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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access March 1, 2016

General numerical radius inequalities for matrices of operators

Mohammed Al-Dolat, Khaldoun Al-Zoubi, Mohammed Ali and Feras Bani-Ahmad
From the journal Open Mathematics


Let AiB(H), (i = 1, 2, ..., n), and T=[00A1A200An00]. In this paper, we present some upper bounds and lower bounds for w(T). At the end of this paper we drive a new bound for the zeros of polynomials.

MSC 2010: 47A05; 47A10; 47A12

1 Introduction

Let B(H) be the C*– algebra of all bounded linear operators on a complex Hilbert space H with inner product 〈.,.〉. The numerical range of TB(H); denoted by W(T), is the subset of complex numbers given by


The numerical radius of T, denoted by w(T), is given by


It is well-known that w(.) defines a norm on B(H), which is equivalent to the usual operator norm ||T||=sup||x||=1||Tx||. In fact for TB(H), we have


Several numerical radius inequalities that provide alternative lower and upper bounds for w(.) have received much attention from many authors. We refer the readers to [1, 2] for the history and significance, and [37] for recent developments this area. Kittaneh [5] proved that for TB(H).


So it is clear that if T2 = 0; then


Also it is known that w(.) is weakly unitarily invariant, that is


for every unitary UB(H).

A fundamental inequality for the numerical radius is the power inequality, which says that for TB(H), we have


for n = 1, 2, 3, ... (see, e.g. [1, p. 118]).

Although some open problems related to the numerical radius inequalities for bound linear operators still remain open, the investigation to establish numerical radius inequalities for several bound linear operators has been started. For example, if T1, T2B(H), it is known [1] that


Moreover, in the case that T1, T2 = T2T1,


However, the sharp inequality


still has not been reached. A useful result in this direction, which can be found in [8], says that for T1, T2B(H),


If T1, T2B(H), and T1 is positive operator, Kittaneh in [9] showed that


Recently, the authors of [10] applied a different approach to obtain a new numerical radius inequality for commutators of Hilbert space operators. They showed that for T1, T2, T3, T4B(H),


The following numerical radius inequality for certain 2 × 2 operator matrices is obtained in [11],


where X, YB(H). Another results in the direction can be found in [12].

The purpose of this work is to present new numerical radius inequalities for n × n operator matrices. Also we deduce (3) and (4) as special cases. At the end of this paper, we give some new bounds for the zeros of any monic polynomial with complex coefficients.

2. Numerical radius inequalities for the n × n operator matrix

The aim of this section is to establish new numerical radius inequalities for matrices of operators and to generalize some known inequalities. In order to do this, we need the following well-known lemma.

Lemma 2.1

([3]). LetX1, X2, ..., XnB(H). Then


Let us use this lemma to generalize the inequality (4).

Theorem 2.2.

Let AiB(H), i = 1, 2,..., n andT=[0A1A2An0]

Then if n is even,

and if n is odd,

Let X1=[0A1000],X2=[00A2000],...,Xn=[000An0].

Then if n is an even number we have Xi2=0 for all i = 1, 2,..., n and so


On the other hand, if n is an odd number, then following the same manner used above we achieve that


Applying Theorem 2.2 with n = 2, A1 = X, A2 = Y we reach the inequality (4). Let us use (2) to prove the following theorem.

Theorem 2.3.

Let AiB(H), (i = 1, 2, ..., n) where n ≥ 3 and 1, α, α2,..., αn − 1are the roots of unity andT=[0A1A2An0]. Then


Let U1=[0α2n2Iα2n3Iαn1I0],U2=[0IαIα2Iαn1I0] and [00I0αIα2I0αn2I000αn1I]

Then it is easy to show that U1, U2 and U3 are unitary operators so by (2) we have w(T)=w(U1TU1*)=w(U2TU2*)=w(U3TU3*), which completes the proof. □

As a direct consequence of Theorem 2.3 we obtain the following corollary

Corollary 2.4.

LetX1, X2, X3B(H) and 1; α, α2be the roots of x3 = 1. Then


For T=[X2α2X1αX3αX3X2α2X1α2X1αX3X2], we have


Another relation for the numerical radius of the operator [0A1A2An0] is as follows.

Theorem 2.5.

LetXiB(H), (i = 1, 2, ..., n) where n ≥ 3, T=[0A1A2An0],T1=[0A2A3An10]andT1t=[0An1An2A20]Then


An application of Theorem 2.5 yields

Corollary 2.6.

LetX1, X2, X3B(H). Then


Using a straightforward technique we derive the following lemma.

Lemma 2.7

LetAi, XiB(H), i = 1, 2, ..., n. Then


Assume that xH is a unit vector. Then


We attain our theorem by taking the supremum over all unit vectors xH.

Our next result can be stated as follows.

Theorem 2.8.

Let Ai, Bi, Xi, YiB(H), i = 1, 2, ..., n. Then




Assume that C=[A1A2AnB1B2Bn00] and Z=[0X1XnY1Yn0]

Applying Lemma 2.1 and Lemma 2.7 we get


Now, if we replace Ai and Bi by t Ai and 1tBi, t > 0, respectively, then we have


Also, since t>0min(i=1nt4||Ai||+||Bi||t2)=2i=1n||Ai||2i=1n||Bi||2 so we have


Finally, replace Yi by −Yi to get


As an application of Theorem 2.8, we obtain the following result.

Corollary 2.9

IfTiB(H), i = 1, 2,..., n, then


LetXi = Yi = Ti and Ai = Bi = I. □

Based on the inequality (1) and Lemma 2.1, an upper bound for the numerical radius of the general n × n operator matrix can be derived.

Theorem 2.10

LetAijB(H) where 1 ≤ i, jn. Then


For 1 ≤ x, yn where xy. Define the operator matrix Txy = [tij] where


Then Txy2=0 so


An application of Theorem 2.10 and Corollary 2.9 yields

Corollary 2.11

LetA, B, C, D = ∈ B(H). Then


3 A bound for the zeros of polynomials

Let p(z) = zn + anzn-1 + ... + a2z + a1 be a monic polynomial of degree n ≥ 3 with complex coefficients a1, a2, ..., an. Then the Frobenius companion matrix of p is the matrix


It is well-known that the zero of p are exactly the eigenvalues of C(p) (see, e.g., [13, 14]). Since the spectral radius of a matrix is dominated by its numerical radius, it follows that if z is any zero of p, then


by the inequality (5) and Theorem 2.10 we can derive a new bound for the zeros of p.

Theorem 3.1

If z is any zero ofp(z), then


Partition C(p) as


where A=[anan110],B=[an2a2a100],C=[010000] and D=[000100100010]

Then we have


It should be mentioned here that other bounds for the zeros of p(z) can be obtained by considering different partitions of C(p). Related bounds for the zeros of p that are based on the inequality (5) and various estimates of w(C(p)) can be found in [13, 14], and references therein.


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Received: 2015-9-2
Accepted: 2016-2-8
Published Online: 2016-3-1
Published in Print: 2016-1-1

© 2016 Al-Dolat et al., published by De Gruyter Open.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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