On a ρ n -Dilation of Operator in Hilbert Spaces

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1 E extracta mathematcae Vol. 31, Núm. 1, (2016) On a ρ n -Dlaton of Operator n Hlbert Spaces A. Salh, H. Zeroual PB 1014, Departement of Mathematcs, Sences Faculty, Mohamed V Unversty n Rabat, Rabat, Morocco rad237@gmal.com, zeroual@fsr.ac.ma Presented by Mostafa Mbekhta Receved March 21, 2016 Abstract: In ths paper we defne the class of ρ n dlatons for operators on Hlbert spaces. We gve varous propertes of ths new class extendng several known results ρ contractons. Some applcatons are also gven. Key words: ρ n-dlaton, ρ-dlaton. AMS Subject Class. (2010): 47A Introducton Sz-Nagy and Foas ntroduced n [8], the subclass C ρ of the algebra L(H) of all bounded operators on a gven complex Hlbert space H. More precsely, for each fxed ρ > 0, an operator T C ρ f there exsts a Hlbert space K contanng H as a subspace and a untary transformaton U on K such that; T n = ρp ru n H for all n N. (1) Where P r : K H s the orthogonal projecton on H. The untary operator U s then called a untary ρ-dlaton of T, and the operator T s a ρ-contracton. Recall that T s power bounded f T n M for some fxed M and every nonegatve nteger n. From Equaton (1), t follows that every ρ-contracton s power bounded snce T n ρ for all n N. Computng the spectral radus of T, t comes that the spectrum of the operator T satsfes σ(t) D, where D = D(0, 1) s the open unt dsc of the set of complex numbers C. Operators n the class C ρ enjoy several nce propertes, we lst below the most known, we refer to [7] for proofs and further nformaton. Ths work s partally supported by Hassan II Academy of Sences and the CNRST Project URAC

2 12 a. salh, h. zeroual (1) The functon ρ C ρ s nondecreasng, that s C ρ C ρ f ρ < ρ. We wll denote by C = C ρ. ρ>0 (2) C 1 concdes whth the class of contractons (see [6]) and C 2 s the class of operators T havng a numercal radus less or equal to 1 (see [1]). The numercal radus s gven by the expresson, w(t) = sup{ Th; h : h = 1}. (3) If T C ρ so s T n. It s however not true n general that the product of two operators n C ρ s n C ρ. Also t s not always true that ξt belongs to C ρ when T C ρ for ξ = 1. (4) For any M a T nvarant subspace, the restrcton of T to the subspace M s n the class C ρ whenever T s. (5) Any operator T such that σ(t) D belongs to C. Numerous papers where devoted the the study of dfferents aspects of C ρ ; we refer to [2, 4, 5] for more nformaton. The next theorem provdes a useful characterzaton of the class C ρ n term of some postvty condtons, Theorem 1.1. Let T be a bounded operator on the Hlbert space H and ρ be a nonnegatve real. The followng are equvalent (1) The operator T belongs to the class C ρ ; (2) for all h H; z D(0; 1) ( 2 ρ 1) zth 2 + (2 2 ρ )Re (zth, h) h 2 ; (2) (3) for all h H; z D(0; 1) (ρ 2) h 2 + 2Re ((I zt) 1 h, h) 0. (3) 2. Untary ρ n -dlaton We extend the noton of ρ-contractons to a more general settng. More precsely, let (ρ n ) n N be a sequence of nonnegatve numbers. We wll say that the operator T on a complex Hlbert space H belongs to the class C ρn f, there

3 on a ρ n -dlaton of operator n hlbert spaces 13 exsts a Hlbert space K contanng H as a subspace and a untary operator U such that T n = ρ n P ru n H for all n N. (4) We say n ths case that the untary operator U s a ρ n -dlaton for the operator T and the operator T wll be called a ρ n -contracton. Remark T (1) For any bounded operator T, the operator hence admts a untary dlaton. We deduce that, T s a contracton and T C ρn for ρ n = T n for all n N. We notce at ths level that, wthout addtonal restrctve assumptons on the sequence (ρ n ) n N, there s no hope to construct a reasonable ρ n -dlaton theory. Our goal wll be to extend the most usefull propertes of ρ contracton to ths more general settng. (2) From Equaton 4, for T C ρn wth U a ρ n -dlaton, we obtan T n ρ n P ru n H ρ n. Therefore the condton lm n (ρ n ) 1 n 1 wll ensure that σ(t) D(0; 1). (3) In contrast wth the class C ρ, the class C (ρn ) s not stable by powers. However, f T C ρn and k 1 s a gven nteger, we obtan T k C ρkn. Ths latter fact can be seen as a trval extenson of the case ρ n = ρ 0 for every n. In the remanng part of ths paper, we wll assume that (ρ n ) n N s a sequence of nonnegatve numbers satsfyng lm (ρ n) 1 n 1. (5) n We assocate wth the sequence (ρ n ) n N, the followng functon, ρ(z) = n 0 z n ρ n. It s easy to see that condton lm n (ρ n ) 1 n 1 mples that ρ H(D). Here H(D) s the set of holomorphc functons on the open unt dsc D. Also, the valued-operators functon ρ(zt) = n 0 z n T n ρ n

4 14 a. salh, h. zeroual s well defned and converges n norm for every z < 1. We gve next a necessary and suffcent condton to the membershp to the class C ρn ; Theorem 2.2. Let T be an operator on a Hlbert space H and (ρ n ) n N s a sequence of nonnegatve numbers. The operator T has a ρ n -dlaton f and only f (1 2 ρ 0 ) h 2 + 2Re ρ(zt)(h); h 0 for all h H; z D(0; 1). (6) We recall frst the next well known lemma from [7, Theorem 7.1] that wll be needed n the proof of the prevous theorem. Lemma 2.3. Let H be a Hlbert space, G be a multplcatve group and Ψ be an operator valued functon s G Ψ(s) L(H) such that Ψ(e) = I, e s the dentty element of G Ψ(s 1 ) = Ψ(s) s G t G (Ψ(t 1 s)h(s); h(t)) 0 for fntely non-zero functon h(s) from G. Then, there exsts a Hlbert space K contanng H as a subspace and a untary representaton U of G, such that and Ψ(s) = P r(u(s)) (s G) K = U(s)H s G Proof of Theorem 2.2. Let T be a bounded operator n the class C ρn and U be the untary ρ n -dlaton of T, gven by the expresson 4. We have clearly, I + 2 z n U n converges to (I + zu)(i zu) 1 for all complex numbers z such that z < 1. And P r(i + 2 z n U n ) = I + 2 z n T n. ρ n

5 on a ρ n -dlaton of operator n hlbert spaces 15 By wrtng, I + 2 z n T n = (1 2 )I + 2 z n T n ρ n ρ 0 ρ n n 0 = (1 2 ρ 0 )I + 2ρ(zT), we get P r((i + zu)(i zu) 1 ) = (1 2 ρ 0 )I + 2ρ(zT). On the other hand, (I + zu)k; (I zu)k = k 2 + zuk; k k; zuk zuk 2 It follows that for every k K, we have Re (I + zu)k; (I zu)k = k 2 zuk 2 Now f we take h = (I zu)k we wll fnd, = k 2 z 2 k 2 = k 2 (1 z 2 ) 0 snce z < 1. Re (I + zu)(i zu) 1 h; h = Re P r(i + zu)(i zu) 1 h; h and hence for every h H, we obtan or equvalently, = Re (1 2 ρ 0 )h + 2ρ(zT)(h); h, Re (1 2 ρ 0 )h + 2ρ(zT)(h); h 0 (1 2 ρ 0 ) h 2 + 2Re ρ(zt)(h); h 0 for every h H and all complex number z such that z < 1. Conversely, let us show that condton (6) mples that the operator T belongs to the class C ρn. To ths am, assume that (6) s satsfed and take 0 r < 1 and 0 ϕ < 2π. We ntroduce the next operator valued functon Q(r; ϕ) = I + r n ρ n (e nϕ T n + e nϕ T n ).

6 16 a. salh, h. zeroual Then Q(r; ϕ) converges n the norm operator for every r and ϕ. Moreover, from the nequalty 6, we have for every l H. Therefore J = 1 2π 2π 0 Q(r; ϕ)l; l 0 Q(r; ϕ)h(ϕ); h(ϕ) dϕ 0 for every h(ϕ) = + h ne nϕ where (h n ) n Z s a sequence wth only fnte number of nonzero elements n H. We have J =: + h n 2 + m n>m r n m ρ n m T n m h n ; h m + m n<m for every 0 r < 1. Now takng r 1 wll mply Ψ (ρn)(n m)h n ; h m 0, n Z m Z r m n ρ m n T (m n) h n ; h m where Ψ (ρn) : Z L(H) s defned by Ψ (ρn)(0) = I, Ψ (ρn)(n) = 1 ρ n T n and Ψ (ρn )( n) = 1 ρ n T n for every n > 0. It s mmedate that Ψ (ρn )(n) s nonnegatve on the addtve group Z of ntegers. Usng Lemma 2.3, there exsts a untary operator U on a Hlbert space K contanng H as a subspace and such that Ψ (ρn )(n) = P ru(n) for all n Z. Therefore for all n N The proof s completed. T n = ρnp ru n H. Remark 2.4. In the case where (ρ n ) n N s a constant sequence, that s ρ n = ρ for all n N wth ρ > 0, we obtan and hence, the nequalty 6 becomes ρ(z) = 1 1 z (1 2 ρ ) h 2 + 2Re (I zt) 1 h; h 0 for all h H and z D. We substtute h by l = (I zt) 1 h to retreve relaton 3 and by Theorem (2.1) we obtan T s a ρ-contracton.

7 on a ρ n -dlaton of operator n hlbert spaces 17 The next two corollares are mmedate consequences of Equaton (6) and are related to analogous results of ρ contracton. Corollary 2.5. Let T C ρn and M a T nvarant subspace. Then T M C ρn. Proof. It suffces to see that Equaton 6 s close to restrctons. Corollary 2.6. Let T be n the class C ρn then T s n the class C rρn. and r 1 be a real number, Proof. The nequalty 6 s equvalent to Plugng rρ n nstead of ρ n, we get (ρ 0 2) h 2 + 2ρ 0 Re ρ(zt)(h); h 0. and thus Therefore T C (rρn ). We also have, (rρ 0 2) h 2 + 2rρ 0 Re 1 ρ(zt)(h); h 0, r (1 2 ) h 2 + 2Re 1 ρ(zt)(h); h 0. rρ 0 r Proposton 2.7. Let T be a bounded operator on a Hlbert space H. Then for every α > 2, there exsts Γ(α) > 0 such that the operator T belongs to C (ρn ), where ρ n s a sequence gven by ρ n = Γ(α). T n (1 + n α ). Proof. Let Γ > 0 and ρ α (z) = ρ α (zt) = For every vector h n H, we set z n Γ. T n (1+n α ) z n T n Γ. T n (1 + n α ) A(z) = ρ α (zt )h; h for all z 1. Then for all z 1.

8 18 a. salh, h. zeroual Settng a n = z n A(z) = n 0 Γ. T n (1 + n α ) T n h; h T n h; h Γ. T n (1 + n α ) zn. n 0 T n h;h Γ. T n (1+n α ), we have a n T n h 2 Γ. T n (1 + n α ) = h 2 Γ.(1 + n α ) <. We conclude that A(z) s holomorphc n the unt dsc and contnuous on the boundary. Snce the maxmun s attaned of the crcle z = 1, we obtan A(z) = n 0 a n z n Now, snce n 0 Γ = 2 n 0 and then 1 1+(n) α 1 1+n α wll leads to a n z n = n 0 n 0 h 2 Γ.(1 + n α ) a n s a convergent sequence (α > 2), then choosng A(z) 1 2 h 2, h 2 + 2Re ρ α (zt )h; h 0 for all h H and z D. Fnally, Inequalty (6) s satsfed and the operator T belongs to the class C (ρn ). 3. The Bergmann shft We devote ths secton to the membershp of the Bergmann shft to the class C (ρn ) for some sutable sequence ρ n. Let H be a Hlbert space and (e ) N be an orthonormal bass of H. Recall that for a gven sequence (ω n ) n N of non negatve numbers; the weghted shft S ω assocated wth ω n s defned on the

9 on a ρ n -dlaton of operator n hlbert spaces 19 bass by S ω (e n ) = ω n e n+1. A detaled study on weghted shfts can be found n the survey [9]. On the other hand; the membershp of weghted shfts to the class C ρ s nvestgated n [3]. The Bergman shft s the weghted shft defned on the bass by the expresson T e n = w n e n+1, where w n = n + 1 n for all nteger n N. It s easy to see that, T = sup(w n ) n N = 2. The weght (w n ) n N s decreasng and then T n = n w = n + 1. =1 In partcular T s not power bounded and hence does not belong to the class C ρ for any ρ > 0. We have Proposton 3.1. Let T be the Bergmann shft and ρ n be the sequence gven by ρ n = n α for some α > 0. Then for every α > 2, there exsts Γ(α) such that T C Γ(α)ρn. Proof. Let Γ > 0 and ρ α (z) = ρ(zt) = zn Γn α for all z 1 n that z n T n for all z 1. Γnα We set, S = ρ(zt) and let x be a vector n H. Therefore S(x) = ρ(zt)(x) = 1 z n T n x Γn α. Wrtng x = 1 x e, we get S(x) = S(x); e e = ( x j Se j ; e )e, 1 1 j 1

10 20 a. salh, h. zeroual and Se j ; e = z n T n Γn α (e j); e = z n Γn α Tn (e j ); e It follows that and then = z n j+n 1 Γn α ( w p )e j+n ; e. p=j Se j ; e = z j Γ 1 ( j)α w p, p=j S(x) = 1 1 z j ( w p )x j Γ( j) 2( α )e. p=j For the Bergman shft, we have 1 p=j w p = j and thus ρ(zt)(x) = 1 Γj( j) 2( α x jz j )e. Fnally, we conclude that the nequalty (6) s equvalent to x 2 + 2Re ( 1 2 If we consder the functon A(z) = and we wrte n = j, we wll obtan, A(z) = n=1 Γj( j) α x x j z j ) 0. Γj( j) α x x j z j Γ( n)γn α x x n z n = 2 n+1 Γ( n)n α x x n z n. We denote by (Â(n)) n = (a n ) n N the sequence of coeffcents of A, a n = 1 2 n+1 Γn α ( n) x x n

11 on a ρ n -dlaton of operator n hlbert spaces 21 Snce a n = n( n) = 1 n + 1 n 2 for every n + 1, we obtan 1 2 n+1 Γn α ( n) x x n 1 Γn α 1 n+1 and by the Cauchy-Schwartz nequalty, t follows, a n 1 Γn α 1 x 2. x x n 1 Γn α 1 n+1 x x n ; We deduce that A(z) s holomorphc n the open unt dsc and contnuous on the closed unt dsc. As the maxmum s attaned on the crcle z = 1, we have A(z) = 1 2 ( n+1 a n z n = 1 Γn ( n) x x n )z n α a n. 1 Now, snce s a convergente sequence (α 2), choosng Γ = (n) α 1 would lead us to 1 (n) α 1 A(z) x 2 = 1 x 2. We derve that, Re (A(z)) A(z) x 2 = 1 x 2, and hence 1 2 Re ( 2 1 jγ( j) α x x j z j ) x 2. 1 Therefore for all x H and a complex z such that z 1 we have x 2 + 2Re ( jγ( j) α x x j z j ) 0. We conclude that the weghted shft {w n } s a ρ n -contracton wth ρ n = Γn α.

12 22 a. salh, h. zeroual Remark 3.2. We clam that for every α 1 the Bergmann shft belongs to a class C,n α. A proof s not avalable for ths clam; however t s motvated by the ncomplete computatons below. Let us set, for exemple, ρ n = 4.n for all nteger n 1, Let H be a Hlbert space and (e ) N be a an orthonormal bass for the Hlbert space H. Consder the Bergmann shft defned on the bass by Te n = n+1 n e n+1 for all n N. Then as n the proof of the prevous proposton, we show that nequalty (6) s equvalent to the next We wrte x 2 +Re ( 1 2 x 2 + Re ( j( j) x x j z j ) 1 2j( j) x x j z j ) 0. (7) x j( j) x x j, and x j( j) x x j =,j 1 a,j x x j, wth { a; = 1 for all 1 a ;j = 4j ( j) for all j Then to show nequalty (7), t suffces to prove that the nfnte symmetrc matrx wth the real entres M = [a ;j ] s nonnegatve. To ths am, we compute the determnant of the frst n n-corner, to check f t s nonnegatve. An attempt on classcal softwares allow to show ths fact for n 150. It s hence reasonable to conjecture that the Bergman shft belongs to C,n. References [1] C.A. Berger, A strange dlaton theorem, Notces Amer. Math. Soc. 12 (1965), 590. [2] G. Casser, H. Zeroual, Operator matrces n class C ρ, Lnear Algebra and ts Applcatons 420 (1-2) (2007), [3] G. Ecksten, A. Racz, Weghted shfts of class C ρ, Acta Sc. Math. (Szeged) 35 (1973),

13 on a ρ n -dlaton of operator n hlbert spaces 23 [4] G. Ecksten, Sur les opérateurs de la classe C ρ, Acta Sc. Math.(Szeged) 33 (3-4) (1972), [5] H. Mahzoul, Vecteurs cyclques, opérateurs de Toepltz généralsés et régularté des algèbres de Banach, Thèse, Unversté Claude Bernard, Lyon 1, [6] B. Sz.-Nagy, Sur les contractons de l éspace de Hlbert, Acta Sc. Math. (Szeged) 15 (1-1) (1953), [7] B. Sz.-Nagy, C. Foas, Harmonc Analyss of Operators on Hlbert Space, North-Holland, Amsterdam, [8] B. Sz Nagy, C. Foas, On certan class of power bounded operators n Hlbert space, Acta. Sc. Math. (Szeged) 27 (1-2) (1996), [9] A.L. Shelds, Weghted shft operators and analytc functon theory, n Topcs n Operator Theory, Mathematcal Survey and Monograph 13, Amer. Math. Soc., Provdence, RI, 1974,