Available online at www.sciencedirect.com Linear Algebra its Applications 48 008 980 994 www.elsevier.com/locate/laa Inequalities for the numerical radius, the norm the maximum of the real part of bounded linear operators in Hilbert spaces S.S. Dragomir School of Computer Science Mathematics, Victoria University, P.O. Box 448, Melbourne City, VIC 800, Australia Received 3 May 007; accepted 3 January 008 Available online 0 March 008 Submitted by M. Tsatsomeros Abstract Some inequalities for the numerical radius, the operator norm the maximum of the real part of bounded linear operators in Hilbert spaces, under suitable assumptions for the involved operator, are given. 008 Elsevier Inc. All rights reserved. AMS classification: 47A; 47A30; 47A63 Keywords: Numerical radius; Operator norm; Semi-inner products; Maximum minimum of the real part of bounded linear operators; Banach algebra. Introduction Let H ;, be a complex Hilbert space. The numerical range of an operator A is the subset of the complex numbers C givenby[8,p.] WA ={ Ax, x, x H, x =}. It is well known see for instance [8] that: E-mail address: sever.dragomir@vu.edu.au URL: http://rgmia.vu.edu.au/dragomir. 004-3795/$ - see front matter 008 Elsevier Inc. All rights reserved. doi:0.06/j.laa.008.0.08
S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 98 p The numerical range of an operator is a convex subset of C the Toeplitz Hausdorff theorem; pp The spectrum of an operator is contained in the closure of its numerical range; ppp A is self-adjoint if only if W is real. The numerical radius wa of an operator A on H is defined by [8, p.8] wa = sup{,λ W A} =sup{ Ax, x, x =}. It is well known that w is a norm on the Banach algebra BH of all bounded linear operators defined on the real or complex Hilbert space H the following inequality holds true [8, p.9] wa wa for any A BH. For classical results on numerical ranges of operators on normed spaces of elements of normed algebras, see the books [,] the original references therein. For a bounded linear operator A on the complex Hilbert space, consider the maximum the minimum of the spectrum of the real part of A denoted by v si A := sup inf Re Ax, x =λ maxminrea. x = x = The following properties are obvious by the definition: a v s A = v i A, A BH; aa v i A 0 for accretive operators on H ; aaa v si A + B v si A + v si B for any A, B BH; av max{ v i A, v s A } = wrea wa for all A BH. More properties which connect these functionals with the semi-inner products generated by the operator norm the numerical radius are outlined in the next section. An improvement of Lumer s classical result [, Lemma ] some bounds are also given. In the previous work [4], in order to estimate how close the numerical radius is from the operator norm, the following reverse inequalities have been obtained under appropriate conditions for the involved operator A BH. If λ C\{0}, r>0 A λi r,. then 0 wa r. In addition, if >r. holds true, then r wa, which provides a refinement of the general inequality wa, in the case when r λ satisfy the assumption r/ 3/.
98 S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 With the same assumption on λ r, i.e. >r,we also have the inequality 0 w r A + r wa provided. holds true. We recall that the bounded linear operator B BH is called strongly m-accretive with m>0 if Re By, y mfor any y H, y =. For m = 0 the operator is called accretive. In general, we then can call the operator m-accretive for m [0,. In the same paper, on assuming that A ϕiφi A is accretive or sufficiently, selfadjoint nonnegative in the operator order of BH, where ϕ,φ C, φ/= ϕ,ϕ, wehave proved the following inequality as well: 0 wa φ ϕ 4 φ + ϕ. If we assume more, i.e., Reφ ϕ > 0 which implies φ/= ϕ, then for A as above, we also have: Reφ ϕ φ + ϕ wa 0 w A [ φ + ϕ Reφ ϕ]wa. Motivated by the above results, we establish in the present paper some upper bounds for the nonnegative quantities v s μa wa 0 wa v s μa 0, for some μ C, μ =under suitable assumptions on the involved operator A BH.Lower bounds for the quantities v sμa wa v sμa wa are also given. They improve some results from the earlier paper [4]. Inequalities in terms of the semi-inner products that can naturally be associated with the operator norm the numerical radius are provided as well. For other recent results concerning inequalities between the operator norm numerical radius see the papers [5,6,7,0,]. Lower bounds for wa are in the finite-dimensional case studied in [3]. For classical results, see the books [8,9] the references therein.. Preliminary results for semi-inner products In any normed linear space E,, since the function f : E R, fx= x is convex, one can introduce the following semi-inner products see for instance [3]: x,y i := lim t 0 y + tx y t, x,y s := lim t 0+ y + tx y,. t where x,y are vectors in E. The mappings, s, i are called the superior respectively the inferior semi-inner product associated with the norm. For the sake of completeness we list here some properties of, si that will be used in the sequel. We have, for p, q {i, s} p/= q, that i x,x p = x for any x E. ii ix,x p = x,ix p = 0 for any x E. iii λx, y p = λ x,y p = x,λy p for any λ 0 x,y E.
S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 983 iv x,y p = x, y p = x,y q for any x,y E. v ix,y p = x,iy p for any x,y E. vi The following Schwarz type inequality holds: x,y p x y for any x,y E. vii The following identity holds: αx + y,x p = α x + y,x p for any α R x,y E. viii The following subsuper-additivity property holds: x + y,z p x,z p + y,z p for any x,y,z E, where the sign applies for the superior semi-inner product, while the sign applies for the inferior one. ix The following continuity property is valid: x + y,z p y,z p x z for any x,y,z E. x From the definition we have the inequality x,y i x,y s for any x,y E. In the Banach algebra BH we can associate to both the operator norm the numerical radius w the following semi-inner products: A, B si,n := A, B si,w := B + ta B lim t 0+ t lim t 0+. w B + ta w B,.3 t respectively, where A, B BH. It is obvious that the semi-inner products, si,nw defined above have the usual properties of such mappings defined on general normed spaces some special properties that will be specified in the following. As a specific property that follows by the well known inequality between the norm the numerical radius of an operator, i.e., wa for any A BH, wehave A, I i,n A, I i,w A, I s,w A, I s,n.4 for any A BH, where I is the identity operator on H. We also observe that A, I si,n = A, I si,w = for any A BH. lim t 0+ I + ta t wi + ta lim t 0+ t
984 S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 It may be of interest to note that A, I s,n A, I s,w are also called the logarithmic norms of A corresponding to w respectively. Logarithmic norms corresponding to a given norm have been rather widely studied mainly in the finite-dimensional case; see [4]. The following result is due to Lumer see [, Lemma ] was obtained originally for the numerical radius of operators in Banach spaces: Theorem []. If A BH, then A, I p,n = v p A, p {s, i}. The following simple result provides a connection between the semi-inner products generated by the operator norm by the numerical radius as follows: Theorem. For any A BH, we have: A, I p,n = A, I p,w, where p {s, i}..5 Proof. Let us give a short proof for the case p = s. Suppose x H, x =. Then for t>0we obviously have: Re x + tax,x Re Ax, x = t.6 x + tax,x wi + ta. t t Taking the supremum over x H, x =, we get wi + ta v s A = sup Re Ax, x x = t for any t>0, which implies, by letting t 0+ that sup x = Re Ax, x A, I s,w.7 for any A BH. By Lumer s theorem we deduce then A, I s,n A, I s,w since, by.4 wehave A, I s,w A, I s,n the equality.5 is obtained. Now, on employing the properties of the semi-inner products outlined above, we can state the following properties as well: va v si A = A, I si,w for any A BH; vaa v si A = v si αi + A α for any α R A BH; vaaa v si A + B v si B wa for any A, B BH. The following inequalities may be stated as well: Theorem 3. For any A BH λ C we have { [ + [ + ] ] v s λa A λi, 4 [ A + λi A λi ],.8
S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 985 { [w A + [w A + ] ] v s λa w A λi, 4 [w A + λi w A λi], respectively..9 Proof. Let x H, x =. Then, obviously 0 Ax Re[ λax, x ]+ = A λix A λi, which is equivalent with [ Ax + ] A λi Re λax, x [ Ax + ].0 for any x H, x =. Taking the supremum over x = we get the first inequality in.8 the one from the first branch in the second. For x H, x =we also have that Ax + λx = Ax λx + 4Re λax, x,. which, on taking the supremum over x =, will produce the second part of the second inequality in.8. The second inequality may be proven in a similar way. The details are omitted. It is well known, in general, that the semi-inner products, si defined on Banach spaces are not commutative. However, for the Banach algebra BH we can point out the following connection between I,A si,nw the quantities v i A v s A, where A BH. Corollary. For any A BH we have v i A = A, I i,n [ I,A s,n + I,A i,n ] A, I s,n = v s A.. v i A = A, I i,w [ I,A s,w + I,A i,w ] A, I s,w = v s A..3 Proof. We have from the second part of the second inequality in.8 that [ A + ti A ti ] v s A.4 t t for any t>0. Taking the limit over t 0+ noticing that A ti lim = I,A t 0+ t s,n = I,A i,n, we get the second inequality in..
986 S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 Now, writing the second inequality in. for A, weget v s A [ I, A s,n + I, A i,n ] = [ I,A s,n + I,A i,n ], which is equivalent with the first part of.. Since wa, hence the first inequality in.9 provides a better upper bound for v s λa than the first inequality in.8. 3. Reverse inequalities in terms of the operator norm The following result concerning reverse inequalities for the maximum of the spectrum of the real part the operator norm of A BH may be stated: Theorem 4. For any A BH\{0} λ C\{0} we have the inequality: λ 0 wa v s A A λi. 3. In addition, if A λi, then we have: λ A I v s λ A wa λ 0 w A vs A λ A λi v s A A λi wa, respectively. Proof. Utilizing the property av, we have λ wa = w A λ λ v s A v s A for any λ C\{0} the first inequality in 3. is proved. By the arithmetic mean geometric mean inequality we have [ + ], which, by.8 provides v s λa A λi, that is equivalent with the second inequality in 3.. 3. 3.3
S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 987 Utilizing the second part of the inequality.8 under the assumption that A λi we can also state that [ v s λa ] + A λi. 3.4 By the arithmetic mean geometric mean inequality we have now: [ ] + A λi A λi, 3.5 which, together with 3.4 implies the first inequality in 3.. The second part of 3. follows from av. From the proof of Theorem 3 we can state that Ax + Re λax, x +r, x =, 3.6 where we denoted r := A λi. We also observe, from 3.6, that Re λax, x > 0 for x H, x =. Now, if we divide 3.6byRe λ Ax, x > 0, we get Ax + Re λ Ax, x r Re λ Ax, x for x =. 3.7 Re λ Ax, x If in this inequality we subtract from both sides the quantity Re λ Ax, x, then we get Ax λ Re Re λ Ax, x Ax, x + r λ Re Re λ Ax, x Ax, x = r λ Re Re λ Ax, x r Ax, x r, which obviously implies that Ax λ λ Re Ax, x + r Re Ax, x for any x H, x =. Now, taking the supremum in 3.8 over x H, x =, we deduce the second inequality in 3.3. The other inequalities are obvious the theorem is proved. 3.8
988 S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 The following lemma is of interest in itself. Lemma. For any A BH γ,γ C we have: v i [A γiγi A] = 4 Γ γ A γ + Γ I. 3.9 Proof. We observe that, for any u, v, y H we have: Re u y,y v = 4 u v y u + v. 3.0 Now, choosing u = Γx, y = Ax, v = γx with x H, x =weget Re Γx Ax, Ax γx = 4 Γ γ Ax γ + Γ x, giving inf x = Re A γiγi Ax, x = 4 Γ γ sup Ax γ + Γ x x =, which is equivalent with 3.9. The following result providing a characterization for a class of operators that will be used in the sequel is incorporated in: Lemma. For A BH, γ,γ C with Γ /= γ q R, the following statements are equivalent: i The operator A γiγi A is q -accretive; ii We have the norm inequality: A γ + Γ I 4 Γ γ q. 3. The proof is obvious by Lemma the details are omitted. Since the self-adjoint operators B satisfying the condition B mi in the operator partial under, are m-accretive, then, a sufficient condition for C γ,γ A := A γiγi A to be q -accretive is that C γ,γ A is self-adjoint C γ,γ A q I. Corollary. Let A BH, γ,γ C with Γ /= ±γ q R. If the operator C γ,γ A is q - accretive, then Γ + γ 0 wa v s Γ + γ A 3. [ ] γ + Γ 4 Γ γ q. If M,mare positive real numbers with M>m the operator C m,m A = A mimi A is q -accretive, then
S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 989 [ ] 0 wa v s A M + m 4 M m q. 3.3 We observe that for q = 0, i.e., if C γ,γ A respectively C m,m A are accretive, then we obtain from 3. 3.3 the inequalities: Γ + γ 0 wa v s Γ + γ A Γ γ 3.4 4 Γ + γ M m 0 wa v s A 4M + m, 3.5 respectively, which provide refinements of the corresponding inequalities.7.34 from [4]. For any bounded linear operator A we know that wa useful result only if λ A I, which is equivalent with 3 A λi. 3.6 In conclusion, for A BH\{0} λ C\{0} satisfying the condition 3.6, the inequality 3. provides a refinement of the classical result:, therefore 3. would produce a wa, A BH. 3.7 Corollary 3. Assume that λ/= 0 or A/= 0.If A λi, then we have λ 0 w A vs A A λi. 3.8 The proof follows by the inequality 3.. The details are omitted. The following corollary providing a sufficient condition in terms of q -accretive property may be stated as well: Corollary 4. Let A BH\{0} γ,γ C, Γ /= γ, q R so that ReΓ γ+ q 0. If C γ,γ A is q -accretive, then ReΓ γ+ q Γ + γ v s Γ+ γ Γ+γ A wa 3.9 0 w A vs Γ + γ Γ + γ A [ Γ + γ 4 Γ γ q ] Γ γ. 3.0 Γ + γ
990 S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 If γ,γ q are such that Γ + γ 4 ReΓ γ+ q, then 3.9 will provide a refinement of the classical result 3.7. If M>m 0 the operator C m,m A is q -accretive, then Mm + q m + M v sa wa 3. [ ] 0 w A vs A m + M 4 M m q. 3. We also observe that, for q = 0, i.e., if C γ,γ A respectively C m,m A are accretive, then we obtain: ReΓ γ Γ + γ Mm m + M v sa v s Γ+ γ Γ+γ A wa 0 w A v s wa, 3.3, 3.4 Γ + γ Γ + γ A Γ γ 3.5 Γ + γ 0 w A vs A M m, 3.6 m + M respectively, which provides refinements of the inequalities.7,.3.0 from [4], respectively. The inequality between the first the last term in 3.6 was not stated in [4]. Corollary 5. Let A BH, γ,γ C, Γ /= γ, q R so that ReΓ γ+ q 0.IfC γ,γ H is q -accretive, then 0 w A vs Γ + γ Γ + γ A Γ + γ Re Γ γ + q Γ + γ v s Γ + γ A 3.7 Γ + γ Re Γ γ + q wa. The proof follows by the last part of Theorem 4. The details are omitted. If M>m 0 the operator C m,m A is q -accretive, then 0 w A vs A M + m Mm + q v s A M + m Mm + q wa. 3.8 Finally, for q = 0, i.e., if C γ,γ A respectively C m,m A are accretive, then we obtain from 3.7 3.8 some refinements of the inequalities.9.33 from [4].
S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 99 4. Reverse inequalities in terms of the numerical radius It is well known that the following lower bound for the numerical radius wa holds see av from Section v p A wa, p {s, i} 4. for any A a bounded linear operator, where, as in Section, v si A = A, I si = sup inf Re Ax, x =λ maxminrea. 4. x = x = It is then a natural problem to investigate how far the left side of 4. from the numerical radius is? We start with the following result: Theorem 5. For any A BH\{0} λ C\{0} we have 0 wa λ λ v s A wa v s A w A λi A λi. Moreover, if wa λi, then we have: v λ s w λ A I A wa v λ s A wa 4.3 4.4 λ 0 w A vs A w A λi w A λi A λ v s wa, 4.5 respectively. Proof. The argument is similar with the one from Theorem 4 the details are omitted. The following lemma is of interest. Lemma 3. For any A BH γ,γ C we have inf Re[ ΓI Ax, x x,a γix ] = x = 4 Γ γ w A γ + Γ I. 4.6 Proof. We observe that for any u, v, y complex numbers, we have the elementary identity: Re[u yȳ v] = 4 u v y u + v. 4.7
99 S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 If we choose in 4.7 u = Γ, y = Ax, x v = γ with x H, x =, then by 4.7 we have: Re[ ΓI Ax, x x,a γix ] = 4 Γ γ A γ + Γ I x,x 4.8 for any x H, x =. Now, taking the infimum over x =in4.8 we deduce the desired identity 4.6. We observe that for any x H, x =wehave μa; γ,γx := Re[ ΓI Ax, x x,a γix ] = ReΓ Re Ax, x Re Ax, x Reγ + ImΓ Im Ax, x Im Ax, x Imγ therefore a sufficient condition for μa; γ,γx to be nonnegative for any x H, x = is that: { ReΓ Re Ax, x Reγ, x H, x =. 4.9 ImΓ Im Ax, x Imγ Now, if we denote by μ i A; γ,γ := inf x = μa; γ,γx, then we can state the following lemma. Lemma 4. For A BH, φ,φ C, the following statements are equivalent: i μ i A; φ; Φ 0; ii wa φ+φ I Φ φ. Utilizing the above results we can provide now some particular reverse inequalities that are of interest. Corollary 6. Let A BH φ,φ C with Φ /= ±φ such that either i or ii of Lemma 4 holds true. Then φ + Φ φ 0 wa v s φ + Φ A + Φ wa v s φ + Φ A 4 Φ φ Φ + φ. 4.0 If N>n>0 are such that either μ i A; n, N 0orw A n+n I N n for a given operator A BA, then 0 wa v s A wa v s A N n 4 N + n. 4. An equivalent additive version of 4.4 is incorporated in the following:
S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 993 Corollary 7. Assume that λ/= 0 or A/= 0. If wa λi, then we have λ 0 w A ws A w Aw A λi A λi. 4. { w A λi w A A λi The following variant of 4.4 can be perhaps more convenient: Corollary 8. Let A BH\{0} φ,φ C with Φ /= φ. If ReΦ φ > 0 either the statement i or equivalently ii from Lemma 4 holds true, then: ReΦ φ v φ+ Φ s φ+φ A v φ+ Φ s φ+φ A 4.3 φ + Φ wa wa φ 0 w A vs + Φ φ + Φ A Φ φ φ + Φ 4 ReΦ φ v s φ + Φ A 4 Φ φ ReΦ φ w A 4 Φ φ ReΦ φ. 4.4 The proof follows by Theorem 5 the details are omitted. If N>n>0are such that either μ i A; n, N 0 or, equivalently, w A n + N I N n, 4.5 then nn n + N v sa, wa 4.6 N N n n 0 wa v s A v s A nn wa 4.7 nn 0 w A v s N n A 4nN v s A Finally, we can state the following result as well: N n 4nN w A. 4.8 Corollary 9. Let A BH, φ,φ C such that ReΦ φ > 0. If either μ i A; φ,φ 0 or, equivalently w A Φ + φ I Φ φ,
994 S.S. Dragomir / Linear Algebra its Applications 48 008 980 994 then φ 0 w A vs + Φ φ + Φ A [ φ + Φ Re Φ φ ] φ + Φ v s φ + Φ A [ ] φ + Φ ReΦ φ wa. 4.9 Moreover, if N>n>0are such that μ i A; n, N 0, then we have the simpler inequality: 0 w A vs A N n vs A N n wa. 4.0 Acknowledgments The author would like to thank the anonymous referee for his/her valuable suggestions that have been incorporated in the final version of the manuscript. References [] F.F. Bonsall, J. Duncan, Numerical Ranges of Operators on Normed Spaces of Elements of Normed Algebras, London Math. Soc. Lecture Notes Series, Cambridge University Press, 97. [] F.F. Bonsall, J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Notes Series, Cambridge University Press, 973. [3] S.S. Dragomir, Semi Inner Products Applications, Nova Science Publishers, New York, 004. [4] S.S. Dragomir, Reverse inequalities for the numerical radius of linear operators in Hilbert spaces, Bull. Austral. Math. Soc. 73 006 55 6. [5] S.S. Dragomir, Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces,linear Algebra Appl. 49 006 56 64. [6] S.S. Dragomir, Inequalities for the norm the numerical radius of linear operators in Hilbert spaces, Demonstratio Math. 40 007 4 47. [7] S.S. Dragomir, Inequalities for the norm the numerical radius of composite operators in Hilbert spaces, RGMIA Res. Rep. Coll. 8 Suppl. 005, Article preprint <http://rgmia.vu.edu.au/v8e.html>. [8] K.E. Gustafson, D.K.M. Rao, Numerical Range, Springer-Verlag, New York, 997. [9] P.R. Halmos, A Hilbert Space Problem Book, second ed., Springer-Verlag, New York, 98. [0] F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Stud. Math. 68 005 73 80. [] F. Kittaneh, A numerical radius inequality an estimate for the numerical radius of the Frobenius companion matrix, Stud. Math. 58 003 7. [] G. Lumer, Semi-inner-product spaces, Trans. Am. Math. Soc. 00 96 9 43. [3] J.K. Merikoski, R. Kumar, Lower bounds for the numerical radius, Linear Algebra Appl. 40 005 35 4. [4] G. Söderlind, The logarithmic norm, Hist. Modern Theory BIT 46 006 63 65.