SEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES

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1 SEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. The main aim of this paper is to establish some connections that exist between the numerical radius w A, operator norm A the semiinner products A, I p,n A, I p,w with p {i, s} that can be naturally defined on the Banach algebra B H of all bounded linear operators defined on a Hilbert space H. Reverse inequalities that provide upper bounds for the nonnegative quantities A w A w A A, I p,n under various assumptions for the operator A are also given.. Introduction In any normed linear space E,, since the function f : E R, f x = x is convex, one can introduce the following semi-inner products see for instance 3]:. x, y i := lim s 0 y + sx y y + tx y, x, y s := lim t 0 t s 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. In the Banach algebra B H of all bounded linear operators defined on the real or complex Hilbert space H we can associate to both the operator norm the numerical radius w the following semi-inner products: B + ta B. A, B si,n := lim t 0+ t.3 A, B si,w := lim t 0+ w B + ta w B t respectively, where A, B B H. 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. For the sake of completeness we list here some properties of, si,nw that will be used in the sequel. We have: i A, A si,n = A, A, A si,w = w A for any A B H ; ii ia, A p,nw = A, ia p,nw = 0 for each A B H ; Date: July 7, Mathematics Subject Classification. 47A, 47A30, 47A63. Key words phrases. Numerical radius, Operator norm, Semi-inner products, Bounded linear operators, Banach algebra.

2 S.S. DRAGOMIR iii λa, B p,nw = λ A, B p,nw = A, λb p,nw for any λ 0 A, B B H ; iv A, B p,nw = A, B p,nw = A, B q,nw for any A, B B H ; v ia, B p,nw = A, ib p,nw for each A, B B H ; vi The following Schwarz type inequalities hold true:.4 A, B p,n A B.5 A, B p,w w A w B for any A, B B H ; vii The following identities hold true.6 αa + B, A p,n = α A + B, A p,n.7 αa + B, A p,w = αw A + B, A p,w for each α R A, B B H ; viii The following subsuper-additivity property holds:.8 A + B, C si,pw A, C si,pw + B, C si,pw, where the sign applies for the superior semi-inner product, while the sign applies for the inferior one; ix The following continuity properties are valid:.9 A + B, C p,n B, C p,n A C.0 A + B, C p,w B, C p,w w A w C for each A, B, C B H ; x From the definition we have the inequality. A, B i,nw A, B s,nw for A, B B H ; where everywhere above p, q {i, s} p q. As a specific property that follows by the well known inequality between the norm the numerical radius of an operator, i.e., w A A for each A B H, we have. A, I i,n A, I i,w A, I s,w A, I s,n for any A B H, where I is the identity operator on H. We also observe that for each A B H. A, I si,n = A, I si,w = I + ta lim t 0+ t w I + ta lim t 0+ t

3 BOUNDED LINEAR OPERATORS IN HILBERT SPACES 3 Motivated by the natural connection that exists between the semi-inner products A, I p,n, A, I p,w with p {i, s}, the numerical radius w A the operator norm A outlined above, the aim of this paper is to establish deeper relationships between these concepts. Amongst others, we show, in fact, that the semi-inner product A, I p,n is equal to A, I p,w for p {i, s} as a consequence the nu- merical radius w A is bounded below by the maximum of the quantities A, I i,n A, I s,n. Also, on utilising these quantities various reverse inequalities for the fundamental fact that A w A A, A B H, are pointed out, improving the recent results of the author from 4], where some upper bounds for the nonnegative differences A w A A w A have been established under appropriate conditions for the bounded linear operator A. For recent results concerning inequalities between the operator norm numerical radius see the papers ], 5], 6], 7], 8], 9], the books ], 0], ] the references therein.. The Functionals v si on B H The following representation result, which plays a crucial role in deriving various inequalities for numerical radius may be stated: Theorem. For any A B H, we have:. A, I s,n = A, I s,w = sup Re Ax, x x =. A, I i,n = A, I i,w = inf Re Ax, x. x = Proof. Let x H, x =. Then for t > 0 we obviously have: Re x + tax, x.3 Re Ax, x = t x + tax, x w I + ta. t t Taking the supremum over x H, x =, we get w I + ta sup Re Ax, x x = t for each t > 0, which implies, by letting t 0+ that.4 sup Re Ax, x A, I s,w, x = for each A B H. Now, let δ := sup x = Re Ax, x. If x H, x =, then for any α > 0 we have I αa x Re I αa x, x = α Re Ax, x αδ. Therefore, by putting x = y y y 0 with y H, we have.5 I αa y αδ y

4 4 S.S. DRAGOMIR for each α > 0 y H. Now, if in.5 we replace y by I + αa z with z H, then we get.6 I α A z αδ I + αa z for each z H. Taking the supremum over z = in.6, we get the operator norm inequality: I α A αδ I + αa which is equivalent with.7 I α A + αδ I + αa I + αa, α > 0. If we subtract from both sides of.7 divide by α > 0, we get I α A I + αa.8 + δ I + αa. α α Taking the limit over α 0+ in.8 noticing that lim I + αa =, lim α 0+ lim α 0+ I α A α then, by.8, we have: I + αa = A, I α 0+ α s,n = lim α 0+ α lim α 0+.9 δ A, I s,n. I α A α = 0, Since, by., we always have A, I s,n A, I s,w, hence.4.9 imply the desired equality.. Now, on utilising iv., we have for all A B H that A, I i,n = A, I s,n = sup Re Ax, x = inf Re Ax, x x = x = the identity. is also obtained. It is well known that a lower bound for the numerical radius w A is A. The following corollary of the above theorem provides the following lower bounds as well: Corollary. For any A B H we have { } A,.0 max I s,n, A, I i,n w A. Proof. By Schwarz s inequality for the semi-inner products, s,w, i,w we have A, I s,w, A, I i,w w A, A B H which, by.., imply the desired inequality.0. Motivated by the representation Theorem, we can introduce the following functionals defined on B H :. v s A := sup Re Ax, x v i A := inf Re Ax, x. x = x = Now, on employing the properties of the semi-inner products outlined above, we can state that: a v s A = v i A, A B H ;

5 BOUNDED LINEAR OPERATORS IN HILBERT SPACES 5 aa v i A 0 for accretive operators on H; aaa v si A + B v si A + v si B for each A, B B H ; av v si A = A, I si,n = A, I si,w for any A B H ; v vsi A w A for all A B H ; va v si A = v si αi + A α for any α R A B H ; vaa v si A + B v si B w A for any A, B B H. The following inequalities may be stated as well: Proposition. For any A B H λ C we have. A + λ ] v s λa Proof. Let x H, x =. Then, obviously A + λ ] A λi, 4 A + λi A λi ]. 0 Ax Re λax, x ] + λ = A λi x A λi, which is equivalent with.3 Ax + λ ] A λi Re λax, x Ax + λ ], for any x H, x =. Taking the supremum over x = we get the first inequality in. the one from the first branch in the second. For x H, x = we also have that.4 Ax + λx = Ax λx + 4 Re λax, x, which, on taking the supremum over x =, will produce the second part of the second inequality in.. It is well known, in general, that the semi-inner products, si are not commutative. However, for the Banach algebra B H we can point out the following connection between A, I si,nw = v si A the quantities I, A s,n I, A i,n, where A B H. Corollary. For any A B H we have.5 v i A ] I, A s,n + I, A i,n v s A. Proof. We have from the second part of the second inequality in. that ] A + ti A.6 A ti A v s A t t for any t > 0. Taking the limit over t 0+ noticing that lim t 0+ A ti A t = I, A s,n = I, A i,n, we get the second inequality in.5.

6 6 S.S. DRAGOMIR Now, writing the second inequality in.5 for A, we get 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.5. Utilising a similar approach for the numerical radius instead of the operator norm, we can state the following result: Proposition. For any A B H λ C we have the double inequality: w A + λ ].7 v s λa The above result has the interesting consequence: w A + λ ] w A λi, 4 w A + λi w A λi ]. Corollary 3. For any A B H we have.8 v i A ] I, A s,w + I, A i,w v s A. Remark. Since w A A, hence the first inequality in.7 provides a better upper bound for v s λa than the first inequality in.. 3. Reverse Inequalities in Terms of Operator Norm The following result concerning reverse inequalities for the numerical radius operator norm may be stated: Theorem. For any A B H \ {0} λ C\ {0} we have the inequality: λ 3. 0 A w A A v s λ A λ A λi. In addition, if A λi λ, then we have: 3. λ A I v λ s λ A w A A A 3.3 respectively. λ 0 A w A A vs λ A λ λ λ A λi v s λ λ A λi w A λ A,

7 BOUNDED LINEAR OPERATORS IN HILBERT SPACES 7 Proof. Utilising the property v, we have λ w A = w λ A v s λ λ λ A v s λ A, for each λ C\ {0} the first inequality in 3. is proved. By the arithmetic mean-geometric mean inequality we have which, by. provides A + λ ] λ A, v s λa λ A A λi that is equivalent with the second inequality in 3.. Utilising the second part of the inequality. under the assumption that A λi λ we can also state that ] 3.4 v s λa A + λ A λi. By the arithmetic mean-geometric mean inequality we have now: 3.5 ] A + λ A λi A λ A λi, which, together with 3.4 implies the first inequality in 3.. The second part of 3. follows from v. From the proof of Proposition we can state that 3.6 Ax + λ Re λax, x + r, x = where we denoted r := A λi λ. We also observe, from 3.6, that Re λax, x > 0 for x H, x =. Now, if we divide 3.6 by Re λ λ Ax, x > 0, we get 3.7 Ax Re λ λ Ax, x λ + r Re λ λ Ax, x λ Re λ λ Ax, x for x =.

8 8 S.S. DRAGOMIR If in this inequality we subtract from both sides the quantity Re λ λ Ax, x, then we get Ax λ λ Ax, x Re λ λ Ax, x Re λ + r λ λ Re Re λ λ Ax, x λ Ax, x λ r = λ Re Re λ λ Ax, x λ λ r, which obviously implies that λ 3.8 Ax Re λ Ax, x + λ λ Ax, x λ λ r λ λ r Re λ 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. The following lemma is of interest in itself. Lemma. For any A B H γ, Γ K we have: 3.9 v i A γi ΓI A] = 4 Γ γ A γ + Γ I. Proof. We observe that, for any u, v, y H we have: 3.0 Re u y, y v = 4 u v y u + v. Now, choosing u = Γx, y = Ax, v = γx with x H, x = we get Re Γx Ax, Ax γx = 4 Γ γ Ax γ + Γ x, giving inf Re x = A γi ΓI A x, x = 4 Γ γ sup which is equivalent with 3.9. x = Ax γ + Γ x, We recall that the bounded linear operator B B H is called strongly m accretive with m > 0 if Re By, y m for 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,. The following result providing a characterisation for a class of operators that will be used in the sequel is incorporated in:

9 BOUNDED LINEAR OPERATORS IN HILBERT SPACES 9 Lemma. For A B H, γ, Γ K with Γ γ q R, the following statements are equivalent: i The operator A γi ΓI A is q accretive; ii We have the norm inequality: 3. A γ + Γ I 4 Γ γ q. The proof is obvious by Lemma the details are omitted. Remark. 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 γi ΓI A to be q accretive is that C γ,γ A is self-adjoint C γ,γ A q I. Corollary 4. Let A B H, γ, Γ K with Γ ±γ q R. If the operator C γ,γ A is q accretive, then Γ + γ 3. 0 A w A A v s Γ + γ A γ + Γ 4 Γ γ q Remark 3. If M, m are positive real numbers with M > m the operator C m,m A = A mi MI A is q accretive, then A w A A v s A M + m ]. 4 M m q Remark 4. We observe that for q = 0, i.e., if C γ,γ A respectively C m,m A are accretive, then we obtain from the inequality: Γ + γ A w A A v s Γ + γ A Γ γ 4 Γ + γ A w A A v s A ]. M m 4 M + m respectively, which provide refinements of the corresponding inequalities.7.34 in 4]. Remark 5. For any bounded linear operator A we know that wa A 3. would produce a useful result only if which is equivalent with 3.6 A λi λ A I, 3 λ., therefore

10 0 S.S. DRAGOMIR In conclusion, for A B H \ {0} λ C\ {0} satisfying the condition 3.6, the inequality 3. provides a refinement of the classical result: 3.7 w A A, A B H. Corollary 5. If A λi λ, then we have A w A A v s λ λ A A A λi λ. 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 6. Let A B H \ {0} γ, Γ K, Γ γ, q R so that Re Γ γ + q 0. If C γ,γ A is q accretive, then Re Γ γ + q Γ + γ v s Γ+ γ Γ+γ A A w A A Γ 0 A w A A + γ vs Γ + γ A ] A Γ + γ 4 Γ γ q A Γ γ. Γ + γ Proof. By Lemma, the fact that C γ,γ A is q accretive implies that 3. A γ + Γ I 4 Γ γ q. Since, obviously 4 Γ + γ hence A γ+γ I to get: 3. since, by 3., γ + Γ ] 4 Γ γ q = Re Γ γ + q 0, Γ + γ we can apply the inequality 3. for λ = γ+γ γ+γ A γ+γ I γ+γ A γ + Γ I γ + Γ v s Γ+ γ Γ+γ A A = Re Γ γ + q, 4 Γ γ + q,

11 BOUNDED LINEAR OPERATORS IN HILBERT SPACES hence by 3. we deduce the desired inequality in 3.9. The inequality 3.0 follows from 3.9. We omit the details. Remark 6. If γ, Γ q are such that Γ + γ 4 Re Γ γ + q, then 3.9 will provide a refinement of the classical result 3.7. Remark 7. If M > m 0 the operator C m,m A is q accretive, then Mm + q m + M v s A A w A A 0 A w A A v s A A m + M 4 M m q Remark 8. We also observe that, for q = 0, i.e., if C γ,γ A respectively C m,m A are accretive, then we obtain: Re Γ γ Γ + γ v s Γ+ γ Γ+γ A A Mm m + M v s A A w A A, 0 A w A A v s A Γ γ Γ + γ w A A, Γ + γ Γ + γ A 0 A w A A v s A A M m m + M respectively, which provides refinements of the inequalities.7,.3.0 in 4], respectively. The inequality between the first the last term in 3.8 was not stated in 4]. Corollary 7. Let A B H, γ, Γ K, Γ γ, q R so that Re Γ γ + q 0. If C γ,γ H is q accretive, then Γ 0 A w A A + γ 3.9 vs Γ + γ A Γ + γ Γ + γ Re Γ γ + q v s Γ + γ A Γ + γ Re Γ γ + q w A. The proof follows by the last part of Theorem on utilising a similar argument to the one employed in Corollary 6. The details are omitted. ].

12 S.S. DRAGOMIR Remark 9. If M > m 0 the operator C m,m A is q accretive, then A w A A vs A M + m Mm + q v s A M + m Mm + q w A. Remark 0. Finally, for q = 0, i.e., if C γ,γ A respectively C m,m A are accretive, then we obtain from some refinements of the inequalities.9.33 from 4]. 4. Reverse Inequalities in Terms of Numerical Radius In Section we established amongst others the following lower bound for the numerical radius w A 4. vsi A w A for any A a bounded linear operator, where 4. v si A = A, I si = sup x = inf x = Re Ax, x. It is then a natural problem to investigate how far the left side of 4. is from the numerical radius w A. We start with the following result: Theorem 3. For any A B H \ {0} λ C\ {0} we have 0 w A λ λ 4.3 v s λ A w A v s λ A λ w A λi A λi. λ Moreover, if w A AI λ, then we have: λ v λ s 4.4 w A I λ A v λ s λ A w A w A 4.5 respectively λ 0 w A vs λ A λ λ w A λi λ λ w A λi λ v s λ A w A, Proof. From 4., we obviously have that λ w A = w λ A λ λ v s λ A v s λ A for A B H λ C\ {0}.

13 BOUNDED LINEAR OPERATORS IN HILBERT SPACES 3 Now, utilising the inequality.7 the elementary arithmetic-geometric mean inequality, we have v s λa w A + λ ] w A λi λ w A w A λi which is clearly equivalent with the third inequality in 4.3. Under the assumption that w A AI λ, on making use of.7 the arithmetic mean-geometric mean inequality, we can also state that v s λa w A w A + λ w A λi λ w A λi, ] which is clearly equivalent to 4.4. Let us denote ρ := w A λi λ. Then for any x H, x = we have which yields that ρ Ax, x λ = Ax, x Re λ Ax, x ] + λ 4.6 Ax, x + λ Re λ Ax, x ] + ρ for any x H, x =. Making use of an argument similar to that in the proof of Theorem, we can get out of 4.6 the following inequality: 4.7 Ax, x Re λ λ Ax, x + λ λ λ ρ Re λ Ax, x for any x H, x =. Taking the supremum over x H, x =, we deduce the desired inequality in 4.5. Then following lemma is of interest in itself. Lemma 3. For any A B H γ, Γ K we have 4.8 inf x = Re ΓI A x, x x, A γi x ] = 4 Γ γ w A γ + Γ I. Proof. We observe that for any u, v, y complex numbers, we have the elementary identity: 4.9 Re u y ȳ v] = 4 u v y u + v. If we choose in 4.9 u = Γ, y = Hx, x v = γ with x H, x =, then by 4.9 we have: 4.0 Re ΓI A x, x x, A γi x ] = 4 Γ γ A γ + Γ I x, x for each x H, x =. Now, taking the infimum over x = in 4.0 we deduce the desired identity 4.8.

14 4 S.S. DRAGOMIR Remark. We observe that for any x H, x = we have µ A; γ, Γ x := Re ΓI A x, x x, A γi x ] = Re Γ Re Ax, x Re Ax, x Re γ + Im Γ Im Ax, x Im Ax, x Im γ therefore a sufficient condition for µ A; γ, Γ x to be nonnegative for each x H, x = is that: Re Γ Re Ax, x Re γ 4., x H, x =. Im Γ Im Ax, x Im γ Now, if we denote by µ i A; γ, Γ := inf x = µ A; γ, Γ x, then we can state the following lemma. Lemma 4. For A B H, φ, Φ K, the following statements are equivalent: i µ i A; φ; Φ 0; ii w A φ+φ I Φ φ. Utilising the above results we can provide now some particular reverse inequalities that are of interest. Corollary 8. Let A B H φ, Φ K with Φ ±φ such that either i or ii of Lemma 4 holds true. Then 0 w A φ + v Φ φ + s φ + Φ A Φ 4. w A v s φ + Φ A 4 Φ φ Φ + φ. Remark. If N > n > 0 are such that either µ i A; n, N 0 or w A n+n I N n for a given operator A B A, then w A v s A w A v s A 4 N n N + n. From a different perspective, we can state the following multiplicative reverse of the inequality 4.. An equivalent additive version of 4.4 is incorporated in the following: Corollary 9. If w A λi λ, then we have λ 0 w A ws λ A w A w A λi 4.4 λ A w A λi λ w A A λi λ A A λi λ. In applications, the following variant of 4.4 can be perhaps more convenient:

15 BOUNDED LINEAR OPERATORS IN HILBERT SPACES 5 Corollary 0. Let A B H \ {0} φ, Φ K with Φ φ. If Re Φ φ > 0 either the statement i or equivalently ii from Lemma 4 holds true, then: Re Φ φ v φ+ Φ s φ+φ A v φ+ Φ s φ+φ A 4.5 φ + Φ w A w A 4.6 φ + Φ 0 w A vs φ + Φ A Φ φ φ + 4 Re Φ Φ φ v s φ + Φ A 4 Φ φ Re Φ φ w A Φ φ 4 Re Φ φ A The proof follows by Theorem 3 on utilising a similar argument to the one incorporated in the proof of Corollary 6 the details are omitted. Remark 3. If N > n > 0 are such that either µ i A; n, N 0 or, equivalently 4.7 w A n + N N n, then w A v s A w A v s A nn n + N v s A, w A N n nn v s A N n 4nN v s A Finally, we can state the following result as well: N n. nn w A N n 4nN w A. Corollary. Let A B H, φ, Φ K such that Re Φ φ > 0. If either µ i A; φ, Φ 0 or, equivalently w A Φ + φ I Φ φ, then 4. φ + Φ 0 w A vs φ + Φ A φ + Φ Re Φ φ ] φ + Φ v s φ + Φ A φ + Φ Re Φ φ ] w A.

16 6 S.S. DRAGOMIR Remark 4. If N > n > 0 are as in Remark 3, then we have the inequality: 4. 0 w A vs A N n vs A N n w A. References ] R. BHATIA, Matrix Analysis, Springer-Verlag, New York, 997. ] R. BHATIA F. KITTANEH, Norm inequalities for positive operators, Lett. Math. Phys., , ] S.S. DRAGOMIR, Semi Inner Products Applications, Nova Science Publishers Inc., N.Y., ] S.S. DRAGOMIR, Reverse inequalities for the numerical radius of linear operators in Hilbert spaces, Bull. Austral. Math. Soc., , ] S.S. DRAGOMIR, Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces, Lin. Alg. Appl., in press. 6] S.S. DRAGOMIR, Inequalities for the norm the numerical radius of linear operators in Hilbert spaces, Preprint available in RGMIA Res. Rep. Coll., 8 005, Supplement, Article 0. ONLINE 7] S.S. DRAGOMIR, Inequalities for the norm the numerical radius of composite operators in Hilbert spaces, Preprint available in RGMIA Res. Rep. Coll., 8 005, Supplement, Article. ONLINE 8] F. KITTANEH, Numerical radius inequalities for Hilbert space operators, Studia Math., , ] F. KITTANEH, A numerical radius inequality an estimate for the numerical radius of the Frobenius companion matrix, Studia Math., , -7. 0] K.E. GUSTAFSON D.K.M. RAO, Numerical Range, Springer Verlag, New York, 997. ] P.R. HALMOS, A Hilbert Space Problem Book, Second Ed., Springer-Verlag, New York, 98. ] J.K. MERIKOSKI P. KUMAR, Lower bounds for the numerical radius, Lin. Alg. Appl., , School of Computer Science Mathematics, Victoria University, PO Box 448, Melbourne City, VIC, Australia address: sever.dragomir@vu.edu.au URL: