operator. Proof. Now, for ρ(a), ad Re > ω we have : R(, A) f (s) e t T (t) f (s)dt ( s 2 + 2 + is ) s 2 + 2 f (s). Hece, R(, A) is a multiplicatio operator with (, s) R( : A) sup s So, R(, A) is ot a uitary operator. 1 (s 2 + 2 ) 1 2 s 2 + 2 + 1 1. is s 2 + 2, ad Ackowledgmet. The authors would like to thak the referee for his soud commets that shaped up the proof of Theorem 1 ad other good commets that improved the fial form of the paper. Refereces [1] BALAKRISHNAN A.V., Applied fuctioal aalysis, Spriger-Verlag, New York 1981. [2] COHEN J. S., Absolutely p-summig, p-uclear operators ad their cojugates, Math. A. 21 (1973), 177 2. [3] DIESTEL J. AND UHL J. R, Vector measures, Amer. Math. Soc. 1977. [4] GOHBERG I. AND KREIN M., Itroductio to the theory of liear oselfadjoit operators, Amer. Math. Soc. 1969. [5] HEIKKI-APIOLA, Duality betwee spaces of p-summable sequeces, (p,q)-summig operators ad characterizatio of uclearity, Math. A. 219 (1976), 53 64. [6] HILLE E. AND PHILLIPS R. S., Fuctioal aalysis ad semigroups, Amer. Math. Soc. Colloq Publi. 31, Providece, Rhode Islad 1957. [7] JARCHOW H., Locally covex spaces, B. G. Teuber Stuttgart, Germay 1981. [8] KHALIL R. AND DEEB W., Oe parameter semigroups of operators of Schatte class C p., Fukcialaj Ekvacioj, 32 (1989), 389 394. [9] KWAPIEN S., Some remarks o (p,q)-absolutely summig operators i l p spaces, Studia Math. 29 (1968), 327 337. [1] PAZY A., O the differetiability ad compactess of semigroups of liear operators, J. Math. ad Mech. 17 (1968), 1131-1141. [11] PAZY A., Semigroups of liear operators ad applicatios to partial differetial equatios, Lecture otes, Uiversity of Marylad, 1974.

c -Semigroup, Ideals 177 [12] PAZY A., Semigroups of liear operators ad applicatios to partial differetial equatios, Spriger- Verlag, New York 1983. [13] PIETSCH A., Operator ideals, North Hollad Publishig Compay, New York 198. [14] RUDIN W., Fuctioal aalysis, McGraw-Hill, Ic. New York 1991. [15] WEIDMANN J., Liear operators i Hilbert spaces, Spriger-Verlag, New York 198. AMS Subject Classificatio: Primary 47A63, Secodary 47A35. Key words: semigroups, ifiitesimal geerator, ideal orms. Sharifa AL-SHARIF, Mathematics Departmet, Yarmouk Uiversity, Irbid, JORDAN e-mail: Sharifa@yu.edu.jo Roshdi KHALIL, Mathematics Departmet, Jorda Uiversity, Amma, JORDAN e-mail: Roshdi@Ju.edu.jo Lavoro perveuto i redazioe il 13.4.23 e, i forma defiitiva, il 12.3.24.