1. Fundamental Principles
Let denote the -algebra of all bounded linear operators on a Hilbert space H. In the case when , we identify with the matrix algebra of all matrices with entries in the complex field. The numerical radius of is defined by
It is well-known that defines a norm on , which is equivalent to the usual operator norm. Namely, for , we have
These inequalities are sharp. The first inequality becomes an equality if , and the second inequality becomes an equality if T is normal (see  ).
An important inequality for is the power inequality stating that for see (  : p. 118).
An important property of the numerical radius norm is its weak unitary invariance, that is, for ,
for every unitary . For further information about the properties of numerical radius inequalities we refer the reader to  -  and references therein.
Let be Hilbert spaces, and consider the direct sum . By considering this decomposition, every operator has a operator matrix representation with entries .
Hirzallah, Kittaneh and Shebrawi have proved in  that:
If , then:
also, they proved that:
If , then:
Moreover, they showed that:
if , then:
Shebrawi and Albadawi have proved in  that:
If and be nonnegative continuous functions on satisfying the relation , then:
for all .
In the special case, where and , , they proved that:
In particular, they proved the following inequalities:
The main purpose of this paper is to give considerable improvements of the inequalities (7), (8), (9), (10), (11), and (12). In order to achieve our goal, we need the following three lemmas which are essential in our analysis.
The first lemma was proved in  .
Lemma 1 If and , then:
If , the inequality (13) becomes an equality where
The second lemma follows from the spectral theorem for positive operators and Jensen’s inequality (see  ).
Lemma 2 Let , and such that . Then:
1) for .
2) for .
The third lemma was proved in 
Lemma 3 Let and be any vectors. If are nonnegative continuous functions on which are satisfying the relation , then:
and more general,
3. Main Results
The first result in this paper is numerical radius inequality which is sharper than the inequality (7).
Theorem 3.1 Let , , and be nonnegative continuous functions on satisfying the relation . Then:
Taking the supremum over all unit vectors , we get
Remark 1 In view of the inequalities (7) and (17), it clears that the inequality (17) is sharper than the inequality (7).
As special case of the inequality (17), let and , , we will get the following inequality which is sharper than the inequality (8).
Corollary 4 Let , , , and . Then:
In particular, if , we get the following inequality which is charper than the inequality (9),
By letting in the inequality (19), we obtain the following inequality which is sharper than the inequality (10).
Corollary 5 Let , , , and . Then:
Letting in the inequality (23), we obtain the following inequality which is sharper than the inequality (11).
Corollary 6 Let , , and . Then:
In the inequality (24), replacing by 0, we have the following inequality which is sharper than the inequality (12).
Corollary 7 Let . Then:
Now, we will prove the following inequality which is another version of the inequality (6).
Theorem 3.2 Let . Then:
Proof. Let , then U is unitary, and
((by Equation (3))
since ( , so ),
and ( , so ).
Chaining the inequality (27) with the inequality (4) yields the following inequality.
Corollary 8 Let . Then:
Proof. In Theorem 3.2, apply the inequality (4) on the right side, we get the result.
Chaining the inequality (27) with the inequality (5) yields the following inequality.
Corollary 9 Let . Then:
Proof. In Theorem 3.2, apply the inequality (5) on the right side, we get the result.
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 Hirzallah, O., Kittaneh, F. and Shebrawi, K. (2011) Numerical Radius Inequalities for Certain 2 × 2 Operator Matrices. Integral Equations and Operator Theory, 71, 129-147.
 Kittaneh, F. and Manasrah, Y. (2010) Improved Young and Heinz Inequalities for Matrices. Journal of Mathematical Analysis and Applications, 361, 262-269.
 Kittaneh, F. (1988) Notes on Some Inequalities for Hilbert Space Operators. Publications of the Research Institute for Mathematical Sciences, 24, 283-293.