SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES SEVER S DRAGOMIR Abstract Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert spaces are given Their connection with Kittaneh s recent results which provide sharp upper and lower bounds for the numerical radius of linear operators are also established Introduction Let B H) denote the C algebra of all bounded linear operators on a complex Hilbert space H with inner product, For A B H), let w A) and A denote the numerical radius and the usual operator norm of A, respectively It is well known that w ) defines a norm on B H), and for every A B H), ) A w A) A For other results concerning the numerical range and radius of bounded linear operators on a Hilbert space, see ] and 3] In ], F Kittaneh has improved ) in the following manner: ) A A + AA w A) A A + AA, with the constants and as best possible Following Popescu s work 5], we consider the Euclidean operator radius of a pair C, D) of bounded linear operators defined on a Hilbert space H;, ) Note that in 5], the author has introduced the concept for an n tuple of operators and pointed out its main properties Let C, D) be a pair of bounded linear operators on H The Euclidean operator radius is defined by: 3) w e C, D) := sup Cx, x + Dx, x ) / x = As pointed out in 5], w e : B H) 0, ) is a norm and the following inequality holds: ) C C + D D / w e C, D) C C + D D /, where the constants and are best possible in ) Date: February, 006 000 Mathematics Subject Classification Primary 7A, 7A30, 7A63 Key words and phrases Numerical radius, Euclidean radius, Bounded linear operators, Operator norm, Self-adjoint operators
SEVER S DRAGOMIR We observe that, if C and D are self-adjoint operators, then ) becomes 5) C + D / w e C, D) C + D / We observe also that if A B H) and A = B + ic is the Cartesian decomposition of A, then we B, C) = sup Bx, x + Cx, x ] x = By the inequality 5) and since see ]) = sup Ax, x = w A) x = 6) A A + AA = B + C ), then we have 7) 6 A A + AA w A) A A + AA We remark that the lower bound for w A) in 7) provided by Popescu s inequality ) is not as good as the first inequality of Kittaneh from ) However, the upper bounds for w A) are the same and have been proved using different arguments The main aim of this paper is to extend Kittaneh s result to Euclidean radius of two operators and investigate other particular instances of interest Related results connecting the Euclidean operator radius, the usual numerical radius of a composite operator and the operator norm are also provided Some Inequalities for the Euclidean Operator Radius The following result concerning a sharp lower bound for the Euclidean operator radius may be stated: Theorem Let B, C : H H be two bounded linear operators on the Hilbert space H;, ) Then ) w B + C )] / we B, C) B B + C C /) The constant is best possible in the sense that it cannot be replaced by a larger constant Proof We follow a similar argument to the one from ] For any x H, x =, we have ) Bx, x + Cx, x Bx, x + Cx, x ) Taking the supremum in ), we deduce B ± C) x, x 3) w e B, C) w B ± C)
INEQUALITIES FOR THE EUCLIDEAN RADIUS 3 Utilising the inequality 3) and the properties of the numerical radius, we have successively: we B, C) w B + C) + w B C) ] { w B + C) ] + w B C) ]} { w B + C) + B C) ]} = w B + C ), which gives the desired inequality ) The sharpness of the constant will be shown in a particular case, later on Corollary For any two self-adjoint bounded linear operators B, C on H, we have ) B + C / w e B, C) B + C /) The constant is sharp in ) Remark The inequality ) is better than the first inequality in 5) which follows from Popescu s first inequality in ) It also provides, for the case that B, C are the self-adjoint operators in the Cartesian decomposition of A, exactly the lower bound obtained by Kittaneh in ) for the numerical radius w A) Moreover, since is a sharp constant in Kittaneh s inequality ), it follows that is also the best possible constant in ) and ), respectively The following particular case may be of interest: Corollary For any bounded linear operator A : H H and α, β C we have: w α A + β A ) ] 5) α + β ) w A) α A A + β AA ) Proof If we choose in Theorem, B = αa and C = βa, we get we B, C) = α + β ) w A) and w B + C ) = w α A + β A ) ], which, by ) implies the desired result 5) Remark If we choose in 5) α = β 0, then we get the inequality 6) A + A ) w A) ) A A + AA, for any bounded linear operator A B H) If we choose in 5), α =, β = i, then we get 7) w A A ) ] w A), for every bounded linear operator A : H H
SEVER S DRAGOMIR The following result may be stated as well Theorem For any two bounded linear operators B, C on H we have: 8) max {w B + C), w B C)} w e B, C) w B + C) + w B C) ] / The constant is sharp in both inequalities Proof The first inequality follows from 3) For the second inequality, we observe that 9) Cx, x ± Bx, x w C ± B) for any x H, x = The inequality 9) and the parallelogram identity for complex numbers give: 0) Bx, x + Cx, x ] = Bx, x Cx, x + Bx, x + Cx, x w B + C) + w B C), for any x H, x = Taking the supremum in 9) we deduce the desired result 8) The fact that is the best possible constant follows from the fact that for B = C 0 one would obtain the same quantity w B) in all terms of 8) Corollary 3 For any two self-adjoint operators B, C on H we have: ) max { B + C, B C } w e B, C) B + C + B C ] / The constant is best possible in both inequalities Corollary Let A be a bounded linear operator on H Then { max i) A + + i) A, + i) A + i) A } ) w A) i) A + + i) A + + i) A + i) A Proof Follows from ) applied for the Cartesian decomposition of A The following result may be stated as well: ] /
INEQUALITIES FOR THE EUCLIDEAN RADIUS 5 Corollary 5 For any A a bounded linear operator on H and α, β C, we have: 3) max {w αa + βa ), w αa βa )} α + β ) / w A) w αa + βa ) + w αa βa ) ] / Remark 3 The above inequality 3) contains some particular cases of interest For instance, if α = β 0, then by 3) we get ) max { A + A, A A } A + A + A A ] /, w A) since, obviously w A + A ) = A + A and w A A ) = A A, A A being a normal operator Now, if we choose in 3), α = and β = i, and taking into account that A + ia and A ia are normal operators, then we get 5) max { A + ia, A ia } A + ia + A ia ] / w A) The constant is best possible in both inequalities ) and 5) The following simple result may be stated as well Proposition For any two bounded linear operators B and C on H, we have the inequality: 6) w e B, C) w C B) + w B) w C) ] / Proof For any x H, x =, we have ] Cx, x Re Cx, x Bx, x + Bx, x giving 7) = Cx, x Bx, x w C B), ] Cx, x + Bx, x w C B) + Re Cx, x Bx, x w C B) + Cx, x Bx, x for any x H, x = Taking the supremum in 7) over x =, we deduce the desired inequality 6) In particular, if B and C are self-adjoint operators, then / 8) w e B, C) B C + B C ) Now, if we apply the inequality 8) for B = A+A and C = A A i, where A B H), then we deduce:
6 SEVER S DRAGOMIR + i) A + i) A w A) / + A + A A A ] The following result provides a different upper bound for the Euclidean operator radius than 6) Proposition For any two bounded linear operators B and C on H, we have 9) w e B, C) min { w B), w C) } + w B C) w B + C) ] / Proof Utilising the parallelogram identity 0), we have, by taking the supremum over x H, x =, that 0) w e B, C) = w e B C, B + C) Now, if we apply Proposition for B C, B + C instead of B and C, then we can state w e B C, B + C) w C) + w B C) w B + C) giving ) w e B, C) w C) + w B C) w B + C) Now, if in ) we swap the C with B then we also have ) w e B, C) w B) + w B C) w B + C) The conclusion follows now by ) and ) A different upper bound for the Euclidean operator radius is incorporated in the following Theorem 3 Let H;, ) be a Hilbert space and B, C two bounded linear operators on H Then { 3) we B, C) max B, C } + w C B) The inequality 3) is sharp Proof Firstly, let us observe that for any y, u, v H we have successively ) y, u u + y, v v = y, u u + y, v v + Re ] y, u y, v u, v y, u u + y, v v + y, u y, v u, v y, u u + y, v v + y, u + y, v ) u, v y, u + y, v ) { max u, v } ) + u, v On the other hand, y, u + y, v ) 5) = y, u u, y + y, v v, y ] for any y, u, v H = y, y, u u + y, v v ] y y, u u + y, v v
INEQUALITIES FOR THE EUCLIDEAN RADIUS 7 Making use of ) and 5) we deduce that { 6) y, u + y, v y max u, v } ] + u, v for any y, u, v H, which is a vector inequality of interest in itself Now, if we apply the inequality 6) for y = x, u = Bx, v = Cx, x H, x =, then we can state that { 7) Bx, x + Cx, x max Bx, Cx } + Bx, Cx for any x H, x =, which is of interest in itself Taking the supremum over x H, x =, we deduce the desired result 3) To prove the sharpness of the inequality 3) we choose C = B, B a self-adjoint operator on H In this case, both sides of 3) become B If information about the sum and the difference of the operators B and C are available, then one may use the following result: Corollary 6 For any two operators B, C BH) we have 8) we B, C) { { max B C, B + C } } + w B C ) B + C)] The constant is best possible in 8) Proof Follows by the inequality 3) written for B + C and B C instead of B and C and by utilising the identity 0) The fact that is best possible in 8) follows by the fact that for C = B, B a self-adjoint operator, we get in both sides of the inequality 8) the quantity B Corollary 7 Let A : H H be a bounded linear operator on the Hilbert space H Then: 9) w A) { max A + A, A A } ] + w A A) A + A )] The constant is best possible Proof If B = A+A, C = A A i and Utilising 3) we deduce 9) is the Cartesian decomposition of A, then w e B, C) = w A) w C B) = w A A) A + A )] Remark If we choose in 3), B = A and C = A, A BH) then we can state that 30) w A) A + w A )] The constant is best possible in 30) Note that this inequality has been obtained in ] by the use of a different argument based on the Buzano s inequality Finally, the following upper bound for the Euclidean radius involving different composite operators also holds:
8 SEVER S DRAGOMIR Theorem With the assumptions of Theorem 3, we have 3) w e B, C) B B + C C + B B C C ] + w C B) The inequality 3) is sharp Proof We use 7) to write that 3) Bx, x + Cx, x Bx + Cx + Bx Cx ] + Bx, Cx for any x H, x = Since Bx = B Bx, x, Cx = C Cx, x, then 3) can be written as 33) Bx, x + Cx, x B B + C C) x, x + B B C C) x, x ] + Bx, Cx x H, x = Taking the supremum in 33) over x H, x = and noticing that the operators B B ± C C are self-adjoint, we deduce the desired result 3) The sharpness of the constant will follow from the one of 36) pointed out below Corollary 8 For any two operators B, C BH), we have 3) w e B, C) { B B + C C + B C + C B + w B C ) B + C)]} The constant is best possible Proof If we write 3) for B + C, B C instead of B, C and perform the required calculations then we get w e B + C, B C) B B + C C + B C + C B ] + w B C ) B + C)], which, by the identity 0) is clearly equivalent with 3) Now, if we choose in 3) B = C, then we get the inequality w B) B, which is a sharp inequality Corollary 9 If B, C are self-adjoint operators on H then 35) we B, C) B + C + B C ] + w CB) We observe that, if B and C are chosen to be the Cartesian decomposition for the bounded linear operator A, then we can get from 35) that 36) w A) { A A + AA + A + A ) } + w A A) A + A )] The constant is best possible This follows by the fact that for A a self-adjoint operator, we obtain in both sides of 36) the same quantity A
INEQUALITIES FOR THE EUCLIDEAN RADIUS 9 Now, if we choose in 3) B = A and C = A, A BH), then we get 37) w A) { A A + AA + A A AA } + w A ) This inequality is sharp The equality holds if, for instance, we assume that A is normal, ie, A A = AA In this case we get in both sides of 37) the quantity A, since for normal operators, w A ) = w A) = A Acknowledgement The author would like to thank the anonymous referee for his/her comments that have been implemented in the final version of the paper References ] SS DRAGOMIR, Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces, RGMIA Res Rep Coll, 8005), Supplement, Article 0 ONLINE http://rgmiavueduau/v8e)html ] ] KE GUSTAFSON and DKM RAO, Numerical Range, Springer-Verlag, New York, Inc, 997 3] PR HALMOS, A Hilbert Space Problem Book, Springer-Verlag, New York, Heidelberg, Berlin, Second edition, 98 ] F KITTANEH, Numerical radius inequalities for Hilbert space operators, Studia Math, 68) 005), 73 80 5] G POPESCU, Unitary invariants in multivariable operator theory, Preprint, ArχivmathOA/009 School of Computer Science and Mathematics, Victoria University, PO Box 8, Melbourne City, VIC, 800, Australia E-mail address: severdragomir@vueduau URL: http://rgmiavueduau/dragomir