For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality w(T)≤||T||cos(π/(n+1)), where n≥2 is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes.Хаагерун и Харн для числового радіуса довільного нільпотентного оператора T у гільбертовому простторі H довели нерівність w(T)≤||T||cos(π/(n+1)), де n≥2 — порядок нільпотентності оператора T. У даній статті доведено нерівність типу нерівності Хаагеруна-Харпа для числового радіуса стиснень із більш загальних класів

Karaev, M.T.

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 517.97M. T. Karaev (Inst. Math. and Mech. Azerbaijan Nat. Acad. Sci., Baku, Azerbaijan)ON ESTIMATE OF NUMERICAL RADIUSOF SOME CONTRACTIONSPRO OCINKU ÇYSLOVOHO RADIUSA DEQKYX STYSNEN\For a numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpeproved the inequality w(T ) ≤ ‖T‖ cos πn + 1, where n ≥ 2 is the nilpotency order of the operator T. In thepresent paper, we prove a Haagerup – de la Harpe-type inequality for a numerical radius of contractions frommore general classes.Xaaherun y Xarp dlq çyslovoho radiusa dovil\noho nil\potentnoho operatora T u hil\bertovomu pros-tori H dovely nerivnist\ w(T ) ≤ ‖T‖ cos πn + 1, de n ≥ 2 — porqdok nil\potentnosti operatora T.U danij statti dovedeno nerivnist\ typu nerivnosti Xaaheruna – Xarpa dlq çyslovoho radiusa stysnen\iz bil\ß zahal\nyx klasiv.1. Introduction. Let B(H) be an algebra of bounded linear operators acting on a com-plex Hilbert space H. The numerical range of the operator T ∈ B(H) is called the setW (T ) ={(Tx, x) : x ∈ (H)1},where (H)1 ={x ∈ H : ‖x‖ = 1} is a unit sphere in the space H, and the numericalradius of an operator T is defined by equalityw(T ) = sup{ |λ| : λ ∈W (T )}.It is known that‖T‖2≤ w(T ) ≤ ‖T‖ (1)for any T ∈ B(H) (see, for instance, [1, 2]). For concrete operators, obtaining for theirnumerical radius more subtle estimates than (1) is of special interest. So, in paper [3]Haagerup and de la Harpe proved the inequalityw(T ) ≤ ‖T‖ cos πn+ 1(2)for the numerical radius of an arbitrary nilpotent operator T ∈ B(H) with the powerof nilpotency n ≥ 2 (i.e., Tn = 0, but Tn−1 = 0). In what follows other proofs ofinequality (2) of Haagerup and de la Harpe are given in the works [4 – 8] (in [7], anevident description of the numerical range of any quadric operator on a Hilbert space isalso given).In the present paper, we prove the inequalities of the type of inequality (2) of Haagerupand de la Harpe for the numerical radius of contractions from the more general classes.2. Notations and preliminaries. The notations in the paper are more or less standard.C.0 denotes the class of all contractions T on H, for which limn T ∗nx = 0 for all x ∈ H.C00 is the class of all contractions T on H, for which limn Tnx = limn T ∗nx = 0 forall x ∈ H. It is clear that C00 ⊂ C.0. It is well known (see, for instance, [9, 10] ) that ifT ∈ C.0, then it is unitary equivalent to the model operator MΘ,MΘf = PΘzf, f ∈ KΘ,where PΘ is the orthogonal projection onto KΘ,KΘ = H2(E)ΘH2(E′),c©M. T. KARAEV, 2006ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10 13351336 M. T. KARAEVwhere E = clos(I − TT ∗)H,E′ is a subspace of E, H2(E) is the Hardy space of E-valued analytic functions on the unit circle D = {z ∈ C : |z| < 1} , and Θ is B(E′, E)-valued bounded analytic function on D (the characteristic function of a contraction T ) forwhich Θ(ξ)Θ(ξ)∗ = IE for almost all ξ ∈ T = ∂D, i.e., Θ is a ∗-inner function (see [9]).In particular, if T ∈ C00, then its characteristic function Θ ∈ B(E) is a two-sided innerfunction, that is Θ(ξ)∗Θ(ξ) = Θ(ξ)Θ(ξ)∗ = IE for almost all ξ ∈ T.The proof of the following key lemma is contained, for instance, in [11].Lemma 1. Let k ≥ 2 be an integer. Thenmax{k−2∑m=0amam+1 :k−1∑m=0a2m = 1, am ≥ 0 (m = 0, 1, ..., k − 1)}= cosπk + 1.3. Estimates of a numerical radius. As is notied in [3], inequality (2) is sharp.However, the following simple example shows that in some cases one can improve in-equality (2):Example. Let NS = S(I−SS∗), where S, Sf = zf, is a shift operator on the Hardyspace H2 = H2(D), and let N be a nilpotent operator on a H, with power of nilpotencyn, n ≥ 3, and ‖N‖ < 12. We consider their orthogonal sum:A = NS ⊕N.It is clear that N2S = 0, An = 0 and ‖A‖ = 1. Since W (NS) = D1/2 (see [12, 13]),w(N) <12and W (A) is a convex hull of numerical ranges of addendums, we haveW (A) = D1/2, and therefore w(A) =12< cosπn+ 1. The latter shows that for thenumerical radius of the considered nilpotent operator A, a stronger estimate is valid.Note that in the considered example, the numerical radius of the operatorA is achievedin the element 1 + z√2⊕ 0 belonging to ker(A2) (to make sure of that, it is sufficient tonote that |(NSf, f)| =∣∣∧f(0)∧f(1)∣∣ ≤ 12for any f ∈ (H2)1). This suggests an ideathat conditions of achievability of the numerical radius of the operator on root subspaceshave to play decisive role in obtaining more subtle estimates for a numerical radius ofoperators. The theorems stated below affirm such point of view.The following our result can be also considered as an extension of already mentionedHaagerup – de la Harpe’s result to some operators from the class C.0 that are not nilpotent(recall that the unitary equivalance of operators preserves the numerical radius).Theorem 1. Let Θ be a ∗-inner function (i.e., Θ(ξ)Θ(ξ)∗ = IE for a.a. ξ ∈ T), letKΘ = H2(E)ΘH2(E′) (E is some auxillary Hilbert space and E′ ⊂ E) be a modelsubspace and let MΘ = PΘz | KΘ be a corresponding model operator. Suppose that forsome n ≥ 2, there exists x ∈ kerMnΘ ∩ (KΘ)1 such that w(MΘ) =∣∣(MΘx, x)∣∣. Thenw(MΘ) ≤ cos πn+ 1.Proof. In fact, the assertion x ∈ kerMnΘ is equivalent to the assertionΘ∗x ∈ H2−(E) ∩ znH2(E′),that is Θ∗x =∑nk=1 akzk, and since ‖x‖ = 1 and Θ is an ∗-inner function,we have‖Θ∗x‖ = 1, that is ∑nk=1 ‖ak‖2 = 1; here, H2−(E) df= L2(E)  H2(E). Now anelementary argument together with Lemma 1 yields a required estimate:ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10ON ESTIMATE OF NUMERICAL RADIUS OF SOME CONTRACTIONS 1337w(MΘ) = |(MΘx, x)| = |(zx, x)| == |(zΘ∗x,Θ∗x)| =∣∣∣∣∣n−1∑k=1(ak+1, ak)∣∣∣∣∣ ≤≤n−1∑k=1‖ak‖ ‖ak+1‖ ≤ cos πn+ 1.The theorem is proved.In particular, it follows from this theorem that, for a numerical radius of the nilpotentmodel operator MΘ with power of nilpotency n (n ≥ 3) satisfying the condition ofTheorem 1 for some k, 2 ≤ k < n, there is the estimatew(MΘ) ≤ cos πk + 1,which is more subtle than the estimate w(MΘ) ≤ cos πn+ 1.There are some more results on this direction.Proposition 1. Let T ∈ B(H) be a contraction such that, for some integer n > 0,there exists x ∈ kerTn ∩ (H)1 such that w(T ) =∣∣(Tx, x)∣∣. Then w(T ) ≤ cos πn+ 1.Proof. Actually, the assertion of this proposition is a “nonmodel variant” of Theo-rem 1. Really, for arbitrary r < 1, the operator rT has two-sided inner characteristicfunction (because ‖rT‖ < 1, and therefore rT ∈ C00). Therefore, taking into accountthat Theorem 1 is also true for the C00-class model operators, we obtain thatw(T ) = w(rT ) ≤ 1rcosπn+ 1.It remains to pass to the limit as r → 1.Proposition 2. Let T ∈ B(H) be a contraction such that, for any ε > 0, thereexists xε ∈ (H)1 such that ‖Tnxε‖ < ε for some n ≥ 2 and w(T ) ≤ |(Txε, xε)| + ε.Then w(T ) ≤ cos πn+ 1.Proof. In fact, using the Berberian construction [14], H → H0, T → T 0 (T ∈∈ B(H), T 0 ∈ B(H0)), one can reduce the condition of a proposition to the conditionof Proposition 1. We recall that the space H0 is constructed by the following way:let ustake a quatient-space of l∞(H) of H-valued bounded sequences with respect to the linearsubspaceN(H) df={x = (xm) : L.i.m.(xm, ym) = 0 for all y = (ym) ∈ l∞(H)}with the scalar product(x+N(x), y +N(y)) = L.i.m.(xm, ym)for x = (xm), y = (ym), where “L.i.m.” is generalized Banach limit on l∞. Then H0 isthe completion of the last prehilbertian space. Here, T 0 is defined asT 0x = (Txn +N(H)).It is well known (and easily verified) thatISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 101338 M. T. KARAEV‖T‖ = ∥∥T 0∥∥ , σ(T ) = σ(T 0), w(T ) = w(T 0). (3)Now pass to the reducing the condition of proposition to the condition of Proposition 1.Choose the sequence of positive numbers εm tending to zero and a sequence of vectorsxεm satisfying the condition of proposition forε = εm such that∣∣(Txεm , xεm)∣∣ → w(T ).Then for x = (xεm) we have∣∣(T 0x, x)∣∣ = |L.i.m.(Txεm , xεm)| == |lim(Txεm , xεm)| = lim |(Txεm , xεm)| ≥ w(T ) = w(T 0).Since the inequality∣∣(T 0x, x)∣∣ ≤ w(T 0) is always true, it implies the equality w(T 0) ==∣∣(T 0x, x)∣∣. Moreover,((Tn)0x, (Tn)0x) = L.i.m.(Tnxεm , Tnxεm) == L.i.m. ‖Tnxεm‖2 ≤ L.i.m.ε2m = 0,that is, (Tn)0x = 0, whence subject to (Tn)0 = (T 0)n we get (T 0)nx = 0. Thus,(T 0)nx = 0 and w(T 0) =∣∣(T 0x, x)∣∣ . Considering equality (3) and Proposition 1, weget the desired inequalityw(T ) ≤ cos πn+ 1.Proposition 3. Let Θ1, Θ2 be an inner functions. We consider the following ope-rator:KΘ1,Θ2df=[TΘ¯1 , TΘ¯2]Θ1(MΘ2),where[TΘ¯1 , TΘ2]= TΘ¯1TΘ2 − TΘ2TΘ¯1 is the commutator of antianalytic Toeplitz ope-rator TΘ¯1 and analytic Toeplitz operator TΘ2 , and Θ1(MΘ2) = PΘ2Θ1 | KΘ2 is thefunction of model operator MΘ. Then w(KΘ1,Θ2) ≤12.Proof. The simple calculations show that the operator KΘ1,Θ2 is the projection of theoperator TΘ¯1NΘ2TΘ1 on a subspace KΘ2 = H2 ΘH2, i.e.,KΘ1,Θ2 = PΘ2(TΘ¯1NΘ2TΘ1) | KΘ2 ,where NΘ2df= TΘ2PΘ2 = TΘ2(I − TΘ2TΘ¯2). In fact, considering that∣∣Θ2(ξ)∣∣ = 1 a.a.ξ ∈ T (i.e., TΘ2 is an isometry ), for each f ∈ KΘ2 we havePΘ2(TΘ¯1NΘ2TΘ1)f = PΘ2TΘ¯1TΘ2PΘ2Θ1f == (I − TΘ2TΘ¯2)TΘ¯1TΘ2Θ1(MΘ2)f == (TΘ¯1TΘ2 − TΘ2TΘ¯2TΘ¯1TΘ2)Θ1(MΘ2)f == (TΘ¯1TΘ2 − TΘ2TΘ2Θ1 Θ2)Θ1(MΘ2)f == (TΘ¯1TΘ2 − TΘ2TΘ¯1)Θ1(MΘ2)f == [TΘ¯1 , TΘ2 ]Θ1(MΘ2)f = KΘ1,Θ2f.Then for any x ∈ (KΘ2)1 we haveISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10ON ESTIMATE OF NUMERICAL RADIUS OF SOME CONTRACTIONS 1339(KΘ1,Θ2x, x) = (PΘ2(TΘ¯1NΘ2TΘ1)x, x) = (NΘ2Θ1x,Θ1x).Taking into account that Θ1x ∈ (H2)1, N2Θ2 = 0, and∥∥NΘ2∥∥ = 1, we get the followingrelation from the last equality and inequality (2):w(KΘ1,Θ2) ≤ cosπ3=12.The proof is completed.I am grateful to Yu. V. Turowski for useful discussions.1. Halmos P. R. A Hilbert space problem book. – New York: Springer, 1982.2. Gustafson K. E., Rao D. K. M. Numerical range. The field of values of linear operators and matrices. –New York: Springer, 1997.3. Haagerup U., de la Harpe P. The numerical radius of a nilpotent operator on a Hilbert space // Proc. Amer.Math. Soc. – 1992. – 115. – P. 371 – 379.4. Pop C. On a result of Haagerup and de la Harpe // Rev. roum. math. pures et appl. – 1998. – 93. – P. 9 – 10.5. Badea C, Cassier G. Constrained von Neumann inequalities // Adv. Math. – 2002. – 166. – P. 260 – 297.6. Suen C. Y. WA contactions // Positivity. – 1998. – 2. – P. 301 – 310.7. Karaev M. T. The numerical range of a nilpotent operator on a Hilbert space // Proc. Amer. Math. Soc. –2004. – 132, # 8. – P. 2321 – 2326.8. Karaev M. T. New proofs of de la Haagerup – Harpe’s inequality // Mat. Zametki. – 2004. – 75, # 5. –P. 787 – 788.9. Szo¨kefalvi-Nagy B., Foias C. Harmonic analysis of operators on Hilbert space. – Amsterdam: North-Holland, 1970.10. Nikolski N. K. Treatise on the shift operator. – New York: Springer, 1986.11. Polya G., Szego G. Problems and theorems in analysis. – Berlin: Springer, 1976. – Vol 2.12. Karaev M. T. On numerical characteristics of some operators associated with isometries // Spectral TheoryOperators and Appl. – 1997. – 11. – P. 90 – 98.13. Tso S-H., Wu P. Y. Matrical ranges of quadratic operators // Rocky Mountain J. Math. – 1999. – 29. –P. 1139 – 1152.14. Berberian S. K. Approximate proper vectors // Proc. Amer. Math. Soc. – 1962. – 13. – P. 111 – 114.Received 05.11.2003,after revision — 03.05.2006ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10

On estimate for numerical radius of some contractions

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On estimate for numerical radius of some contractions

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)