## Abstract

Let *A _{i}* ∈

*B*(

*H*), (

*i*= 1, 2, ...,

*n*), and

*w*(

*T*). At the end of this paper we drive a new bound for the zeros of polynomials.

## 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 *T* ∈ *B*(*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* ∈ *B*(*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 [3–7] for recent developments this area. Kittaneh [5] proved that for *T* ∈ *B*(*H*).

So it is clear that if *T*^{2} = 0; then

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

for every unitary *U* ∈ *B*(*H*).

A fundamental inequality for the numerical radius is the power inequality, which says that for *T* ∈ *B*(*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 *T*_{1}, *T*_{2} ∈ *B*(*H*), it is known [1] that

Moreover, in the case that *T*_{1}, *T*_{2} = *T*_{2}*T*_{1},

However, the sharp inequality

still has not been reached. A useful result in this direction, which can be found in [8], says that for *T*_{1}, *T*_{2} ∈ *B*(*H*),

If *T*_{1}, *T*_{2} ∈ *B*(*H*), and *T*_{1} 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 *T*_{1}, *T*_{2}, *T*_{3}, *T*_{4} ∈ *B*(*H*),

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

where *X*, *Y* ∈ *B*(*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.

([3]). *Let**X*_{1}, *X*_{2}, ..., *X _{n}* ∈

*B*(

*H*).

*Then*

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

*Let A _{i}* ∈

*B*(

*H*),

*i*= 1, 2,...,

*n and*

*Then if n is even,*

*and if n is odd,*

*Proof.*

Let

Then if *n* is an even number we have *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, *A*_{1} = *X*, *A*_{2} = *Y* we reach the inequality (4). Let us use (2) to prove the following theorem.

*Let A _{i}* ∈

*B*(

*H*), (

*i*= 1, 2, ...,

*n*)

*where n*≥ 3

*and*1,

*α*,

*α*

^{2},...,

*α*

^{n − 1}

*are the roots of unity and*

*Then*

*Proof.*

Let

Then it is easy to show that *U*_{1}, *U*_{2} and *U*_{3} are unitary operators so by (2) we have

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

*Let**X*_{1}, *X*_{2}, *X*_{3} ∈ *B*(*H*) *and* 1; *α*, *α*^{2}*be the roots of x*^{3} = 1. *Then*

*Proof.*

For

Another relation for the numerical radius of the operator

*Let**X _{i}* ∈

*B*(

*H*), (

*i*= 1, 2, ...,

*n*)

*where n*≥ 3,

*and*

*Then*

An application of Theorem 2.5 yields

*Let**X*_{1}, *X*_{2}, *X*_{3} ∈ *B*(*H*). *Then*

Using a straightforward technique we derive the following lemma.

*Let**A _{i}*,

*X*∈

_{i}*B*(

*H*),

*i*= 1, 2, ...,

*n. Then*

*Proof.*

Assume that *x* ∈ *H* is a unit vector. Then

We attain our theorem by taking the supremum over all unit vectors *x* ∈ *H*.

Our next result can be stated as follows.

*Let A _{i}, B_{i}, X_{i}, Y_{i}* ∈

*B*(

*H*),

*i*= 1, 2, ...,

*n. Then*

*where*

*Proof.*

Assume that

Applying Lemma 2.1 and Lemma 2.7 we get

Now, if we replace *A _{i}* and

*B*by

_{i}*t A*and

_{i}*t*> 0, respectively, then we have

Also, since

Finally, replace *Y _{i}* by −

*Y*to get

_{i}As an application of Theorem 2.8, we obtain the following result.

*If**T _{i}* ∈

*B*(

*H*),

*i*= 1, 2,...,

*n*,

*then*

*Proof.*

*Let**X _{i}* =

*Y*=

_{i}*T*and

_{i}*A*=

_{i}*B*=

_{i}*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.

*Let**A _{ij}* ∈

*B*(

*H*)

*where*1 ≤

*i*,

*j*≤

*n*.

*Then*

*Proof.*

For 1 ≤ *x*, *y* ≤ *n* where *x* ≠ *y*. Define the operator matrix *T _{xy}* = [

*t*] where

_{ij}Then

□

An application of Theorem 2.10 and Corollary 2.9 yields

*Let**A*, *B*, *C*, *D* = ∈ *B*(*H*). *Then*

## 3 A bound for the zeros of polynomials

Let *p*(*z*) = *z ^{n}* +

*a*+ ... +

_{n}z^{n-1}*a*+

_{2}z*a*be a monic polynomial of degree

_{1}*n*≥ 3 with complex coefficients

*a*

_{1},

*a*

_{2}, ...,

*a*. Then the Frobenius companion matrix of

_{n}*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*.

*If z is any zero of**p*(*z*), *then*

*Proof.*

Partition *C*(*p*) as

where

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.