SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES. 1. Introduction

Transcription

1 SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES SEVER S. DRAGOMIR 1 AND MOHAMMAD SAL MOSLEHIAN Abstract. An oerator T is called (α, β)-normal (0 α 1 β) if α T T T T β T T. In this aer, we establish various ineualities between the oerator norm and its numerical radius of (α, β)-normal oerators in Hilbert saces. For this urose, we emloy some classical ineualities for vectors in inner roduct saces. 1. Introduction An oerator T acting on a Hilbert sace (H;, ) is called (α, β)-normal (0 α 1 β) if Then α T T T T β T T. α T T x, x T T x, x β T T x, x, (1.1) α T x T x β T x, for all x H. If T is invertible, then so is the bounded oerator T T 1. Hence T T 1 is bounded below and so T is (α, β)-normal for some α and β. Normal and hyonormal oerators are trivially (α, β)-normal for some aroriate values of α and β. There are however oerators which are neither normal nor 3 hyonormal. The following examle of an (α, β)-normal with α = 5 and 3+ β = 5 is due to M. Mirzavaziri [ 1 ] B(C ). 000 Mathematics Subject Classification. 47A1. Key words and hrases. Numerical radius, Bounded linear oerator, Hilbert sace, (α, β)- normal oerator. 1

2 S.S. DRAGOMIR, M.S. MOSLEHIAN Let (H;, ) be a comlex Hilbert sace. The numerical radius w(t ) of an oerator T on H is given by (1.) w(t ) = su{ T x, x, x = 1}. Obviously, by (1.), for any x H one has (1.3) T x, x w(t ) x. It is well known that w( ) is a norm on the Banach algebra B(H) of all bounded linear oerators. Moreover, we have w(t ) T w(t ) (T B(H)). For other results and historical comments on the numerical radius see [10]. In this aer, we establish various ineualities between the oerator norm and its numerical radius of (α, β)-normal oerators in Hilbert saces. For this urose, we emloy some classical ineualities for vectors in inner roduct saces due to Buzano, Dunkl Williams, Dragomir Sándor, Goldstein Ryff Clarke and Dragomir.. Ineualities Involving Numerical Radius In this section we study some ineualities concerning the numerical radius and norm of (α, β)-normal oerators. Our first result reads as follows, see also [6]: Theorem.1. Let T B(H) be an (α, β)-normal oerator. Then β r w(t ) + r β r βt T, if r 1, (.1) (α r + β r ) T β r w(t ) + βt T, if r < 1. Proof. We use the following ineuality for vectors in inner roduct saces due to Goldstein, Ryff and Clarke [9]: (.) a r + b r a r b r Re a, b a b rovided r R and a, b H with a b. r a r a b if r 1, b r a b if r < 1, Suose that r 1. Let x H with x = 1. Noting to (1.1) and alying (.) for the choices a = βt x, b = T x we get (.3) βt x r + T x r βt x r 1 T x r 1 Re βt x, T x r βt x r βt x T x

3 SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES 3 for any x H, x = 1 and r 1. Using (1.1) and (.3) we get (.4) (α r + β r ) T x r β r T x r 1 T x r 1 T x, x + r β r T x r βt x T x. Taking the suremum in (.4) over x H, x = 1, we deduce (α r + β r ) T r β r T r T r 1 w(t ) + r β r T r βt T, which is the first ineuality in (.1). If r < 1, then one can similarly rove the second ineuality in (.1). Theorem.. Let T B(H) be an (α, β)-normal oerator. Then (.5) w(t ) 1 [ β T + w(t ) ]. Proof. The following ineuality is known in the literature as the Buzano ineuality [1]: (.6) a, e e, b 1 ( a b + a, b ), for any a, b, e in H with e = 1. Let x H with x = 1. Put e = x, a = T x, b = T x in (.6) to get T x, x x, T x 1 ( T x T x + T x, T x ) Taking the suremum over x H, x = 1, we obtain (.5). 1 (β T x + T x, x ). Theorem.3. Let T B(H) be an (α, β)-normal oerator and λ C. Then (.7) α T w(t ) + β T λt (1 + λ α). Proof. Using the Dunkl Williams ineuality [8] 1 ( a + b ) a a b b a b (a, b H \ {0}) we get Re a, b a b = a a b b 4 a b ( a + b ) (a, b H \ {0}) a b a b a b ( a + b ) + a, b (a, b H \ {0}). Put a = T x and b = λt to get T x T x T x, x + T x T x T x λt x ( T x + λ T x )

4 4 S.S. DRAGOMIR, M.S. MOSLEHIAN so that (.8) α T x T x, x + β T x T x λt x ( T x + λ α T x ) T x, x + β (T λt )x (1 + λ α). Taking the suremum in (.8) over x H, x = 1, we get the desired result (.7). Theorem.4. Let T B(H) be an (α, β)-normal oerator and λ C\{0}. Then [ ( ) ] 1 (.9) α λ + β T 4 w(t ). Proof. We aly the following reverse of the uadratic Schwarz ineuality obtained by Dragomir in [5] (.10) (0 ) a b a, b 1 λ a a λb rovided a, b H and λ C\{0}. Set a = T x, b = T x in (.10), to get α T x 4 T x, T x + 1 λ T x T x λt x (.11) [ α T x, x + 1 λ T x (1 + λ β) T x ( ) ] 1 λ + β T x 4 T x, x. Taking the suremum in (.11) over x H, x = 1, we get the desired result (.9). Theorem.5. Let T B(H) be an (α, β)-normal oerator, r 0 and λ C {0}. If λt T r and r λ inf{ T x : x = 1}, then (.1) α T 4 w(t ) + r λ T. Proof. We use the following reverse of the Schwarz ineuality obtained by Dragomir in [3] (see also [4,. 0]): (.13) (0 ) y a [Re y, a ] r y, rovided y a r a.

5 SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES 5 By the assumtion of theorem T x λt x r λt x. Setting a = λt x and y = T x, with x = 1 in (.13) we get T x λt x [Re T x, λt x ] + r T x (.14) α λ T x 4 λ T x, x + r T x. Taking the suremum in (.14) over x H, x = 1, we get the desired result (.1). Finally, the following result that is less restrictive for the involved arameters r and λ (from the above theorem) may be stated as well: Theorem.6. Let T B(H) be an (α, β)-normal oerator, r 0 and λ C {0}. If λt T r, then (.15) α T w(t ) + r λ. Proof. We use the following reverse of the Schwarz ineuality obtained by Dragomir in [] (see also [4,. 7]): (.16) (0 ) y a Re y, a 1 r, rovided y a r. Setting a = λt x and y = T x, with x = 1 in (.16) we get which gives T x λt x T x, λt x + 1 r α T x T x, x + 1 r. Now, taking the suremum over x = 1 in this ineuality, we get the desired result (.15) 3. Ineualities Involving Norms Our first result in this section reads as follows. Theorem 3.1. Let T B(H) be an (α, β)-normal oerator. If, then (3.1) (1 + α ) T 1 ( T + T + T T ). In general, for each T B(H) and we have (3.) T T + T T / 1 4 ( T + T + T T ).

6 6 S.S. DRAGOMIR, M.S. MOSLEHIAN Proof. We use the following ineuality obtained by Dragomir and Sándor in [7] (see also [11,. 544]): (3.3) a + b + a b ( a + b ) for any a, b H and. Now, if we choose a = T x, b = T x in (3.3), then we get (3.4) T x + T x + T x T x ( T x + T x ), (3.5) T x + T x + T x T x ( T x + α T x ), for any x H, x = 1. Taking the suremum in (3.5) over x H, x = 1, we get the desired result (3.1). Now for the general case T B(H), observe that (3.6) T x + T x = ( T x ) + ( T x ) and by alying the elementary ineuality: a + b ( ) a + b, a, b 0 and 1 we have (3.7) ( T x ) + ( T x ) 1 ( T x + T x ) = 1 [ T x, T x + T x, T x ] = 1 [ (T T + T T )x, x ]. Combining (3.4) with (3.7) and (3.6) we get ( 1 (3.8) 4 [ T x T x + T x + T x ] T T + T T ) / x, x for any x H, x = 1. Taking the suremum over x H, x = 1, and taking into account that ( T T + T T ) w = T T + T T, we deduce the desired result (3.). Theorem 3.. Let T B(H) be an (α, β)-normal oerator. λ, µ C, then If (1, ) and (3.9) [( λ + β µ ) + max{ λ µ β, α µ λ }] T λt + µt + λt µt.

7 SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES 7 Proof. We use the following ineuality obtained by Dragomir and Sándor in [7] (see also [11,. 544]) (3.10) ( a + b ) + a b a + b + a b, for any a, b H and (1, ). Put a = λt x, b = µt x in (3.10) to obtain ( λt x + µt x ) + λt x µt x λt x + µt x + λt x µt x, (3.11) ( λ + µ α) T x + (max{ λ µ β, α µ λ }) T x λt x + µt x + λt x µt x, for any x H, x = 1. Taking the suremum in (3.11) over x H, x = 1, we get the desired result (3.9). 4. Other Ineualities for General Oerators Finally, we resent two results holding in the general case of bounded linear oerators in Hilbert saces: Theorem 4.1. Let T, S B(H). Then (4.1) T T + S S T + S w(s T ), in articular, (4.) T T + λ T T T + λt λ w(t ). Proof. We have (4.3) T x ± Sx = T x ± Re T x, Sx + Sx for any x H. Hence T x + Sx (T T + S S)x, x + (S T )x, x. Taking the suremum over x H, x = 1, we get T + S w(t T + S S) + w(s T ) = T T + S S + w(s T ),

8 8 S.S. DRAGOMIR, M.S. MOSLEHIAN It follows from (4.3) that (T T + S S)x, x = T x + Sx = Re T x, Sx + T x Sx T x, Sx + T x Sx. Relacing S by S in the later euality and taking the suremum over x H, x = 1, we get T T + S S = w(t T + S S) w(s T ) + T + S. The desired ineuality (4.1) follows from (1.) and (1.3). The last ineuality can be obtained by utting S = λt in (4.1). Theorem 4.. Let T, S B(H), and, > 0. Then (4.4) w(t + S) In articular, w(t + λt ) [ ( + w T T + + 1/ S S)]. [ ( + w Proof. Utilizing the following elementary ineuality T T + + )] 1/ λ T T. (a + b) + a + b, holding for any real numbers a, b and for the ositive numbers,, we get (T + S)x, x ( T x, x + Sx, x ) ( + ) + ( ) T x, x Sx, x ( + ) + ( ) T x ( + ) + Sx ( T T S ( + ) x, x S + ( + ) x, x Putting S = λt in (4.4), we get the last desired ineuality. T T + + ) S S x, x.

9 SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES 9 References [1] M.L. Buzano, Generalizzatione della disiguaglianza di Cauchy-Schwaz (Italian). Rend. Sem. Mat. Univ. e Politech. Torino, 31 (1971/73), (1974). [] S.S. Dragomir, New reverses of Schwarz, triangle and Bessel ineualities in inner roduct saces, Austral. J. Math. Anal. & Alics., 1(1) (004), Article 1. [3] S.S. Dragomir, Reverses of Schwarz, triangle and Bessel ineualities in inner roduct saces, J. Ineual. Pure & Al. Math., 5(3) (004), Article 76. [4] S.S. Dragomir, Advances in Ineualities of the Schwarz, Grüss and Bessel Tye in Inner Product Saces, Nova Science Publishers, New York, 005. [5] S.S. Dragomir, A otourri of Schwarz related ineualities in inner roduct saces (II), J. Ine. Pure Al. Math., 7(1) (006), Art. 14. [htt://jiam.vu.edu.au/article.h?sid=619]. [6] S.S. Dragomir, Ineualities for the norm and the numerical radius of linear oerators in Hilbert saces, Demonstratio Mathematica (Poland), XL(007), No., Prerint available on line at RGMIA Res. Re. Coll. 8(005), Sulement, Article 10, [ONLINE htt://rgmia.vu.edu.au/v8(e).html ]. [7] S.S. Dragomir and J. Sándor, Some ineualities in rehilbertian saces, Studia Univ. Babeş- Bolyai - Mathematica, 3(1) (1987), [8] C.F. Dunkl and K.S. Williams, A simle norm ineuality, Amer. Math. Monthly, 71(1) (1964), [9] A. Goldstein, J.V. Ryff and L.E. Clarke, Problem 5473, Amer. Math. Monthly, 75(3) (1968), 309. [10] K.E. Gustafson and D.K.M. Rao, Numerical Range, Sringer-Verlag, New York, [11] D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Classical and New Ineualities in Analysis, Kluwer Academic Publishers, Dordrecht, School of Comuter Science and Mathematics, Victoria University, P. O. Box 1448, Melbourne City, Victoria 8001, Australia. address: sever.dragomir@vu.edu.au Deartment of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad 91775, Iran; Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University, Iran. address: moslehian@ferdowsi.um.ac.ir and moslehian@ams.org