Further refinements of generalized numerical radius inequalities for Hilbert space operators
Monire Hajmohamadi, Rahmatollah Lashkaripour, Mojtaba Bakherad
aa r X i v : . [ m a t h . F A ] M a y FURTHER REFINEMENTS OF GENERALIZED NUMERICALRADIUS INEQUALITIES FOR HILBERT SPACE OPERATORS
MONIRE HAJMOHAMADI , RAHMATOLLAH LASHKARIPOUR , MOJTABA BAKHERAD Abstract.
In this paper, we show some refinements of generalized numerical radiusinequalities involving the Young and Heinz inequalities. In particular, we present w pp ( A ∗ T B , ..., A ∗ n T n B n ) ≤ n − r r (cid:13)(cid:13)(cid:13) n X i =1 [ B ∗ i f ( | T i | ) B i ] rp + [ A ∗ i g ( | T ∗ i | ) A i ] rp (cid:13)(cid:13)(cid:13) r − inf k x k =1 η ( x ) , where T i , A i , B i ∈ B ( H ) (1 ≤ i ≤ n ), f and g are nonnegative continuous functionson [0 , ∞ ) satisfying f ( t ) g ( t ) = t for all t ∈ [0 , ∞ ), p, r ≥ N ∈ N and η ( x ) = 12 n X i =1 N X j =1 (cid:16) j q h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i j − − k j h ( B ∗ i f ( | T i | ) B i ) p x, x i k j − j q h ( B ∗ i f ( | T i | ) B i ) p x, x i k j +1 h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i j − − k j − (cid:17) . Introduction
Let B ( H ) denote the C ∗ -algebra of all bounded linear operators on a complex Hilbertspace H with an inner product h · , · i and the corresponding norm k · k . In the casewhen dim H = n , we identify B ( H ) with the matrix algebra M n of all n × n matriceswith entries in the complex field. The numerical radius of T ∈ B ( H ) is defined by w ( T ) := sup {| h T x, x i | : x ∈ H , k x k = 1 } . It is well known that w ( · ) defines a norm on B ( H ), which is equivalent to the usualoperator norm k · k . In fact, for any T ∈ B ( H ), k T k ≤ w ( T ) ≤ k T k ; see [6].The quantity w ( T ) is useful in studying perturbation, convergence and approximationproblems as well as interactive method, etc. For more information see [1, 2, 4, 5, 7, 8,13, 19] and references therein.The classical Young inequality says that if 0 ≤ ν ≤
1, then a ν b − ν ≤ νa + (1 − Mathematics Subject Classification.
Primary 47A12, Secondary 47A63, 47A30 .
Key words and phrases.
Euclidean operator radius; Heinz means; Numerical radius; positive oper-ator; Young inequality. ν ) b ( a, b > a ν b − ν ≤ νa + (1 − ν ) b − r ( a − b ) , (1.1)where r = min { ν, − ν } .Recently, Sababheh and Choi in [15] obtained a refinement of the Young inequality a ν b − ν ≤ νa + (1 − ν ) b − S N ( ν ) , (1.2)in which S N ( ν ) := N X j =1 (cid:18) ( − r j j − ν + ( − r j +1 (cid:20) r j + 12 (cid:21)(cid:19) (cid:16) j p b j − − k j a k j − j p a k j +1 b j − − k j − (cid:17) , where N ∈ N , r j = [2 j ν ] and k j = [2 j − ν ]. Here [ x ] is the greatest integer less than orequal to x . When N = 1, inequality (1.2) reduces to (1.1).It follows from νa + (1 − ν ) b ≤ ( νa r + (1 − ν ) b r ) r ( r ≥
1) and inequality (1.1) that a ν b − ν ≤ ( νa r + (1 − ν ) b r ) r − S N ( ν ) . In particular, for ν = we get a b ≤ ( 12 ) r ( a r + b r ) r − N X j =1 (cid:16) j p b j − − k j a k j − j p a k j +1 b j − − k j − (cid:17) . If N = 1, then we reach to inequality (2 .
1) in [12] as follows: a b ≤ ( 12 ) r ( a r + b r ) r −
12 ( a − b ) . Let T i ∈ B ( H ) (1 ≤ i ≤ n ). The Euclidean operator radius of T , ..., T n is defined in[14] by w e ( T , ..., T n ) := sup k x k =1 n X i =1 |h T i x, x i| ! . In [16], the functional w p of operators T , ..., T n for p ≥ w p ( T , ..., T n ) := sup k x k =1 n X i =1 |h T i x, x i| p ! p . Let T , ..., T n ∈ B ( H ). Recently, Sheikhhosseini et al. in [18] showed w pp ( A ∗ T B , ..., A ∗ n T n B n ) ≤ n − r r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 [ B ∗ i f ( | T i | ) B i ] rp + [ A ∗ i g ( | T ∗ i | ) A i ] rp ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r − inf k x k =1 ζ ( x ) , (1.3) URTHER REFINEMENTS OF GENERALIZED NUMERICAL RADIUS INEQUALITIES 3 where ζ ( x ) = P ni =1 (cid:16) h [ B ∗ i f ( | T i | ) B i ] p x, x i − h [ A ∗ i f ( | T ∗ i | ) A i ] p x, x i (cid:17) . They alsopresented the following inequality w p ( T , ..., T n ) ≤ " n X i =1 (cid:18)(cid:13)(cid:13) | T i | α + | T ∗ i | − α ) (cid:13)(cid:13) − k x k =1 ζ i ( x ) (cid:19) p p , (1.4)in which 0 ≤ α ≤ p ≥ ζ i ( x ) = ( h| T i | α x, x i − h| T ∗ i | − α ) x, x i ) .In the same paper, they showed w pp ( T , ..., T n ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 ( | T i | αp + | T ∗ i | − α ) p ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − inf k x k =1 ζ ( x ) , (1.5)where ζ ( x ) = P ni =1 ( h| T i | αp x, x i − h| T ∗ i | − α ) p x, x i ) .Moreover, they established the inequalities w pp ( T , ..., T n ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 α | T i | p + (1 − α ) | T ∗ i | p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − inf k x k =1 ζ ( x ) , (1.6)and w rp ( | T | , ..., | T n | ) w rq ( | T ∗ | , ..., | T ∗ n | ) ≤ rp k n X i =1 | T i | p k + rq k n X i =1 | T ∗ i | q k − inf k x k = k y k =1 δ ( x, y ) , (1.7)where ζ ( x ) = min { α, − α } P ni =1 ( h| T i | p x, x i − h| T ∗ i | p x, x i ) and δ ( x, y ) = rp (cid:16) vuut n X i =1 h| T i | x, y i p − vuut n X i =1 h| T ∗ i | x, y i q (cid:17) . Assume that X ∈ B ( H ). The mixed Heinz means are defined by H α ( A, B ) = A α XB − α + A − α XB α , in which 0 ≤ α ≤ A, B ≥
0, see [10]. In [17], the authors showed that w r ( A α XB − α ) ≤ k X k r k αA r + (1 − α ) B r k , (1.8)where A, B, X ∈ B ( H ) such that A, B are positive, r ≥ ≤ α ≤ w r ( H α ( A, B )) ≤ k X k r k A r + B r k . (1.9)In this present paper, we refine inequalities (1.3)-(1.9). We also find an upper boundfor the functional w p . M. HAJMOHAMADI, R. LASHKARIPOUR, M. BAKHERAD main results To prove our numerical radius inequalities, we need several known lemmas. Thefirst lemma is a simple result of the classical Jensen, Young and a genaralized mixedCauchy-Schwarz inequalities [9, 11].
Lemma 2.1.
Let a, b ≥ , ≤ ν ≤ and r = 0 . Then ( a ) a ν b − ν ≤ νa + (1 − ν ) b ≤ ( νa r + (1 − ν ) b r ) r for r ≥ . ( b ) If T ∈ B ( H ) and x, y ∈ H be any vectors, then | h T x, y i | ≤ h| T | ν x, x ih| T ∗ | − ν ) y, y i . ( c ) If f , g are nonnegative continuous functions on [0 , ∞ ) which are satisfying therelation f ( t ) g ( t ) = t ( t ∈ [0 , ∞ )) , then | h T x, y i |≤k f ( | T | ) x kk g ( | T ∗ | ) x k for all x, y ∈ H . Lemma 2.2. ( McCarty inequality [11]) . Let T ∈ B ( H ) , T ≥ and x ∈ H be a unitvector. Then ( a ) h T x, x i r ≤ h T r x, x i for r ≥ b ) h T r x, x i ≤ h T x, x i r for < r ≤ . Now, by using inequality (1.2) we get the first result.
Theorem 2.3.
Let
A, B, X ∈ B ( H ) such that A, B are positive, r ≥ and ≤ ν ≤ .Then w r ( A ν XB − ν ) ≤ k X k r (cid:20) k νA r + (1 − ν ) B r k − inf k x k =1 η ( x ) (cid:21) , (2.1) where η ( x ) = N X j =1 (cid:18) ( − r j j − ν + ( − r j +1 (cid:20) r j + 12 (cid:21)(cid:19) × (cid:18) j q h B r x, x i j − − k j h A r x, x i k j − j q h A r x, x i k j +1 h B r x, x i j − − k j − (cid:19) . URTHER REFINEMENTS OF GENERALIZED NUMERICAL RADIUS INEQUALITIES 5
Proof.
Let x ∈ H be unit vector. Then |h A ν XB − ν x, x i| r = |h XB − ν x, A ν x i| r ≤ k X k r k B − ν x k r k A ν x k r = k X k r h B − ν ) x, x i r h A ν x, x i r ≤ k X k r h A r x, x i ν h B r x, x i − ν ( by Lemma 2 . ≤ k X k r [ ν h A r x, x i + (1 − ν ) h B r x, x i ] − k X k r N X j =1 (cid:18) ( − r j j − ν + ( − r j +1 (cid:20) r j + 12 (cid:21)(cid:19) ×× (cid:18) j q h B r x, x i j − − k j h A r x, x i k j − j q h A r x, x i k j +1 h B r x, x i j − − k j − (cid:19) ( by inequality (1.2)) . Taking the supremum over x ∈ H with k x k = 1 in the above inequality we deducethe desired inequality. (cid:3) Remark . Let N = 1 in inequality (2.1). Then w r ( A ν XB − ν ) ≤ k X k r (cid:20) k νA r + (1 − ν ) B r k − inf k x k =1 η ( x ) (cid:21) , (2.2)in which η ( x ) = r (cid:16) h A r x, x i − h B r x, x i (cid:17) and r = min { ν, − ν } . Hence inequality(2.2) is a refinement of inequality (1.8).Using Theorem 2.3 we can find an upper bound for Heinz means of matrices that itis a refinement of (1.9). Theorem 2.5.
Suppose
A, B, X ∈ B ( H ) such that A, B are positive. Then w r ( H ν ( A, B )) ≤ k X k r (cid:20) k A r + B r k −
12 inf ζ ( x ) (cid:21) , where r ≥ , ≤ ν ≤ , n ∈ N and ζ ( x ) = N X j =1 (cid:18) ( − r j j − + ( − r j +1 (cid:20) r j + 12 (cid:21)(cid:19) × (cid:18) j q h B r x, x i j − − k j h A r x, x i k j − j q h A r x, x i k j +1 h B r x, x i j − − k j − (cid:19) . M. HAJMOHAMADI, R. LASHKARIPOUR, M. BAKHERAD
Proof.
For unit vector x ∈ H , we have (cid:12)(cid:12)(cid:12)D A ν XB − ν + A − ν XB ν x, x E(cid:12)(cid:12)(cid:12) r ≤ (cid:18) |h A ν XB − ν x, x i| + |h A − ν XB ν x, x i| (cid:19) r ≤ |h A ν XB − ν x, x i| r + |h A − ν XB ν x, x i| r ≤ k X k r h νA r + (1 − ν ) B r x, x i ] − k X k r N X j =1 (cid:18) ( − r j j − ν + ( − r j +1 (cid:20) r j + 12 (cid:21)(cid:19) × (cid:16) j p b j − − k j a k j − j p a k j +1 b j − − k j − (cid:17) + k X k r h (1 − ν ) A r + νB r x, x i ] − k X k r N X j =1 (cid:18) ( − r j j − ν + ( − r j +1 (cid:20) r j + 12 (cid:21)(cid:19) × (cid:18) j q h B r x, x i j − − k j h A r x, x i k j − j q h A r x, x i k j +1 h B r x, x i j − − k j − (cid:19) = k X k r (cid:20)(cid:28) A r + B r x, x (cid:29)(cid:21) − k X k r N X j =1 (cid:18) ( − r j j − + ( − r j +1 (cid:20) r j + 12 (cid:21)(cid:19) × (cid:18) j q h B r x, x i j − − k j h A r x, x i k j − j q h A r x, x i k j +1 h B r x, x i j − − k j − (cid:19) . If we take the supremum over x ∈ H with k x k = 1, then we deduce the desiredinequality. (cid:3) In the next theorem we show a refinement of inequality (1.3).
Theorem 2.6.
Let T i , A i , B i ∈ B ( H ) (1 ≤ i ≤ n ) and let f and g be nonnegativecontinuous functions on [0 , ∞ ) satisfying f ( t ) g ( t ) = t for all t ∈ [0 , ∞ ) . Then w pp ( A ∗ T B , ..., A ∗ n T n B n ) ≤ n − r r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 [ B ∗ i f ( | T i | ) B i ] rp + [ A ∗ i g ( | T ∗ i | ) A i ] rp (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r − inf k x k =1 η ( x ) , (2.3) where p, r ≥ , N ∈ N and η ( x ) = 12 n X i =1 N X j =1 (cid:16) j q h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i j − − k j h ( B ∗ i f ( | T i | ) B i ) p x, x i k j − j q h ( B ∗ i f ( | T i | ) B i ) p x, x i k j +1 h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i j − − k j − (cid:17) . URTHER REFINEMENTS OF GENERALIZED NUMERICAL RADIUS INEQUALITIES 7
Proof.
Let x ∈ H be any unit vector. Then n X i =1 |h A ∗ i T i B i x, x i| p = n X i =1 |h T i B i x, A i x i| p ≤ n X i =1 k f ( | T i | ) B i x k p k g ( | T ∗ i | ) A i x k p (by Lemma (2.1), (c))= n X i =1 h f ( | T i | ) B i x, f ( | T i | ) B i x i p h g ( | T ∗ i | ) A i x, g ( | T ∗ i | ) A i x i p = n X i =1 h B ∗ i f ( | T i | ) B i x, x i p h A ∗ i g ( | T ∗ i | ) A i x, x i p ≤ n X i =1 h ( B ∗ i f ( | T i | ) B i ) p x, x i h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i (by Lemma (2.2), (a)) ≤ n X i =1 "(cid:18) h ( B ∗ i f ( | T i | ) B i ) pr x, x i + 12 h ( A ∗ i g ( | T ∗ i | ) A i ) pr x, x i (cid:19) r (by (1.2)) − n X i =1 N X j =1 (cid:16) j q h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i j − − k j h ( B ∗ i f ( | T i | ) B i ) p x, x i k j − j q h ( B ∗ i f ( | T i | ) B i ) p x, x i k j +1 h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i j − − k j − (cid:17) ≤ n − r r * n X i =1 (cid:16)(cid:2) B ∗ i f ( | T i | ) B i (cid:3) rp + (cid:2) A ∗ i g ( | T ∗ i | ) A i (cid:3) rp (cid:17)! x, x + r − n X i =1 N X j =1 (cid:16) j q h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i j − − k j h ( B ∗ i f ( | T i | ) B i ) p x, x i k j − j q h ( B ∗ i f ( | T i | ) B i ) p x, x i k j +1 h ( A ∗ i g ( | T ∗ i | ) A i ) p x, x i j − − k j − (cid:17) By taking supremum on unit vector x in H we reach the desired inequality. (cid:3) Corollary 2.7.
Let A i , B i ∈ B ( H ) (1 ≤ i ≤ n ) . Then for r, p ≥ we have w pp ( A ∗ B , ..., A ∗ n B n ) ≤ n − r r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 (cid:0) | B i | rp + | A i | rp (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r − inf k x k =1 η ( x ) , where η ( x ) = 12 n X i =1 N X j =1 (cid:16) j q h ( | A i | p x, x i j − − k j h ( | B i | p x, x i k j − j q h| B i | p x, x i k j +1 h| A i | p x, x i j − − k j − (cid:17) . M. HAJMOHAMADI, R. LASHKARIPOUR, M. BAKHERAD
Proof.
Choosing f ( t ) = g ( t ) = t and T i = I for i = 1 , , ..., n in Theorem 2.6, we getthe desired result. (cid:3) Corollary 2.8.
Let T i ∈ B ( H ) (1 ≤ i ≤ n ) , let f and g be nonnegative continuousfunctions on [0 , ∞ ) such that f ( t ) g ( t ) = t for all t ∈ [0 , ∞ ) and r, p ≥ . Then w pp ( T , ..., T n ) ≤ n − r r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 (cid:0) | f rp ( | T i | ) + g rp ( | T ∗ i | ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r − inf k x k =1 η ( x ) , (2.4) where η ( x ) = 12 n X i =1 N X j =1 (cid:16) j q h g p ( | T ∗ i | ) x, x i j − − k j h f p ( | T i | ) x, x i k j − j q h f p ( | T i | ) x, x i k j +1 h g p ( | T ∗ i | ) x, x i j − − k j − (cid:17) . In particular, w pp ( T , ..., T n ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 ( | T i | αp + | T ∗ i | − α ) p ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − inf k x k =1 η ( x ) , (2.5) where ≤ α ≤ and η ( x ) = 12 n X i =1 N X j =1 (cid:16) j q h| T ∗ i | − α ) p x, x i j − − k j h| T i | αp x, x i k j −− j q h| T i | αp x, x i k j +1 h| T ∗ i | − α ) p x, x i j − − k j − (cid:17) . Proof.
Selecting A i = B i = I for i = 1 , , .., n in Theorem 2.6, we get the first result.Letting f ( t ) = t α , g ( t ) = t − α , r = 1 and B i = A i = I for i = 1 , , ..., n in inequality(2.4), we reach the second inequality. (cid:3) Remark . Note that inequality (2.5) is a refinement of inequality (1.5), since n X i =1 ( h| T i | αp x, x i − h| T ∗ i | − α ) p x, x i ) ≤ n X i =1 N X j =1 (cid:16) j q h| T ∗ i | − α ) p x, x i j − − k j h| T i | αp x, x i k j − j q h| T i | αp x, x i k j +1 h| T ∗ i | − α ) p x, x i j − − k j − (cid:17) . Now by letting n = 2, N = 1, T = B and T = C in Theorem 2.6, we obtain thefollowing consequence. Corollary 2.10.
Let
B, C ∈ B ( H ) . Then for all p ≥ and ≤ α ≤ w pp ( B, C ) ≤ (cid:13)(cid:13) | B | αp + | B ∗ | − α ) p + | C | αp + | C ∗ | − α ) p (cid:13)(cid:13) − inf k x k =1 η ( x ) , where η ( x ) = h ( h| B | αp x, x i − h| B ∗ | − α ) p x, x i ) + ( h| C | αp x, x i − h| C ∗ | − α ) p x, x i ) i . URTHER REFINEMENTS OF GENERALIZED NUMERICAL RADIUS INEQUALITIES 9
Theorem 2.11.
Let T i ∈ B ( H ) (1 ≤ i ≤ n ) . Then w p ( T , ..., T n ) ≤ " n X i = (cid:18)(cid:13)(cid:13) | T i | α + | T ∗ i | − α ) (cid:13)(cid:13) − k x k =1 η i ( x ) (cid:19) p p , (2.6) where p ≥ , ≤ α ≤ and η i ( x ) = 12 N X j =1 (cid:16) j q h| T ∗ i | − α ) p x, x i j − − k j h| T i | αp x, x i k j − j q h| T i | αp x, x i k j +1 h| T ∗ i | − α ) p x, x i j − − k j − (cid:17) . Proof.
By using of Lemma 2.1 and inequality (1.2), for any unit vector x ∈ H we have n X i =1 |h T i x, x i| p ≤ n X i =1 ( h| T i | α x, x i h| T ∗ i | − α ) x, x i ) p (by Lemma (2.1), (b)) ≤ p n X i =1 h h| T i | α x, x i + h| T ∗ i | − α ) x, x i−− N X j =1 (cid:16) j q h| T ∗ i | − α ) p x, x i j − − k j h| T i | αp x, x i k j − j q h| T i | αp x, x i k j +1 h| T ∗ i | − α ) p x, x i j − − k j − (cid:17) i p (by (1.2))= 12 p n X i =1 h h| T i | α + | T ∗ i | − α ) x, x i− N X j =1 (cid:16) j q h| T ∗ i | − α ) p x, x i j − − k j h| T i | αp x, x i k j −− j q h| T i | αp x, x i k j +1 h| T ∗ i | − α ) p x, x i j − − k j − (cid:17) i p . Thus n X i =1 |h T i x, x i| p ! p ≤ h n X i =1 (cid:16) h| T i | α + | T ∗ i | − α ) x, x i−− N X j =1 (cid:16) j q h| T ∗ i | − α ) p x, x i j − − k j h| T i | αp x, x i k j − j q h| T i | αp x, x i k j +1 h| T ∗ i | − α ) p x, x i j − − k j − (cid:17) (cid:17) p i p = 12 " n X i =1 (cid:16) h| T i | α + | T ∗ i | − α ) x, x i − η i ( x ) (cid:17) p p . Now, by taking the supremum over all unit vector x ∈ H we get the desired result. (cid:3) Remark . If N = 1 in inequality (2.6), then we reach to inequality (1.4), it followsfrom
12 ( h| T i | α x, x i − h| T ∗ i | − α ) x, x i ) ≤ N X j =1 (cid:16) j q h| T ∗ i | − α ) p x, x i j − − k j h| T i | αp x, x i k j − j q h| T i | αp x, x i k j +1 h| T ∗ i | − α ) p x, x i j − − k j − (cid:17) , that inequality (2.6) is a refinement of inequality (1.4). Theorem 2.13.
Let T i ∈ B ( H ) (1 ≤ i ≤ n ) . Then for ≤ α ≤ and p ≥ w pp ( T , ..., T n ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 ( α | T i | p + (1 − α ) | T ∗ i | p ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − inf k x k =1 η ( x ) , (2.7) where η ( x ) = n X i =1 (cid:16) N X j =1 (cid:16) ( − r j j − α + ( − r j +1 h r j + 12 i(cid:17) × (cid:16) j q h| T ∗ i | p x, x i j − − k j h| T i | p x, x i k j − j q h| T i | p x, x i k j +1 h| T ∗ i | p x, x i j − − k j − (cid:17) URTHER REFINEMENTS OF GENERALIZED NUMERICAL RADIUS INEQUALITIES 11
Proof.
For every unit vector x ∈ H we have n X i =1 |h T i x, x i| p = n X i =1 ( |h T i x, x i| ) p ≤ n X i =1 ( h| T i | α x, x ih| T ∗ i | − α ) x, x i ) p (by Lemma 2 . , ( b )) ≤ n X i =1 ( h| T i | px, x i α h| T ∗ i | p x, x i − α ) (by Lemma 2 . , ( b )) ≤ n X i =1 ( α h| T i | p x, x i + (1 − α ) h| T ∗ i | p x, x i ) −− n X i =1 (cid:16) N X j =1 (cid:16) ( − r j j − α + ( − r j +1 h r j + 12 i(cid:17) × (cid:16) j q h| T ∗ i | p x, x i j − − k j h| T i | p x, x i k j − j q h| T i | p x, x i k j +1 h| T ∗ i | p x, x i j − − k j − (cid:17) (by (1.2)) ≤ n X i =1 h ( α | T i | p + (1 − α ) | T ∗ i | p ) x, x i − inf k x k =1 η ( x )= h n X i =1 ( α | T i | p + (1 − α ) | T ∗ i | p ) x, x i − inf k x k =1 η ( x ) . Now by taking supremum over unit vector x ∈ H we get. (cid:3) Remark . If we put N = 1 in inequality (2.7), then we get inequality (1.6). Henceinequality (2.7) is refinement of (1.6).In [16, Remark 3.10], the author showed w pp ( B, C ) ≤ k| B | p + | B ∗ | p + | C | p + | C ∗ | p k , (2.8)in which B, C ∈ B ( H ) and p ≥
2. In the following result we show a refinement of(2.8).
Corollary 2.15.
Let
B, C ∈ B ( H ) . Then for p ≥ , w pp ( B, C ) ≤ k| B | p + | B ∗ | p + | C | p + | C ∗ | p k − inf k x k =1 η ( x ) , (2.9) where η ( x ) = (cid:16) ( h| B | p x, x i − h| B ∗ | p x, x i ) − ( h| C | p x, x i − h| C ∗ | p x, x i ) (cid:17) . In particular, if A ∈ B ( H ) , then w ( A ) ≤ k A ∗ A + AA ∗ k . Proof.
If we take N = 1, n = 2, T = B , T = C , and α = in Theorem 2.13, we getthe first inequality.In particular case, let A = B + iC be the Cartesian decomposition of A . Then A ∗ A + AA ∗ = 2( B + C ), and inf k x k =1 η ( x ) = 0. Thus, for p = 2, inequality (2.9) can bewritten as w ( B, C ) ≤ k B + C k = 12 k A ∗ A + AA ∗ k . The desired inequality follows by noting that w ( B, C ) = sup k x k =1 {|h Bx, x i| + |h Cx, x i| } = sup k x k =1 |h Ax, x i| = w ( A ) . (cid:3) Theorem 2.16.
Let T i ∈ B ( H ) (1 ≤ i ≤ n ) , r ≥ , and p ≥ q ≥ with p + q = r .Then w rp ( | T | , ..., | T n | ) w rq ( | T ∗ | , ..., | T ∗ n | ) ≤ rp k n X i =1 | T i | p k + rq k n X i =1 | T ∗ i | q k − inf k x k = k y k =1 λ ( x, y ) , (2.10) where λ ( x, y ) = N X j =1 (cid:16) ( − r j j − ( rp ) + ( − r j +1 h r j + 12 i(cid:17)(cid:16) j vuut ( n X i =1 h| T ∗ i | x, y i q ) j − − k j ( n X i =1 h| T i | x, y i p ) k j − j vuut ( n X i =1 h| T i | x, y i p ) k j +1 ( n X i =1 h| T ∗ i | x, y i q ) j − − k j − (cid:17) . URTHER REFINEMENTS OF GENERALIZED NUMERICAL RADIUS INEQUALITIES 13
Proof.
Let x, y ∈ B ( H ) be unit vectors. Applying inequality (1.2), we get n X i =1 h| T i | x, y i p ! p n X i =1 h| T ∗ i | x, y i q ! q r ≤ rp n X i =1 h| T i | x, y i p + rq n X i =1 h| T ∗ i | x, y i q − N X j =1 (cid:16) ( − r j j − ( rp ) + ( − r j +1 h r j + 12 i(cid:17) × (cid:16) j vuut ( n X i =1 h| T ∗ i | x, y i q ) j − − k j ( n X i =1 h| T i | x, y i p ) k j − j vuut ( n X i =1 h| T i | x, y i p ) k j +1 ( n X i =1 h| T ∗ i | x, y i q ) j − − k j − (cid:17) (by (1.2)) ≤ rp n X i =1 h| T i | p x, y i + rq n X i =1 h| T ∗ i | q x, y i−− N X j =1 (cid:16) ( − r j j − ( rp ) + ( − r j +1 h r j + 12 i(cid:17) × (cid:16) j vuut ( n X i =1 h| T ∗ i | x, y i q ) j − − k j ( n X i =1 h| T i | x, y i p ) k j − j vuut ( n X i =1 h| T i | x, y i p ) k j +1 ( n X i =1 h| T ∗ i | x, y i q ) j − − k j − (cid:17) (by Lemma (2.2), (a))= rp D(cid:16) n X i =1 | T i | p (cid:17) x, y E + rq D(cid:16) n X i =1 | T ∗ i | q (cid:17) x, y E − N X j =1 (cid:16) ( − r j j − ( rp ) + ( − r j +1 h r j + 12 i(cid:17) × (cid:16) j vuut ( n X i =1 h| T ∗ i | x, y i q ) j − − k j ( n X i =1 h| T i | x, y i p ) k j − j vuut ( n X i =1 h| T i | x, y i p ) k j +1 ( n X i =1 h| T ∗ i | x, y i q ) j − − k j − (cid:17) By taking supremum on x, y ∈ H with k x k = k y k = 1, we get desired inequality. (cid:3) Remark . If we put N = 1 in inequality (2.10), then we reach to inequality (1.7).It follows from rp (cid:16) vuut n X i =1 h| T i | x, y i p − vuut n X i =1 h| T ∗ i | x, y i q (cid:17) ≤ N X j =1 (cid:16) ( − r j j − ( rp ) + ( − r j +1 h r j + 12 i(cid:17)(cid:16) j vuut ( n X i =1 h| T ∗ i | x, y i q ) j − − k j ( n X i =1 h| T i | x, y i p ) k j − j vuut ( n X i =1 h| T i | x, y i p ) k j +1 ( n X i =1 h| T ∗ i | x, y i q ) j − − k j − (cid:17) , that inequality (2.10) is refinement of (1.7). Corollary 2.18.
Let T i ∈ B ( H ) (1 ≤ i ≤ n ) . Then w e ( | T | , ..., | T n | ) w e ( | T ∗ | , ..., | T ∗ n | ) ≤ k n X i =1 T ∗ i T i k + k n X i =1 T i T ∗ i k ! − inf k x k = k y k =1 λ ( x, y ) , where λ ( x, y ) = N X j =1 (cid:16) ( − r j j − ( 12 ) + ( − r j +1 h r j + 12 i(cid:17)(cid:16) j vuut ( n X i =1 h| T ∗ i | x, y i ) j − − k j ) ( n X i =1 h| T i | x, y i ) k j − j vuut ( n X i =1 h| T i | x, y i ) k j +1) ( n X i =1 h| T ∗ i | x, y i ) j − − k j − (cid:17) . Proof.
The result obtained by letting p = q = 2 and r = 1 in inequality (2.10). (cid:3) Corollary 2.19.
Let T , ..., T n ∈ B ( H ) be positive operators. Then w e ( T , ..., T n ) ≤ k n X i =1 T i k . References
1. O. Axelsson, H. Lu and B. Polman,
On the numerical radius of matrices and its application toiterative solution methods , Linear Multilinear Algebra. (1994), 225–238.2. M. Bakherad and F. Kittaneh, Numerical radius inequalities involving commutators of G1 opera-tors , Complex Anal. Oper. Theory (2017). https://doi.org/10.1007/s11785-017-0726-9.3. M. Bakherad, M. Krnic and M.S. Moslehian,
Reverses of the Young inequality for matrices andoperators , Rocky Mountain J. Math. (4) (2016), 1089–1105. URTHER REFINEMENTS OF GENERALIZED NUMERICAL RADIUS INEQUALITIES 15
4. M. Bakherad and Kh. Shebrawi,
Upper bounds for numerical radius inequalities involving off-diagonal operator matrices , Ann. Funct. Anal. (2018) (to appear).5. M. Boumazgour, H. Nabwey,
A note concerning the numerical range of a basic elementary oper-ator , Ann. Funct. Anal. (2016), no. 3, 434–441.6. K.E. Gustafson and D.K.M. Rao, Numerical Range, The Field of Values of Linear Operators andMatrices , Springer, New York, 1997.7. M. Hajmohamadi, R. Lashkaripour, M. Bakherad,
Some generalizations of numerical radius onoff-diagonal part of × operator matrices. , J. Math. Inequal. (2) (2018), 447457.8. M. Hajmohamadi, R. Lashkaripour, M. Bakherad, Some extensions of the Young and Heinz in-equalities for matrices , Bull. Iranian Math. Soc. (2017) (to appear).9. G.H. Hardy, J.E. Littlewood and G. Polya, inequalities , 2nd ed., Cambridge Univ. Press, Cam-bridge, ( Further refinements of the Heinz inequality ,Linear Algebra Appl. (2014), 26–37.11. F. Kittaneh,
Notes on some inequalitis for Hilbert space operators , Publ. Res. Inst. Math. Sci. (2) (1988), 283–293.12. F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices , J. Math.Anal. Appl. (2010), 262–269.13. M.S. Moslehian and M. Sattari,,
Inequalities for operator space numerical radius of 2 × ,J. Math. Phys. (1) (2016), 015201, 15 pp.14. G. Popescu, Unitary invariants in multivariable operator theory , Mem. Amer. Math. Soc. (941)(2009).15. M. Sababheh and D. Choi,
A complete refinement of Yongs inequality , J. Math. Anal. Appl. (2016), 379–393.16. M. Sattari, M.S. Moslehian and K. Shebrawi,
Extension of Euclidean operator radius inequalities ,Math. Scand. (2017), 129–144.17. M. Sattari, M.S. Moslehian and T. Yamazaki, Some genaralized numerical radius inequalities forHilbert space operators , Linear Algebra Appl, (2014), 1–12.18. A. Sheikhhosseini, M.S. Moslehian and K. Shebrawi,
Inequalities for generalized Euclidean operatorradius via Young’s inequality , J. Math. Anal. Appl. (2) (2016), 1516–1529.19. A. Zamani,
Some lower bounds for the numerical radius of Hilbert space operators,
Adv. Oper.Theory (2) (2017), 98–107. , , Department of Mathematics, Faculty of Mathematics, University of Sistan andBaluchestan, Zahedan, I.R.Iran.
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