A Numerical Radius Version of the Arithmetic-Geometric Mean of Operators

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1 Filomat 30:8 (2016), DOI /FIL S Published by Faculty of Sciences and Mathematics, University of Niš, Serbia vailable at: htt://wwwmfniacrs/filomat Numerical Radius Version of the rithmetic-geometric Mean of Oerators lemeh Sheikhhosseini a a Deartment of Pure Mathematics, Faculty of Mathematics and Comuter, Shahid Bahonar University of Kerman, Kerman, Iran bstract In this aer, we obtain some numerical radius ineualities for oerators, in articular for ositive definite oerators, B a numerical radius and some oerator norm versions for arithmeticgeometric mean ineuality are obtained, resectively as ω B 2 ( B) ω inf δ(x), where δ(x) = ( B)x, x 2, and where, δ(x, y) = ( y, y Bx, x )2 B 1 2 ( 2 + B 2 ) 1 2 inf δ(x, y), 1 Introduction Let H be a comlex Hilbert sace with inner roduct, and let B(H) denote the algebra of all bounded linear oerators on H Let denote any unitarily invariant norm, ie, a norm with the roerty that UV =, for all B(H) and for all unitary U, V B(H) For B(H), the sectral norm of is defined by It is evident that this norm is unitary invariant The numerical range of a B(H) is defined as = su{ x, y : x = y = 1, x, y H} W() = su{ x, x : x = 1, x H} For any B(H), W() is a convex subset of the comlex lane containing the sectrum of See [5, Chater 2] for this toic The numerical radius of B(H) is defined by ω() = su{ λ : λ W()} We recall the following results that were roved in [6] 2010 Mathematics Subject Classification Primary 4730; Secondary 4712 Keywords Geometric mean, Ineualities, Numerical radius, Oerator norm Received: 15 May 2014; cceted: 24 July 2014 Communicated by Mohammad Sal Moslehian address: sheikhhosseini@ukacir (lemeh Sheikhhosseini)

2 Sheikhhosseini / Filomat 30:8 (2016), Lemma 11 Let B(H) and let ω() be the numerical radius Then (i) ω() is a norm on B(H), (ii) ω(uu ) = ω(), for all unitary oerators U, (iii) ω() = if ( but not only if) is normal, (iv) 1 2 ω() Moreover, ω() is not a unitarily invariant norm and is not submultilicative For ositive real numbers a and b, the most familiar form of the Young ineuality is the following: ab a + b, (1) where, > 1 such that = 1, or euivalently a ν b 1 ν νa + (1 ν)b, with ν [0, 1] Recently, Kittaneh and Manasrah [8] obtained a refinement of (1) ab + r 0 (a /2 b /2 ) 2 a + b, (2) where r 0 = min{ 1, 1 } For ositive definite oerators, B B(H), the oerator geometric mean is defined by B 1/2 ( 1/2 B 1/2 ) 1/2 1/2 The oerator geometric mean has the symmetric roerty ( B = B ) If B = B, then B = (B) 1/2 In this aer we obtain some ineualities (uer bound) for ω(( B)X), where X B(H) is arbitrary Throughout the aer we use the notation > 0 to mean that is ositive definite and M n the sace of all n n matrices 2 Main Results Bhatia and Kittaneh in 1990 [3] established a matrix mean ineuality as follows: B B B, (3) for matrices, B M n In [2] a generalization of (3) was roved, for all X M n, XB 1 2 X + XBB ndo in 1995 [1] established a matrix Young ineuality: B + B (4) for, > 1 with 1/ + 1/ = 1 and ositive matrices, B In [9] we considered the ineualities (3) and (4) with the numerical radius norm as follows:

3 Sheikhhosseini / Filomat 30:8 (2016), Proosition 21 [9, Proosition 1] If, B are n n matrices, then ω( B) 1 2 ω( + B B) lso if and B are ositive matrices and, > 1 with 1/ + 1/ = 1, then ω(b) ω( + B ) Moreover, the authors, in [9, Theorem 2 ] and [10, Theorem 23], showed that the ineuality XB X + X B does not holds for numerical radius and sectral norm for all X M n and ositive matrices, B The following lemma is a conseuence of the sectral theorem for ositive oerators and Jensen s ineuality (see, eg, [7]) Lemma 22 Let be a ositive semidefinite oerator in B(H) and let x H be any unit vector Then for all r 1 x, x r r x, x, and for all 0 r 1 r x, x x, x r Theorem 23 Let, B, X B(H), such that, B > 0 and > 1 where 1/ + 1/ = 1 Then for all r 2 ω r r/2 (( B)X) ω + (X BX) r/2 1 inf δ(x), (6) where δ(x) = ( x, x r/4 X BXx, x r/4) 2 Proof Let x H, with x = 1 By the Schwarz ineuality in the Hilbert sace (H;, ), we have ( B)Xx, x r = 1/2 ( 1/2 B 1/2 ) 1/2 1/2 Xx, x r = ( 1/2 B 1/2 ) 1/2 1/2 Xx, 1/2 x r ( 1/2 B 1/2 ) 1/2 1/2 Xx r 1/2 x r = ( 1/2 B 1/2 ) 1/2 1/2 Xx, ( 1/2 B 1/2 ) 1/2 1/2 Xx r/2 1/2 x, 1/2 x r/2 = x, x r/2 X BXx, x r/2 Now, by Young s ineuality and (2) we have x, x r/2 X BXx, x r/2 1 x, x r/2 + 1 X BXx, x r/2 1 ( x, x r/4 X BXx, x r/4) 2 and by (5) we have 1 x, x r/2 + 1 X BXx, x r/2 1 ( x, x r/4 X BXx, x r/4) 2 1 r/2 x, x + 1 (X BX) r/2 x, x 1 ( x, x r/4 X BXx, x r/4) 2 ( ) r/2 = + (X BX) r/2 x, x 1 ( x, x r/4 X BXx, x r/4) 2 Now, the result follows by taking the suremum over all unit vectors in H (5)

4 Sheikhhosseini / Filomat 30:8 (2016), Remark 24 Let r = = = 2 Then δ(x) 0 if and only if X BX = 0 In general, δ(x) = 0 if and only if x, x r/4 = X BXx, x r/4 The following examle shows that, ineuality (6) does not hold in general for sectral norm [ ] [ ] Examle 25 If we take = = 2, r = 1, = 1, B = I 0 2 and X =, then = ( B)X r > r/2 + (X BX) r/2 = 5 8 Put X = I in Theorem 23, we obtain the following corollary Corollary 26 Let, B B(H), be ositive definite and > 1 such that 1/ + 1/ = 1 Then for all r 2 ω r r/2 ( B) ω + Br/2 1 inf δ(x), where δ(x) = ( x, x r/4 Bx, x r/4) 2 Note that it is enough to relace r with 2r and X = I in the statement of Theorem 23 to obtain the following corollary Corollary 27 Let, B B(H) be ositive definite oerators and > 1 such that 1/ + 1/ = 1 Then for all r 1, ω 2r ( B) ω( r where δ(x) = ( x, x r/2 Bx, x r/2) 2 + Br ) 1 inf δ(x), (7) Remark 28 Note that, if we set r = 1 and = = 2 in (7), then we have ω B 2 ( B) ω inf δ(x), (8) where δ(x) = ( B)x, x 2 Notice that (8) is an oerator numerical radius version for arithmetic-geometric mean and moreover if, 0 W( B), then inf δ(x) > 0 In the roof of Theorem 23, if we ut r = 2 and X = I, then we have the following corollary Corollary 29 Let, B B(H), be ositive definite oerators Then B 2 B Let T, U B(H) The Euclidean radius(see [4]) is defined by ( ω e (T, U) = su Tx, x 2 + Ux, x 2) 1/2 Corollary 210 Let, B B(H), be ositive definite oerators Then 2 B ωe (, B) 2 + B 2 1/2, inarticular, 2ω( B) ωe (, B) ω 1/2 ( 2 + B 2 )

5 Sheikhhosseini / Filomat 30:8 (2016), Proof Same as, in the roof of Theorem 23, if we set r = = = 2, then we have ( B)x, x ( x, x 2 + Bx, x 2 ) (9) and by Lemma 22, 1 2 ( x, x 2 + Bx, x 2 ) 1 2 ( 2 x, x + B 2 x, x ) = 1 ( 2 + B 2 )x, x (10) 2 Now, the result follows by taking the suremum in (9) and (10) over all unit vectors in H 3 dditional Results Proosition 31 Let, B, X B(H) such that, B > 0 and > 1 such that 1/ + 1/ = 1 Then for all r 2 ( B)X r r/2 + BX) r/2 (X 1 where δ(x, y) = ( y, y r/4 X BXx, x r/4) 2 inf δ(x, y), Proof Let x, y H, such that x = y = 1 By the Schwarz ineuality in the Hilbert sace (H;, ), we have ( B)Xx, y r = 1/2 ( 1/2 B 1/2 ) 1/2 1/2 Xx, y r = ( 1/2 B 1/2 ) 1/2 1/2 Xx, 1/2 y r ( 1/2 B 1/2 ) 1/2 1/2 Xx r 1/2 y r = ( 1/2 B 1/2 ) 1/2 1/2 Xx, ( 1/2 B 1/2 ) 1/2 1/2 Xx r/2 1/2 y, 1/2 y r/2 = y, y r/2 X BXx, x r/2 Now, by Young s ineuality and (2) we have r/2 y, y X BXx, x r/2 1 r/2 y, y + 1 X BXx, x r/2 1 ( y, r/4 y X BXx, x r/4) 2 and by (5) we have 1 r/2 y, y + 1 X BXx, x r/2 1 ( y, r/4 y X BXx, x r/4) 2 1 r/2 y, y + 1 (X BX) r/2 x, x 1 ( y, r/4 y X BXx, x r/4) 2 Now, the result follows by taking the suremum over all unit vectors x, y H Corollary 32 Let, B, X B(H) be such that, B > 0 Then for all r 1 2 ( B)X r r + (X BX) r inf where δ(x, y) = ( y, y r/2 X BXx, x r/2) 2 If in relation (11) we set X = I we obtain the following corollary δ(x, y), (11)

6 Sheikhhosseini / Filomat 30:8 (2016), Corollary 33 Let, B B(H) be ositive definite oerators Then for all r 1 2 B r r + B r inf where δ(x, y) = ( y, y r/2 Bx, x r/2) 2 δ(x, y), Proosition 34 Let, B, X B(H) such that, B > 0 and > 1 such that 1/ + 1/ = 1 Then for all r 2/ ( X BX ) r/2 r/2 + BX) r/2 (X 1 where δ(x, y) = ( y, y r/4 X BXx, x r/4) 2 inf δ(x, y), (12) Proof Let x, y H with x = y = 1 By the ineuality (2), we have r/2 y, y X BXx, x r/2 1 r/2 y, y + 1 X BXx, x r/2 1 ( y, r/4 y X BXx, x r/4) 2 and by (5) we have 1 r/2 y, y + 1 X BXx, x r/2 1 ( y, r/4 y X BXx, x r/4) 2 1 r/2 y, y + 1 (X BX) r/2 x, x 1 ( y, r/4 y X BXx, x r/4) 2 Now, the result follows by taking the suremum over all unit vectors x, y H If in relation (12), we set X = I and r = 2, then we obtain the following corollary Corollary 35 Let, B B(H) be ositive definite oerators and > 1 such that 1/ + 1/ = 1 Then B + B 1 inf δ(x, y), (13) where δ(x, y) = ( y, y /2 Bx, x /2) 2 Remark 36 Note that, if we set = = 2 in (13), then we have B 1 2 ( 2 + B 2 ) 1 2 inf δ(x, y), (14) where δ(x, y) = ( y, y Bx, x )2 Notice that (14) is an oerator norm version for arithmetic-geometric mean and moreover if, W() and W(B) are searated, then inf δ(x, y) > 0 Examle 37 Let = = 2 and = dia (1, 2), B = dia (5, 6) in the ineuality (13) Then inf δ(x, y) = 9 > 0 and hence, 12 = B 1 2 ( 2 + B 2 ) 1 2 inf δ(x, y) = 31 2 Whereas, if we set this values in the ineuality (4), with the sectral norm, then we obtain Thus, in this case, we have 12 = B + B = 20 B = B < + B 1 inf δ(x, y) < + B cknowledgement: The author would like to thank the anonymous referee for careful reading and the helful comments imroving this aer

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