Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces

Transcription

1 This is page i Printer: Opaque this Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces Silvestru Sever Dragomir February 03

2 ii ABSTRACT The present monograph is focused on numerical radius inequalities for bounded linear operators on complex Hilbert spaces for the case of one and two operators. The book is intended for use by both researchers in various elds of Linear Operator Theory in Hilbert Spaces and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their speci c areas.

3 Contents This is page iii Printer: Opaque this Preface v Introduction. Basic De nitions and Facts Results for One Operator Results for Two Operators References 9 Inequalities for One Operator. Reverse Inequalities for the Numerical Radius Reverse Inequalities More Inequalities for Norm and Numerical Radius A Result Via Buzano s Inequality Other Related Results Some Associated Functionals Some Fundamental Facts Preliminary Results General Inequalities Reverse Inequalities Inequalities for the Maximum of the Real Part Introduction Preliminary Results for Semi-Inner Products Reverse Inequalities in Terms of the Operator Norm 38

4 iv Contents.4.4 Reverse Inequalities in Terms of the Numerical Radius 44.5 New Inequalities of the Kantorovich Type Some Classical Facts Some Grüss Type Inequalities Operator Inequalities of Grüss Type Reverse Inequalities for the Numerical Range New Inequalities of Kantorovich Type References 59 3 Inequalities for Two Operators General Inequalities for Numerical Radius Preliminary Facts Further Inequalities for an Invertible Operator Other Norm and Numerical Radius Inequalities Other Norm and Numerical Radius Inequalities Related Results Power Inequalities for the Numerical Radius Inequalities for a Product of Two Operators Inequalities for the Sum of Two Products Vector Inequalities for the Commutator A Functional Associated with Two Operators Some Basic Facts General Inequalities Operators Satisfying the Uniform (; )-property The Transform C ; (; ) and Other Inequalities Some Inequalities of the Grüss Type Additive and Multiplicative Grüss Type Inequalities Numerical Radius Inequalities of Grüss Type Some Particular Cases of Interest Some Inequalities for the Euclidean Operator Radius Preliminary Facts Some Inequalities for the Euclidean Operator Radius Other Results References 9

5 Preface This is page v Printer: Opaque this As pointed out by Gustafson and Rao in their seminal book [Numerical Range. The Field of Values of Linear Operators and Matrices. Universitext. Springer-Verlag, New York, 997. xiv89 pp.] the concepts of numerical range and numerical radius play an important role in various elds of Contemporary Mathematics, including Operator Theory, Operator Trigonometry, Numerical Analysis, Fluid Dynamics and others. Since 997 the research devoted to these mathematical objects has grown greatly. A simple search in the database MathSciNet of the American Mathematical Society with the key word "numerical range" in the title reveals more than 300 papers published after 997 while the same search with the key word "numerical radius" adds other 00, showing an immense interest on the subject by numerous researchers working in di erent elds of Modern Mathematics. If no restrictions for the year is imposed the number of papers with those key words in the title exceed 000. However, the size of the areas of applications for numerical ranges and radii is very di cult to estimate. If we perform a search looking for the publications where in a way or another the concept of "numerical range" is used, we can get more then 550 items. The present monograph is focused on numerical radius inequalities for bounded linear operators on complex Hilbert spaces for the case of one and two operators. The book is intended for use by both researchers in various elds of Linear Operator Theory in Hilbert Spaces and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by

6 vi Preface postgraduate students and scientists applying inequalities in their speci c areas. In the introductory chapter we present some fundamental facts about the numerical range and the numerical radius of bounded linear operators in Hilbert spaces. Some classical inequalities due to Berger, Holbrook, Fong & Holbrook and Bouldin are given. More recent and interesting results obtained by Kittaneh, El-Haddad & Kittanek and Yamazaki are provided as well. In Chapter, we present recent results obtained by the author concerning numerical radius and norm inequalities for one operator on a complex Hilbert space. The techniques employed to prove the results are elementary. We also use some special vector inequalities in inner product spaces due to Buzano, Goldstein, Ry & Clarke as well as some reverse Schwarz inequalities and Grüss type inequalities obtained by the author. Numerous references for the Kantorovich inequality that is extended to larger classes of operators than positive operators are provided as well. In Chapter 3, we present recent results obtained by the author concerning the norms and the numerical radii of two bounded linear operators. The techniques in this case are also elementary and can be understood by undergraduate students taking a subject in Operator Theory. Some vector inequalities in inner product spaces as well as inequalities for means of nonnegative real numbers are also employed. For the sake of completeness, all the results presented are completely proved and the original references where they have been rstly obtained are mentioned. The chapters are followed by the list of references used therein and therefore are relatively independent and can be read separately. The Author Melbourne, February 03

7 Introduction This is page Printer: Opaque this In this introductory chapter we present some fundamental facts about the numerical range and the numerical radius of bounded linear operators in Hilbert spaces that are used throughout the book. Some famous inequalities due to Berger, Holbrook, Fong & Holbrook and Bouldin are given. More recent results obtained by Kittaneh, El-Haddad & Kittanek and Yamazaki are provided as well.. Basic De nitions and Facts Let (H; h; i) be a complex Hilbert space. The numerical range of an operator T is the subset of the complex numbers C given by [6, p. ]: W (T ) = fht x; xi ; x H; kxk = g : The following properties of W (T ) are immediate: (i) W (I T ) = W (T ) for ; C; (ii) W (T ) = ; W (T ) ; where T is the adjoint operator of T ; (iii) W (U T U) = W (T ) for any unitary operator U: The following classical fact about the geometry of the numerical range [6, p. 4] may be stated:

8 . Introduction Theorem (Toeplitz-Hausdor ) The numerical range of an operator is convex. An important use of W (T ) is to bound the spectrum (T ) of the operator T [6, p. 6]: Theorem (Spectral inclusion) The spectrum of an operator is contained in the closure of its numerical range. The self-adjoint operators have their spectra bounded sharply by the numerical range [6, p. 7]: Theorem 3 The following statements hold true: (i) T is self-adjoint i W (T ) is real; (ii) If T is self-adjoint and W (T ) = [m; M] (the closed interval of real numbers m; M), then kt k = max fjmj ; jmjg : (iii) If W (T ) = [m; M] ; then m; M (T ) : The numerical radius w (T ) of an operator T on H is given by [6, p. 8]: w (T ) = sup fjj ; W (T )g = sup fjht x; xij ; kxk = g : (.) Obviously, by (.), for any x H one has jht x; xij w (T ) kxk : (.) It is well known that w () is a norm on the Banach algebra B (H) of all bounded linear operators T : H! H; i.e., (i) w (T ) 0 for any T B (H) and w (T ) = 0 if and only if T = 0; (ii) w (T ) = jj w (T ) for any C and T B (H) ; (iii) w (T V ) w (T ) w (V ) for any T; V B (H) : This norm is equivalent with the operator norm. In fact, the following more precise result holds [6, p. 9]: Theorem 4 (Equivalent norm) For any T B (H) one has w (T ) kt k w (T ) : (.3) Let us now look at two extreme cases of the inequality (.3). In the following r (t) := sup fjj ; (T )g will denote the spectral radius of T and p (T ) = f (T ) ; T f = f for some f Hg the point spectrum of T: The following results hold [6, p. 0]:

9 . Results for One Operator 3 Theorem 5 We have (i) If w (T ) = kt k ; then r (T ) = kt k : (ii) If W (T ) and jj = kt k ; then p (T ) : To address the other extreme case w (T ) = kt k ; we can state the following su cient condition in terms of (see [6, p. ]) R (T ) := ft f; f Hg and R (T ) := ft f; f Hg : Theorem 6 If R (T )? R (T ) ; then w (T ) = kt k : It is well-known that the two-dimensional shift 0 0 S = ; 0 has the property that w (T ) = kt k : The following theorem shows that some operators T with w (T ) = kt k have S as a component [6, p. ]: Theorem 7 If w (T ) = kt k and T attains its norm, then T has a twodimensional reducing subspace on which it is the shift S : For other results on numerical radius, see [7], Chapter.. Results for One Operator The following power inequality for one operator is a classical result in the eld (for a simple proof see [4]): Theorem 8 (Berger [], 965) For any operator T B (H) and natural number n we have w (T n ) w n (T ) : Further, we list some recent inequalities for one operator. Theorem 9 (Kittaneh [0], 003) For any operator T B (H) we have the following re nement of the rst inequality in (.3) w (T ) kt k T = : (.4) Utilizing the Cartesian decomposition for operators, F. Kittaneh improves the inequality (.3) as follows:

10 4. Introduction Theorem 0 (Kittaneh [], 005) For any operator T B (H) we have 4 kt T T T k w (T ) kt T T T k : (.5) When more information concerning the angle between the ranges of T and T is available, the the following interesting estimate holds: Theorem (Bouldin [3], 97) If we denote by the angle between the ranges of T and T ; then w (T ) kt k hcos cos = i : (.6) For powers of the absolute value of operators, one can state the following results: Theorem (El-Haddad & Kittaneh [9], 007) If for an operator T B (H) we denote jt j := (T T ) = ; then w r (T ) jt j r jt j ( )r (.7) and w r (T ) jt j r ( ) jt j r (.8) where (0; ) and r : If we take = and r = we get from (.7) w (T ) kjt j jt jk (.9) and from (.8) w (T ) jt j jt j : (.0) For the Cartesian decomposition of T the we have: Theorem 3 (El-Haddad & Kittaneh [9], 007) If T = BiC is the Cartesian decomposition of T then: w r (T ) kjbj r jcj r k (.) for r (0; ]: If r ; then and w r (T ) r kjbj r jcj r k (.) r kjb Cj r jb Cj r k (.3) w r (T ) kjb Cjr jb Cj r k :

11 We observe that for r = we get from (.).3 Results for Two Operators 5 w (T ) kjbj jcjk (.4) while for r = we get from (.) or from (.) w (T ) jbj jcj (.5) and from (.3) 4 jb Cj jb Cj w (T ) jb Cj jb Cj : (.6) Let T = U jt j be the polar decomposition of the bounded linear operator T. The Aluthge transform T e of T is de ned by T e := jt j = U jt j = ; see []. The following properties of T e are as follows: (i) T e kt k ; (ii) w et w (T ) ; (iii) r et = w (T ) ; (iv) w et T = ( kt k) ; [5]. Utilizing this transform one can obtain the following re nement of Kittaneh s inequality (.4). Theorem 4 (Yamazaki [5], 007) For any operator T B (H) we have w (T ) kt k w et kt k T = : (.7) We remark that if e T = 0; then obviously w (T ) = kt k :.3 Results for Two Operators The following general result for the product of two operators holds [6, p. 37]: Theorem 5 (Holbrook [8], 969) If A; B are two bounded linear operators on the Hilbert space (H; h; i) ; then w (AB) 4w (A) w (B) : In the case that AB = BA; then w (AB) w (A) w (B) : The constant is best possible here.

12 6. Introduction The following results are also well known [6, p. 38]. Theorem 6 (Holbrook [8], 969) If A is a unitary operator that commutes with another operator B; then w (AB) w (B) : (.8) If A is an isometry and AB = BA; then (.8) also holds true. We say that A and B double commute if AB = BA and AB = B A: The following result holds [6, p. 38]. Theorem 7 (Holbrook [8], 969) If the operators A and B double commute, then w (AB) w (B) kak : (.9) As a consequence of the above, we have [33, p. 39]: Corollary 8 Let A be a normal operator commuting with B: Then w (AB) w (A) w (B) : (.0) A related problem with the inequality (.9) is to nd the best constant c for which the inequality w (AB) cw (A) kbk holds for any two commuting operators A; B B (H) : It is known that :064 < c < :69; see [4], [] and [3]. In relation to this problem, it has been shown that: Theorem 9 (Fong & Holbrook [5], 983) For any A; B B (H) we have w (AB BA) p w (A) kbk : (.) The following result for several operators holds: Theorem 0 (Kittaneh [], 005) For any A; B; C; D; S; T B (H) we have w (AT B CSD) (.) A jt j ( ) A B jt j B C js j ( ) C D jt j D ; where [0; ] : Following [] we list here some particular inequalities of interest. If we take T = I and S = 0 in (.) we get w (AB) kaa B Bk : (.3)

13 .3 Results for Two Operators 7 In addition to this we have the related inequality w (AB) ka A BB k : (.4) If we choose T = S = I; C = B and D = A in (.) we get w (AB BA) ka A AA BB B Bk (.5) which provides an upper bound for the numerical radius of the commutator AB BA: If we take = in (.) we also can derive the inequality w (AB B A) kjaj ja j B (jaj ja j) Bk : (.6)

14 8. Introduction

15 References This is page 9 Printer: Opaque this [] A. ALUTHGE, Some generalized theorems on p-hyponormal operators, Integral Equations Operator Theory 4 (996), [] C. BERGER, Notices Amer. Math. Soc. (965), 590, Abstract [3] R. BOULDIN, The numerical range of a product. II. J. Math. Anal. Appl. 33 (97), -9. [4] K. R. DAVIDSON and J. A. R. HOLBROOK, Numerical radii of zeroone matricies, Michigan Math. J. 35 (988), [5] C. K. FONG and J. A. R. HOLBROOK, Unitarily invariant operators norms, Canad. J. Math. 35 (983), [6] K. E. GUSTAFSON and D.K.M. RAO, Numerical Range, Springer- Verlag, New York, Inc., 997. [7] P. R. HALMOS, A Hilbert Space Problem Book, Springer-Verlag, New York, Heidelberg, Berlin, Second edition, 98. [8] J. A. R. HOLBROOK, Multiplicative properties of the numerical radius in operator theory, J. Reine Angew. Math. 37 (969), [9] M. EL-HADDAD and F. KITTANEH, Numerical radius inequalities for Hilbert space operators. II, Studia Math. 8 (007), No.,

16 0 References [0] F. KITTANEH, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 58 (003), No., -7. [] F. KITTANEH, Numerical radius inequalities for Hilbert space operators, Studia Math. 68 (005), No., [] V. MÜLLER, The numerical radius of a commuting product, Michigan Math. J. 39 (988), [3] K. OKUBO and T. ANDO, Operator radii of commuting products, Proc. Amer. Math. Soc. 56 (976), [4] C. PEARCY, An elementary proof of the power inequality for the numerical radius. Michigan Math. J [5] T. YAMAZAKI, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 78 (007), No.,

17 Inequalities for One Operator This is page Printer: Opaque this In this chapter we present with complete proofs some recent results obtained by the author concerning numerical radius and norm inequalities for a bounded linear operator on a complex Hilbert space. The techniques employed to prove the results are elementary. We also use some special vector inequalities in inner product spaces due to Buzano, Goldstein, Ry & Clarke, Dragomir & Sándor as well as some reverse Schwarz inequalities and Grüss type inequalities obtained by the author. Many references for the Kantorovich inequality that is extended here to larger classes of operators than positive operators are provided as well.. Reverse Inequalities for the Numerical Radius.. Reverse Inequalities The following results may be stated: Theorem (Dragomir [3], 005) Let T : H! H be a bounded linear operator on the complex Hilbert space H: If Cn f0g and r > 0 are such that kt Ik r; (.) where I : H! H is the identity operator on H; then (0 ) kt k w (T ) r jj : (.)

18 . Inequalities for One Operator Proof. For x H with kxk = ; we have from (.) that giving kt x xk kt Ik r; kt xk jj Re ht x; xi r (.3) jj jht x; xij r : Taking the supremum over x H; kxk = in (.3) we get the following inequality that is of interest in itself: kt k jj w (T ) jj r : (.4) Since, obviously, kt k jj kt k jj ; (.5) hence by (.4) and (.5) we deduce the desired inequality (.). Remark If the operator T : H! H is such that R (T )? R (T ) ; kt k = and kt Ik, then the equality holds in (.). Indeed, by Theorem 6, we have in this case w (T ) = kt k = and since we can choose = ; r = in Theorem, then we get in both sides of (.) the same quantity : The following corollary may be stated: Corollary 3 Let T : H! H be a bounded linear operator and '; C with = f '; 'g : If Re h x T x; T x 'xi 0 for any x H; kxk = (.6) then (0 ) kt k w (T ) 4 j 'j j 'j : (.7) Proof. Utilising the fact that in any Hilbert space the following two statements are equivalent: (i) Re hu x; x zi 0; x; z; u H; (ii) x zu ku zk ; we deduce that (.6) is equivalent to T x ' Ix j 'j (.8)

19 . Reverse Inequalities for the Numerical Radius 3 for any x H; kxk = ; which in its turn is equivalent to the operator norm inequality: T ' I j 'j : (.9) Now, applying Theorem for T = T; = ' and r = j j ; we deduce the desired result (.7). Remark 4 Following [33, p. 5], we say that an operator B : H! H is accretive, if Re hbx; xi 0 for any x H: One may observe that the assumption (.6) above is then equivalent with the fact that the operator (T 'I) ( I T ) is accretive. Perhaps a more convenient su cient condition in terms of positive operators is the following one: Corollary 5 Let '; C with = f '; 'g and T : H! H a bounded linear operator in H: If (T 'I) ( I T ) is self-adjoint and in the operator partial order, then (T 'I) ( I T ) 0 (.0) (0 ) kt k w (T ) 4 j 'j j 'j : (.) Corollary 6 Assume that T; ; r are as in Theorem. If, in addition, for some 0; then Proof. From (.4) of Theorem, we have jjj w (T )j ; (.) (0 ) kt k w (T ) r : (.3) kt k w (T ) r w (T ) w (T ) jj jj (.4) = r (jj w (T )) : The desired inequality follows from (.). Remark 7 In particular, if kt Ik r and jj = w (T ) ; C, then The following result may be stated as well. (0 ) kt k w (T ) r : (.5)

20 4. Inequalities for One Operator Theorem 8 (Dragomir [3], 005) Let T : H! H be a nonzero bounded linear operator on H and C n f0g ; r > 0 with jj > r: If kt Ik r; (.6) then s r jj w (T ) kt k ( ) : (.7) Proof. From (.4) of Theorem, we have kt k jj r jj w (T ) ; which implies, on dividing with q jj r > 0 that kt k q jj qjj r jj w (T ) : (.8) r qjj r By the elementary inequality kt k kt k q jj qjj r (.9) r and by (.8) we deduce kt k w (T ) jj ; qjj r which is equivalent to (.7). Remark 9 Squaring (.7), we get the inequality (0 ) kt k w (T ) r jj kt k : (.0) Remark 30 For any bounded linear operator T : H! H we have the relation w (T ) kt k : Inequality (.7) would produce a re nement of this classic fact only in the case when which is equivalent to r= jj p 3=: The following corollary holds.! r jj ;

21 . Reverse Inequalities for the Numerical Radius 5 Corollary 3 Let '; C with Re ( ') > 0: If T : H! H is a bounded linear operator such that either (.6) or (.0) holds true, then: p Re ( ') j 'j w (T ) kt k ( ) (.) and (0 ) kt k w (T ) ' ' kt k : (.) Proof. If we consider = jj r = ' and r = j 'j ; then ' ' = Re ( ') > 0: Now, on applying Theorem 8, we deduce the desired result. Remark 3 If j 'j p 3 j 'j ; Re ( ') > 0; then (.) is a re nement of the inequality w (T ) kt k : The following result may be of interest as well. Theorem 33 (Dragomir [3], 005) Let T : H! H be a nonzero bounded linear operator on H and Cn f0g ; r > 0 with jj > r: If kt Ik r; (.3) then (0 ) kt k w r (T ) w (T ) : (.4) jj qjj r Proof. From the proof of Theorem, we have kt xk jj Re ht x; xi r (.5) for any x H; kxk = : If we divide (.5) by jj jht x; xij ; (which, by (.5), is positive) then we obtain kt xk jj jht x; xij Re ht x; xi r jj jht x; xij jj jht x; xij for any x H; kxk = : jj jht x; xij (.6)

22 6. Inequalities for One Operator If we subtract in (.6) the same quantity we get jht x;xij jj from both sides, then Since and kt xk jj jht x; xij Re ht x; xi jj jht x; xij = Re ht x; xi jj jht x; xij = Re ht x; xi jj jht x; xij q jj r : jj jht x; xij jj r jj jht x; xij jj r jj jht x; xij 0 q jj p jj jht x; xij jht x; xij jj jht x; xij jj Re ht x; xi jj jht x; xij 0 q jj p jj jht x; xij p jht x; xij p A 0; jj jj jht x; xij p jht x; xij p A jj (.7) by (.7) we get kt xk jj jht x; xij which gives the inequality jht x; xij jj kt xk jht x; xij jht x; xij jj jj qjj r jj qjj r (.8) for any x H; kxk = : Taking the supremum over x H; kxk = ; we get kt k sup jht x; xij jht x; xij jj qjj r n sup jht x; xij o jj qjj r sup fjht x; xijg = w (T ) jj which is clearly equivalent to (.4). qjj r w (T ) ;

23 . More Inequalities for Norm and Numerical Radius 7 Corollary 34 Let '; C with Re ( ') > 0: If T : H! H is a bounded linear operator such that either (.6) or (.0) hold true, then: (0 ) kt k w (T ) h j 'j p i Re ( ') w (T ) : (.9) Remark 35 If M m > 0 are such that either (T mi) (MI T ) is accretive, or, su ciently, (T mi) (MI T ) is self-adjoint and (T mi) (MI T ) 0 in the operator partial order, (.30) then, by (.) we have: which is equivalent to ( ) kt k w (T ) M m p mm ; (.3) pm p m (0 ) kt k w (T ) while from (.4) we have p mm w (T ) ; (.3) (0 ) kt k w (T ) p M p m w (T ) : (.33) Also, the inequality (.7) becomes (0 ) kt k w (T ) 4 (M m) M m : (.34). More Inequalities for Norm and Numerical Radius.. A Result Via Buzano s Inequality The following result may be stated as well: Theorem 36 (Dragomir [6], 007) Let (H; h; i) be a Hilbert space and T : H! H a bounded linear operator on H: Then w (T ) The constant is best possible in (.35). h w T kt k i : (.35)

24 8. Inequalities for One Operator Proof. We need the following re nement of Schwarz s inequality obtained by the author in 985 [4, Theorem ] (see also [0] and [4]): kak kbk jha; bi ha; ei he; bij jha; ei he; bij jha; bij ; (.36) provided a; b; e are vectors in H and kek = : Observing that jha; bi ha; ei he; bij jha; ei he; bij jha; bij ; hence by the rst inequality in (.36) we deduce (kak kbk jha; bij) jha; ei he; bij : (.37) This inequality was obtained in a di erent way earlier by M.L. Buzano in []. Now, choose in (.37), e = x; kxk = ; a = T x and b = T x to get kt xk kt xk T x; x jht x; xij (.38) for any x H; kxk = : Taking the supremum in (.38) over x H; kxk = ; we deduce the desired inequality (.35). Now, if we assume that (.35) holds with a constant C > 0; i:e:; h w (T ) C w T kt k i (.39) for any T B (H) ; then if we choose T a normal operator and use the fact that for normal operators we have w (T ) = kt k and w T = T = kt k ; then by (.39) we deduce that C which proves the sharpness of the constant. Remark 37 From the above result (.35) we obviously have w (T ) hw T kt k i = T = kt k kt k (.40) and w (T ) hw T kt k i = w (T ) kt k = kt k : (.4)

25 .. Other Related Results. More Inequalities for Norm and Numerical Radius 9 The following result may be stated. Theorem 38 (Dragomir [6], 007) Let T : H! H be a bounded linear operator on the Hilbert space H and Cn f0g : If kt k jj ; then kt k r jj r kt k r jj r w (T ) r jj r kt Ik ; (.4) where r : Proof. We use the following inequality for vectors in inner product spaces due to Goldstein, Ry and Clarke [3]: kak r kbk r kak r kbk r Re ha; bi kak kbk 8 < r kak r ka bk if r ; : kbk r ka bk if r < ; (.43) provided r R and a; b H with kak kbk : Now, let x H with kxk = : From the hypothesis of the theorem, we have that kt xk jj kxk and applying (.43) for the choices a = x; kxk = ; b = T x; we get kt xk r jj r kt xk r jj r jht x; xij r jj r kt x xk (.44) for any x H; kxk = and r : Taking the supremum in (.44) over x H; kxk = ; we deduce the desired inequality (.4). The following result may be stated as well: Theorem 39 (Dragomir [6], 007) Let T : H! H be a bounded linear operator on the Hilbert space (H; h; i) : Then for any [0; ] and t R one has the inequality: kt k h ( ) i w (T ) kt tik (.45) ( ) kt itik : Proof. We use the following inequality obtained by the author in [4]: h ktb ak ( ) kitb ak i kbk kak kbk [( ) Im ha; bi Re ha; bi] ( 0)

26 0. Inequalities for One Operator to get: kak kbk [( ) Im ha; bi Re ha; bi] (.46) h ktb ak ( ) kitb ak i kbk h ( ) i jha; bij h ktb ak ( ) kitb ak i kbk for any a; b H; [0; ] and t R. Choosing in (.46) a = T x; b = x; x H; kxk = ; we get h kt xk ( ) i jht x; xij (.47) ktx T xk ( ) kitx T xk : Finally, taking the supremum over x H; kxk = in (.47), we deduce the desired result. The following particular cases may be of interest. Corollary 40 For any T a bounded linear operator on H; one has: 8 inf >< kt (0 ) kt k w tr tik (T ) (.48) >: inf kt tr itik and kt k w (T ) h inf kt tik kt itik i : (.49) tr Remark 4 The inequality (.48) can in fact be improved taking into account that for any a; b H; b 6= 0; (see for instance [6]) the bound actually implies that inf ka C bk = kak kbk jha; bij kbk kak kbk jha; bij kbk ka bk (.50) for any a; b H and C. Now if in (.50) we choose a = T x; b = x; x H; kxk = ; then we obtain kt xk jht x; xij kt x xk (.5) for any C, which, by taking the supremum over x H; kxk =, implies that (0 ) kt k w (T ) inf kt C Ik : (.5)

27 .3 Some Associated Functionals Remark 4 If we take a = x; b = T x in (.50), then we obtain kt xk jht x; xij kt xk kx T xk (.53) for any x H; kxk = and C. Now, if we take the supremum over x H; kxk = in (.53), then we get (0 ) kt k w (T ) kt k inf C ki T k : (.54) Finally and from a di erent view point we may state: Theorem 43 (Dragomir [6], 007) Let T : H! H be a bounded linear operator on H: If p ; then: for any C. kt k p jj p (kt Ikp kt Ik p ) ; (.55) Proof. We use the following inequality obtained by Dragomir and Sándor in [0]: ka bk p ka bk p (kak p kbk p ) (.56) for any a; b H and p : Now, if we choose a = T x; b = x; then we get kt x xk p kt x xk p (kt xk p jj p ) (.57) for any x H; kxk = : Taking the supremum in (.57) over x H; kxk = ; we get the desired result (.55). Remark 44 For p = ; we have the simpler result: kt k jj kt Ik kt Ik (.58) for any C. This can easily be obtained from the parallelogram identity as well..3 Some Associated Functionals.3. Some Fundamental Facts Replacing the supremum with the in mum in the de nitions of the operator norm and numerical radius, we can also consider the quantities ` (T ) := inf kxk= kt xk and m (T ) = inf kxk= jht x; xij : By the Schwarz inequality, it is obvious that m (T ) ` (T ) for each T B (H) :

28 . Inequalities for One Operator We can also consider the functionals v s ; s : B (H)! R introduced in [7] and given by v s (T ) := sup Re ht x; xi and s (T ) := sup Im ht x; xi (.59) kxk= kxk= where s stands for supremum, while the corresponding ones for in mum are de ned as: v i (T ) := inf Re ht x; xi and i (T ) := inf Im ht x; xi : (.60) kxk= kxk= We notice that the functionals v p ; p with p fs; ig are obviously connected by the formula p (T ) = v q (it ) for any T B (H) ; (.6) where p 6= q and the i in front of T represents the imaginary unit. Also, by de nition, v s and s are positive homogeneous and subadditive while v i and i are positive homogeneous and superadditive. Due to the fact that for any x H; kxk = we have w (T ) jht x; xij Re ht x; xi ; Im ht x; xi jim ht x; xij w (T ) ; then, by taking the supremum and the in mum respectively over x H; kxk = ; we deduce the simple inequality: max fjv p (T )j ; j p (T )jg w (T ) ; T B (H) (.6) where p fs; ig : The main aim of this section is two fold. Firstly, some natural connections amongst the functionals v p ; p, the operator norm and the numerical ranges w; m; w e and m e are pointed out. Secondly, some new inequalities for operators T B (H) for which the composite operator C ; (T ) with ; K is assumed to be c -accretive with c R are also given. New upper bounds for the nonnegative quantity kt k w (T ) ; are obtained as well..3. Preliminary Results In the following we establish an identity connecting the numerical radius of an operator with the other functionals de ned in the introduction of this section. Lemma 45 Let T B (H) and ; K. Then for any x H; kxk = we have the equality: Re [h( I T ) x; xi hx; (T I) xi] (.63) = T 4 j j I x; x :

29 .3 Some Associated Functionals 3 Proof. We use the following elementary identity for complex numbers: Re a b = h ja bj ja bj i ; a; b C; (.64) 4 for the choices a = h( I T ) x; xi = ht x; xi and b = h(t I) x; xi = ht x; xi to get h i Re h( I T ) x; xi h(t I) x; xi (.65) = hj j jh ht x; xi ( )ij i 4 for x H; kxk = ; which is clearly equivalent with (.63). Corollary 46 For any T B (H) and ; K we have and inf Re [h( I T ) x; xi hx; (T I) xi] (.66) kxk= = 4 j j w T I sup Re [h( I T ) x; xi hx; (T I) xi] (.67) kxk= = 4 j j m T I : The proof is obvious from the identity (.63) on taking the in mum and the supremum over x H; kxk = ; respectively. If we denote by S H := fx Hj kxk = g the unit sphere in H and, for T B (H) ; ; K we de ne [7] (T ; ; ) (x) := Re [h( I T ) x; xi hx; (T I) xi] ; x S H ; then, on utilizing the elementary properties of complex numbers we have (T ; ; ) (x) = (Re Re ht x; xi) (Re ht x; xi Re ) (.68) (Im Im ht x; xi) (Im ht x; xi Im ) for any x S H : If we denote [7]: s(i) (T ; ; ) := sup kxk= inf kxk= (T ; ; ) (x) then (.66) can be stated as: i (T ; ; ) w T I = 4 j j (.69)

30 4. Inequalities for One Operator while (.67) can be stated as: s (T ; ; ) m T for any T B (H) and ; K. I = 4 j j (.70) Remark 47 Utilising the equality (.68), a su cient condition for the inequality i (T ; ; ) 0 or, equivalently, w T I j j to hold is that Re Re ht x; xi Re and Im Im ht x; xi Im (.7) for each x H; kxk = : The following identity that links the norm with the inner product also holds. Lemma 48 Let T B (H) and ; K. The for each x H; kxk = ; we have the equality: Re h(t I) ( I T ) x; xi = T 4 j j I x : (.7) Proof. We utilize the simple identity in inner product spaces Re hu y; y vi = 4 ku u v vk y ; (.73) u; v; y H; for the choices u = x; y = T x; v = x with x H; kxk = to get Re h x T x; T x xi = T 4 j j I x ; x H; kxk = ; which is clearly equivalent with (.7). Corollary 49 For any T B (H) and ; K we have v i [(T I) ( I T )] (.74) = 4 j j T I and v s [(T I) ( I T )] (.75) = 4 j j ` T I :

31 .3 Some Associated Functionals 5 We recall that a bounded linear operator T : H! H is called strongly c -accretive (with c 6= 0) if Re ht y; yi c for each y H; kyk = : For c = 0; the operator is called accretive. Therefore, and for the sake of simplicity, we can call the operator c -accretive for c R and understand the statement in the above sense. Utilising the identity (.7) we can state the following characterization result that will be useful in the sequel: Lemma 50 (Dragomir [7], 007) For T B (H) and ; K, c R; the following statements are equivalent: (i) The operator C ; (T ) := (T I) ( I T ) is c accretive; (ii) We have the inequality: T I 4 j j c : (.76) Remark 5 Since the self-adjoint operator T : H! H satisfying the condition: T c I in the operator partial order is c accretive, then a su cient condition for C ; (T ) to be c accretive is that C ; (T ) is self-adjoint and C ; (T ) c I:.3.3 General Inequalities We can state the following result that provides some inequalities between di erent numerical radii: Theorem 5 (Dragomir [7], 007) For any T B (H) and ; we have the inequalities 4 j j m T 8 I < w e ( I T; T I) : w ( I T ) w (T I) K (.77) and 4 j j w T I m e ( I T; T I) : (.78) Proof. Utilising the elementary inequality Re a b h jaj jbj i ; a; b C (.79)

32 6. Inequalities for One Operator we can state that h i Re h( I T ) x; xi h(t I) x; xi hjh( I T ) x; xij jh(t I) x; xij i (.80) for any x H; kxk = : Taking the supremum over x H; kxk = in (.80) and utilizing the representation (.67) in Corollary 46, we deduce 4 j j m sup kxk= T I hjh( I T ) x; xij jh(t I) x; xij i = w e ( I T; T I) ; which is clearly equivalent to the rst inequality in (.77). Now, by the elementary inequality we can also state that Re a b jaj jbj for each a; b C; 4 j j m T I sup [jh(t I) x; xij jh(t I) x; xij] kxk= sup jh(t kxk= = w ( I T ) w (T I) I) x; xij sup jh(t kxk= I) x; xij and the second part of (.77) is also proved. Taking the in mum over x H; kxk = in (.80) and making use of the representation (.66) from Corollary 46, we deduce the inequality in (.78). Remark 53 If the operator T B (H) and the complex numbers ; are such that i (T ; ; ) 0 or, equivalently w T I j j ; then we have the reverse inequalities 0 4 j j m T I (.8) 8 < w e ( I T; T I) : w ( I T ) w (T I)

33 .3 Some Associated Functionals 7 and 0 4 j j w T I (.8) m e ( I T; T I) : Since, in general, w (B) kbk ; B B (H) ; hence a su cient condition for (.8) and (.8) to hold is that T I j j holds true. We also notice that this last condition is equivalent with the fact that the operator C ; (T ) = (T I) ( I T ) is accretive. From a di erent perspective and as pointed out in Remark 47, a su cient condition for i (T ; ; ) 0 to hold is that (.7) holds true and, therefore, if (.7) is valid, then both (.8) and (.8) can be stated. The following reverse inequality of (.8) is incorporated in the following result: Proposition 54 (Dragomir [7], 007) Let T B (H) and ; such that (.7) holds true. Then K be (0 ) (Re v s (T )) (v i (T ) Re ) (.83) (Im s (T )) ( i (T ) Im ) 4 j j w T Proof. Taking the in mum for x H; kxk = in the identity (.68) and utilizing the representation (.66) and the properties of in mum, we have: 4 j j w T I inf [(Re Re ht x; xi) (Re ht x; xi Re )] kxk= I : inf [(Im Im ht x; xi) (Im ht x; xi Im )] kxk= inf (Re Re ht x; xi) inf (Re ht x; xi Re ) kxk= kxk= inf (Im Im ht x; xi) inf (Im ht x; xi Im ) kxk= kxk= = Re sup Re ht x; xi! inf Re ht x; xi Re kxk= kxk= Im sup Im ht x; xi! inf Im ht x; xi Im kxk= kxk= which is exactly the desired result (.83).

34 8. Inequalities for One Operator The representation in Lemma 48 has its natural consequences relating the numerical values ` (T ) and w (T ) of certain operators as described in the following: Theorem 55 (Dragomir [7], 007) For any T B (H) and ; we have: T I and 4 j j ` 8 4 j j 8 respectively. >< w I T ( I T ) (T I) (T I) ; w [(T I) ( I T )] ; >: 4 k(t I) ( I T ) Ik T I m I T ( I T ) (T I) (T I) ; >< m [(T I) ( I T )] ; >: 4` [(T I) ( I T ) I] ; Proof. Utilising the elementary inequality in inner product spaces we can state that Re hu; vi K (.84) (.85) h kuk kvk i ; u; v H; (.86) Re h( I T ) x; (T I) xi (.87) hk( I T ) xk k(t I) xk i = I T ( I T ) x; x h(t I) (T I) x; xi = I T ( I T ) (T I) (T I) x; x for each x H; kxk = : Taking the supremum in (.87) over x H; kxk = and utilizing the representation (.75), we deduce the rst inequality in (.84). Now, by the elementary inequality Re (T ) jt j ; T C we have Re h(t I) ( I T ) x; xi jh(t I) ( I T ) x; xij ; (.88)

35 .3 Some Associated Functionals 9 which provides, by taking the supremum over x H; kxk = ; the second inequality in (.84). Finally, on utilizing the inequality we also have Re hu; vi 4 ku vk ; u; v H; Re h(t I) ( I T ) x; xi 4 k[(t I) ( I T ) I] xk (.89) for any x H; kxk = ; which gives, by taking the supremum, the last part of (.84). The proof of (.85) follows by the representation (.74) in Corollary 49 and by the inequalities (.87) (.89) above in which we take the in mum over x H; kxk = : Corollary 56 Let T B (H) and ; K. If C ; (T ) is accretive, then 0 4 j j ` T I (.90) 8 >< w I T ( I T ) (T I) (T I) ; w [(T I) ( I T )] ; >: 4 k(t I) ( I T ) Ik and respectively. 0 4 j j T 8 I >< m I T ( I T ) (T I) (T I) ; m [(T I) ( I T )] ; >: 4` [(T I) ( I T ) I] ;.3.4 Reverse Inequalities (.9) The inequality kt k w (T ) for any bounded linear operator T B (H) is a fundamental result in Operator Theory and therefore it is useful to know some upper bounds for the nonnegative quantity kt k w (T ) under various assumptions for the operator T. In our recent paper [3] several such inequalities have been obtained. In order to establish some new results that would complement the inequalities outlined in the Introduction, we need the following lemma which provides two simple identities of interest:

36 30. Inequalities for One Operator Lemma 57 (Dragomir [7], 007) For any T B (H) and ; we have K kt xk jht x; xij (.9) = T I x T I x; x = Re [h( I T ) x; xi hx; (T I) xi] Re h( I T ) x; (T I) xi ; for each x H; kxk = : Proof. The rst identity is obvious by direct calculation. The second identity can be obtained, for instance, by subtracting the identity (.7) from (.63). As a natural application of the above lemma in providing upper bounds for the nonnegative quantity kt k w (T ) ; T B (H) ; we can state the following result: Theorem 58 (Dragomir [7], 007) For any T B (H) and ; we have K (0 ) kt k w (T ) (.93) T I m T I = 4 j j m T I v i [(T I) ( I T )] : Proof. From the rst identity in (.9) we have kt xk = jht x; xij T I x T I x; x (.94) for any x H; kxk = : Taking the supremum over x H; kxk = and utilizing the fact that " sup jht x; xij # T I x kxk= T I x; x sup jht x; xij sup T I x kxk= kxk= inf kxk= T I x; x = w (T ) T I x m T I ;

37 we deduce the rst part of (.93). The second part follows by the identity (.74)..3 Some Associated Functionals 3 Remark 59 Utilising the inequality (.77) in Theorem 5 we can obtain from (.93) the following result: (0 ) kt k w (T ) (.95) 8 < v i [(T w e ( I T; T I) ; I) ( I T )] : w ( I T ) w (T I) ; which holds true for each T B (H) and ; K. Since m T I 0, hence we also have the general inequality (0 ) kt k w (T ) (.96) for any T B (H) and ; K. 4 j j v i [(T I) ( I T )] ; Theorem 58 admits the following particular case that provides a simple reverse inequality for kt k w (T ) under some appropriate assumptions for the operator T that have been considered in the introduction and are motivated by earlier results: K, c R: If the composite oper- Corollary 60 Let T B (H) and ; ator C ; (T ) is c accretive, then: (0 ) kt k w (T ) (.97) 4 j j c m T I : The proof is obvious by the rst part of the inequality (.93) and by Lemma 50 which states that C ; (T ) is c accretive if and only if the inequality (.73) holds true. Remark 6 From (.97) we can deduce the following reverse inequalities which are coarser, but perhaps more useful when the terms in the upper bounds are known: (0 ) kt k w (T ) (.98) 8 4 >< j j ; c w e ( I T; T I) ; >: w ( I T ) w (T I) :

38 3. Inequalities for One Operator In particular, if C ; stated: (T ) is accretive, then the following inequalities can be (0 ) kt k w (T ) (.99) 4 j j m T I 8 4 >< j j ; w e ( I T; T I) ; >: w ( I T ) w (T I) : Remark 6 If N n > 0 and the composite operator C n;n (T ) is c accretive or, su ciently, self-adjoint and positive de nite with the constant c 0; then we have the inequalities: (0 ) kt k w (T ) (.00) 4 (N n) c m T I 8 4 >< (N n) ; c w e (NI T; T ni) ; >: w (NI T ) w (T ni) : Remark 63 If the operator T on the scalars ; from the statement of Corollary 60 have in addition the property that T I x; x d for each x H; kxk = ; (.0) where d > 0 is given, then by (.97) we also have (0 ) kt k w (T ) 4 j j c d : (.0) We notice that a su cient condition for (.0) to hold is that the operator T I be d accretive. Remark 64 Finally, we note that if the operator C n;n (T ) is accretive, (or su ciently self-adjoint and positive), then we have the following reverse inequalities: 8 4 >< (N n) ; (0 ) kt k w (T ) w e (NI T; T ni) ; (.03) >: w (NI T ) w (T ni) :

39 .4 Inequalities for the Maximum of the Real Part 33.4 Inequalities for the Maximum of the Real Part.4. Introduction For a bounded linear operator T on the complex Hilbert space, consider the maximum and the minimum of the spectrum of the real part of T denoted by [8] v s(i) (T ) := sup kxk= inf kxk= Re ht x; xi = max(min) (Re (T )) : The following properties are obvious by the de nition: (a) v s ( T ) = v i (T ) ; T B (H) ; (aa) v i (T ) 0 for accretive operators on H; (aaa) v s(i) (A B) () v s(i) (A) v s(i) (B) for any A; B B (H) ; (av) maxfjv i (A)j ; jv s (A)jg = w (Re(A)) w (A) for all A B (H) : More properties which connect these functionals with the semi-inner products generated by the operator norm and the numerical radius are outlined in the next section. An improvement of Lumer s classical result and some bounds are also given. Motivated by the above results, we establish in the present section some upper bounds for the nonnegative quantities kak v s (A) ( kak w (A) 0) and w (A) v s (A) ( 0); for some C, jj = under suitable assumptions on the involved operator A B (H) : Lower bounds for the quantities v s(a) kak w(a) kak and vs(a) w(a) ( ) are also given. They improve some results from the earlier paper [3]. Inequalities in terms of the semi-inner products that can naturally be associated with the operator norm and the numerical radius are provided as well. For other recent results concerning inequalities between the operator norm and numerical radius see the papers [3], [6], [], [39] and [38]. Lower bounds for w (A) are in the nite-dimensional case studied in [43]. For classical results, see the books [33], [34] and the references therein..4. Preliminary Results for Semi-Inner Products In any normed linear space (E; kk) ; since the function f : E! R, f (x) = kxk is convex, one can introduce the following semi-inner products (see for instance [0]): hx; yi i := lim t!0 ky txk kyk ; (.04) t hx; yi s := lim t!0 ky txk kyk t

40 34. Inequalities for One Operator where x; y are vectors in E: The mappings h; i s and h; i i are called the superior respectively the inferior semi-inner product associated with the norm kk : For the sake of completeness we list here some properties of h; i s(i) that will be used in the sequel. We have, for p; q fi; sg and p 6= q; that (i) hx; xi p = kxk for any x E; (ii) hix; xi p = hx; ixi p = 0 for any x E; (iii) hx; yi p = hx; yi p = hx; yi p for any 0 and x; y E; (iv) h x; yi p = hx; yi p = hx; yi q for any x; y E; (v) hix; yi p = hx; iyi p for any x; y E; (vi) The following Schwarz type inequality holds: hx; yi p kxk kyk ; for any x; y E; (vii) The following identity holds: for any R and x; y E; hx y; xi p = kxk hy; xi p, (viii) The following sub(super)-additivity property holds: hx y; zi p () hx; zi p hy; zi p ; for any x; y; z E, where the sign applies for the superior semiinner product, while the sign applies for the inferior one; (ix) The following continuity property is valid: hx y; zi p hy; zi p kxk kzk ; for any x; y; z E; (x) From the de nition we have the inequality for any x; y E: hx; yi i hx; yi s

41 .4 Inequalities for the Maximum of the Real Part 35 In the Banach algebra B (H) we can associate to both the operator norm kk and the numerical radius w () the following semi-inner products: ha; Bi s(i);n := kb tak kbk lim t!0( ) t (.05) and ha; Bi s(i);w := lim t!0( ) w (B ta) t w (B) (.06) respectively, where A; B B (H) : It is obvious that the semi-inner products h; i s(i);n(w) de ned above have the usual properties of such mappings de ned on general normed spaces and some special properties that will be speci ed in the following. As a speci c property that follows by the well known inequality between the norm and the numerical radius of an operator, i.e., w (T ) kt k for any T B (H) ; we have ht; Ii i;n ht; Ii i;w () ht; Ii s;w ht; Ii s;n (.07) for any T B (H) ; where I is the identity operator on H: We also observe that ki tt k ht; Ii s(i);n = lim t!0( ) t and ht; Ii s(i);w = w (I tt ) lim t!0( ) t for any T B (H) : It may be of interest to note that ht; Ii s;n and ht; Ii s;w are also called the logarithmic norms of T corresponding to kk and w respectively. Logarithmic norms corresponding to a given norm have been rather widely studied (mainly in the fnite-dimensional case); see [53]. The following result is due to Lumer and was obtained originally for the numerical radius of operators in Banach spaces: Theorem 65 (Lumer [4], 96) If T B (H) ; then ht; Ii p;n = v p (T ) ; p fs; ig : The following simple result provides a connection between the semi-inner products generated by the operator norm and by the numerical radius as follows: Theorem 66 (Dragomir [8], 008) For any T B (H) ; we have: where p fs; ig : ht; Ii p;n = ht; Ii p;w ; (.08)

42 36. Inequalities for One Operator Proof. Let us give a short proof for the case p = s: Suppose x H; kxk = : Then for t > 0 we obviously have: Re hx tt x; xi Re ht x; xi = t jhx tt x; xij t Taking the supremum over x H; kxk = ; we get w (I tt ) : t w (I tt ) v s (T ) = sup Re ht x; xi kxk= t (.09) for any t > 0; which implies, by letting t! 0 that sup Re ht x; xi ht; Ii s;w ; (.0) kxk= for any T B (H) : By Lumer s theorem we deduce then ht; Ii s;n ht; Ii s;w and since, by (.07) we have ht; Ii s;w ht; Ii s;n the equality (.08) is obtained. Now, on employing the properties of the semi-inner products outlined above, we can state the following properties as well: (va) v s(i) (T ) = ht; Ii s(i);w for any T B (H) ; (vaa) v s(i) (T ) = v s(i) (I T ) for any R and T B (H) ; (vaaa) vs(i) (T B) v s(i) (B) w (T ) for any T; B B (H) : The following inequalities may be stated as well: Theorem 67 (Dragomir [8], 008) For any T B (H) and C we have hkt k jj i v s T (.) 8 >< >: h 4 kt k jj i kt Ik ; h kt Ik kt Ik i ; and respectively. hw (T ) jj i v s T 8 < : h 4 w (T ) jj i w (T I) ; w (T I) w (T I) : (.)

43 .4 Inequalities for the Maximum of the Real Part 37 Proof. Let x H; kxk = : Then, obviously 0 kt xk Re T x; x jj = k(t I) xk kt Ik ; which is equivalent with h kt xk jj i kt Ik Re T x; x h kt xk jj i ; (.3) for any x H; kxk = : Taking the supremum over kxk = we get the rst inequality in (.) and the one from the rst branch in the second. For x H; kxk = we also have that kt x xk = kt x xk 4 Re T x; x ; (.4) which, on taking the supremum over kxk = ; will produce the second part of the second inequality in (.). 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 h; i s(i) de ned on Banach spaces are not commutative. However, for the Banach algebra B (H) we can point out the following connection between hi; T i s(i);n(w) and the quantities v i (T ) and v s (T ) ; where T B (H) : Corollary 68 For any T B (H) we have (v i (T ) =) ht; Ii i;n i hhi; T i s;n hi; T i i;n ht; Ii s;n (= v s (T )) : (.5) and hhi; T i s;w hi; T i i;w i (v i (T ) =) ht; Ii i;w ht; Ii s;w (= v s (T )) : (.6) Proof. We have from the second part of the second inequality in (.) that " # kt tik kt k kt tik kt k v s (T ) (.7) t t for any t > 0: Taking the limit over t! 0 and noticing that kt tik kt k lim = h I; T i t!0 t s;n = hi; T i i;n ;

44 38. Inequalities for One Operator we get the second inequality in (.5). Now, writing the second inequality in (.5) for v s ( T ) = hhi; T i s;n hi; T i i;n i hhi; T i s;n hi; T i i;n i ; T; we get which is equivalent with the rst part of (.5). Since w (T ) kt k ; hence the rst inequality in (.) provides a better upper bound for v s T than the rst inequality in (.)..4.3 Reverse Inequalities in Terms of the Operator Norm The following result concerning reverse inequalities for the maximum of the spectrum of the real part and the operator norm of T B (H) may be stated: Theorem 69 (Dragomir [8], 008) For any T B (H) n f0g and Cn f0g we have the inequality: (0 kt k w (T ) ) kt k v s jj T (.8) jj kt Ik : In addition, if kt s T Ik jj ; then we have: v s jj T I kt k w (T ) kt k (.9) and 0 kt k w (T ) kt k jj jj v s jj T qjj kt Ik v s jj T qjj kt Ik w (T ) ; (.0) respectively. Proof. Utilizing the property (av), we have w (T ) = w jj T v s jj T v s jj T ;

45 .4 Inequalities for the Maximum of the Real Part 39 for any Cn f0g and the rst inequality in (.8) is proved. By the arithmetic mean-geometric mean inequality we have which, by (.) provides h kt k jj i jj kt k ; v s T jj kt k kt Ik that is equivalent with the second inequality in (.8). Utilizing the second part of the inequality (.) and under the assumption that kt Ik jj we can also state that " # v s T kt k qjj kt Ik : (.) By the arithmetic mean-geometric mean inequality we have now: kt " kt k qjj # Ik q kt k jj kt Ik ; (.) which, together with (.) implies the rst inequality in (.9). The second part of (.9) follows from (av). From the proof of Theorem 67 we can state that kt xk jj Re T x; x r ; kxk = (.3) where we denoted r := kt Ik jj : We also observe, from (.3), that Re T x; x > 0 for x H; kxk = D : E Now, if we divide (.3) by Re jj T x; x > 0; we get kt xk Re D jj T x; x E jj r Re D jj T x; x E jj Re D jj T x; x E (.4) for kxk = :

46 40. Inequalities for One Operator D E If in this inequality we subtract from both sides the quantity Re jj T x; x ; then we get kt xk D E Re jj T x; x = jj Re jj T x; x jj r jj D E Re jj T x; x 0 q jj r jj r Re D E jj T x; x qjj r ; which obviously implies that kt xk Re jj Re jj T x; x s Re jj T x; x C A q jj r jj T x; x (.5) qjj r Re jj T x; x for any x H; kxk = : Now, taking the supremum in (.5) over x H; kxk = ; we deduce the second inequality in (.0). The other inequalities are obvious and the theorem is proved. The following lemma is of interest in itself. Lemma 70 (Dragomir [8], 008) For any T B (H) and ; C we have: v i [(T I) ( I T )] = 4 j j T I : (.6) Proof. We observe that, for any u; v; y H we have: Re hu y; y vi = 4 ku u v vk y : (.7) Now, choosing u = x; y = T x; v = x with x H; kxk = we get Re h x T x; T x xi = 4 j j T x x ;