New numerical radius inequalities for certain operaor matrices WATHEQ

In this paper we prove some upper and lower bounds for the numerical radius of the off-diagonal part of 3× 3 operator matrices and some bounds for the numerical radius inequalities of the general 3× 3 operator matrix. Mathematics Subject Classification (2010). 47A12, 47A30, 47A63, 47B15.


Introduction
Let H be a complex Hilbert space with inner product h., .iand let B (H) be the space of all bounded linear operators on H.For A ∈ B (H) , let ω (A) and kAk denote the numerical radius and the usual operator norm, respectively.Recall that ω (A) = sup {|λ| : λ ∈ W (A)} , where W (A) is the numerical range of A which is a subset of the complex numbers, and kAk = sup {kAxk : kxk = 1} .
It is well-known that ω (.) defines a norm on B (H) , which is equivalent to the usual operator norm kAk .In fact, for A ∈ B (H) ,we have 1 2 kAk ≤ ω (A) ≤ kAk . (1.1) These inequalities are sharp.The first inequality becomes an equality if A 2 = 0, and the second inequality becomes an equality if A is normal.
One of the important properties of ω (.) is that it is weakly unitarily invariant, that is, for A ∈ B (H) , we have ω (UAU * ) = ω (A) , (1.2) for every unitary U ∈ B (H) .This improvement of the seconed inequality in (1.1) has been given in [6].It says that for A ∈ B (H) , we have consequently, if A 2 = 0, then New Numerical Radius Inequalities for Certain Operator Matrices 279 ω (A) = 1 2 kAk . (1.4) The equality (1.4) follows from the inequality (1.3) and the first inequality in (1.1).
In this paper, we give some new numerical radius inequalities for certain 3 × 3 operator matrices.In section 2, we establish upper and lower bounds for the numerical radii of the off-diagonal parts of 3 × 3 operator matrices.In section 3, we establish upper and lower bounds for the numerical radii of general 3 × 3 operator matrices.

Numerical radius inequalities for the operator matrix
Our goal in this section is to give bounds for the numerical radius of the off-diagonal part To achieve our goal, we need two basic lemmas.Part (a) of the first lemma is well-known, and it can be found in [3].Part (b) is also known (see, e.g., [1]) and it follows by applying the identity (1.2) to the operator matrix ⎦ and the unitary op- are the cubic roots of unity.

New Numerical Radius Inequalities for Certain Operator Matrices 281
Proof.To prove part (a), let Then U 1 , U 2 , U 3 , U 4 , U 5 , U 6 , and U 7 are unitary operator matrices, where I is the identity operator in B (H) .
Now, it is easy to prove the following identities Hence, from the property (1.2), we obtain the required results.Now, to prove part (b), take (by Lemma 1 (a)).

2
Our first result in this section can be stated as follows.

Now, since
This proves the second inequality in (2.1).Now, we give some inequalities that involve ω ´, and Proof.First, we prove the inequality (2.2).We have and so This completes the proof of the inequality (2.2).Now, to prove the inequality (2.3), let 2) and (2.3) becomes equalities.
In the following two results we give further upper and lower bounds for the numerical radius of In these results, we use the observation that for X ∈ B (H) , we have So by the identity (1.4) we have Then U is unitary.It follows that ω (X) = ω (UXU * ) (by the identity (1.2) (by the identity (2.4) and Lemma 2 (a) and (b)).
3. Upper and lower bounds for the numerical radius of the general 3 × 3 operator matrix.
We start our results by the following lemma which satisfies certain pinching inequalities (see, e.g., [3]).
Then, to prove part (c) for example, it is easy to prove that and from the fact that the numerical radius is a norm, which is weakly unitarily invariant, we have

2
Based on the Lemmas 1 and 8, we have our first result in this section.
Proof.For the second inequality, we have The first inequality follows from Lemma 1 (a), Theorem 4, and Lemma 8.
At the end of this section, we present a general numerical radius inequalities for 3 × 3 operator matrices.These new inequalities are based on the pinching inequalities given in Lemma 8, the triangle inequality for ω (.) , Lemma 1 (a) and Lemma 2 (a), concerning the numerical radii of the diagonal parts of 3 × 3 operator matrices, and our estimates of the numerical radii of the off-diagonal parts of these operator matrices given in Theorem 4. Theorem 3. Let A ij ∈ B(H), for all i, j = 1, 2, 3. Then Proof.