New Inequalities for Numerical Radius of Hilbert Space Operator And New Bounds For The Zeros Of Polynomials

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OPEN ACCESS

New Inequalities for Numerical Radius of Hilbert

Space Operator And New Bounds For The Zeros

Of Polynomials

To cite this article: Mohammad Al-Hawari and Abdullah Ahmed Aldahash 2013 J. Phys.: Conf. Ser. 423 012013

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New Inequalities for Numerical Radius of

Hilbert Space Operator And New Bounds

For The Zeros Of Polynomials

.

Mohammad Al-Hawari 1 and Abdullah Ahmed Aldahash 2

1_{Associate Professor of Mathematics, College of Applied Medical Sciences,}

Majmaah University, Kingdom of Saudi Arabia, P.O. Box 1405, Almajmaah Zipcode 11952

2

Associate Professor of Mathematics Education, Dean of the College community, College of community, Majmaah University

E-mail: [email protected] 1

Abstract: It is shown that if Ar

does not converge to the zero operator on a complex Hilbert space for some r  1 , then

wA  1

2 A

r_1r  A2 1 2 _.

And if there exists r such that Ar

is a zero operator on a complex Hilbert space for all r  2 , then

wA  1

2A,

where w.  and . are the numerical radius and the usual operator norm,respectively.Also, the previous inequalities are better than any other classical inequalities.

Key words: Numerical radius, Operator norm, Cauchy- Schwarz inequality. AMS: 47A12,47A30,47A63.

1. Introduction

Let BH denote the C - algebra of all bounded linear operators on a

complex Hilbert space H with inner product , . For A  BH, let wA

and A denote the numerical radius and the operator norm of A ,

respectively. It is well-known that w defines a norm on BH, and that for every A  BH,

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1

2A  wA  12 A  A

2_12  A. 1

For basic properties of the numerical radius, we refer to 4 and 7. It has been recently shown in 5, that if A2

does not converge to the zero operator in Mn, then

wA  A2_12_,2

Also, it has been proved in 6; that if A,B  BH for 0    1, then

for r  1 we have

wA  21r 1|A|2r  |A|21r1r  1

2A

2_12_.3

In the next section of this paper, we establish inequalities that refines the inequalities (1) and (2) using some classical inequalities for nonnegative real numbers and some operator inequalities. To prove our generalized numerical radius inequalities, we need several well-known lemmas. The following lemma follows from spectral theorem for positive operators and Jensen's inequality 1 .

Lemma 1.1. Let A  BH be positive operator and let x  H be any unit

vector. Then

Ax,xr _{ }_Ar_x_,_x_{ for all}_r _{ 1, 4}

Ar_x_,_x_{  }_Ax_,_x_r _{for all 0} _ _r _{ 1. 5}

Definitions 1.1[ 3] If A  BH , then (I) The spectral norm (or the operator norm) is defined by

A  maxAx

x :x 0,A maxAx: x 1. 6

(II) The numerical radius of A is defined by

wA  max Ax,x :x  H,x 1 7

In the following Lemma we present a sharper numerical radius inequality for any operator.

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Lemma1.2.[6].if A,B  BH for 0    1, then for r  1 we have wA  21r 1|A_|2r  |A_|21r1r  1

2A

2_12. 8

2. Numerical radius inequalities

From Lemma1.2 we present several classical and sharp numerical radius inequalities as follows.

Theorem 2.1.if Ar_{does not converge to the zero operator on a complex}

Hilbert space for some r  1 , then wA  1

2 A

r_1r  A2 1 2 _.9

proof: Letting   12 and r  1 in (8), we obtain the inequality

wA  21r 1|A_|r  |A_|r1r  1 2A 2_12. and since, s₁2_|_A |r  s₁2_|_A_|r_,

where s1A denotes the largest singular value of A and s1A  A.

Hence,

|A|r  |A|r. So, we get the result.

Corollary 2.1. Letting r  1 in (9), we obtain the inequality wA  1

2 A  A

2_12 , 10

(that is Kittaneh's inequality see e.g.[1] )

Corollary 2.2. Letting r  2 in (9), we obtain the inequality wA  A2_12_,11

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A2_{does not converge to the zero operator. (that is Al-Hawari's inequality}

see e.g.[5] ).

If A2_{is a non-zero operator,then the inequality (11) is not true}

To see this ,consider

A  t 1 0 0 , then wA  t 1 t 2 2  12, and A2_{  1}

2, for taking smallt. The inequality (9) refines the inequality (1), because

Ar_1_r _{ }_A_{, 12}

for every A  BH.

Corollary 2.3.If Ar_{is the zero operator for all}_r _{ 2, then}

wA  1

2A

r_1r  1

2A, 13 A is a non-zero operator.

Corollary 2.4. Since wA  12A .So,we get from the inequalities (1) and

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wA  1

2A. 14 If Ar

is the zero operator for all r  2. (e.g.see [1]).

It should be mentioned that it is not true if Ar

is the zero operator for some

r  3, then we get the inequality (14) .

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To see this, consider A  0 0 0 1 0 0 0 1 0 , then wA  1 2  12A, and A  A2_{  1,}

where A3 _{is the zero matrix operator.}

Open Problems

The first open problem is possible to complement the bound (9) by giving an upper bound estimate for the zeros of

pz  zn __a

nzn1 a_n_1zn2 ....a2za1withai   for 1  i nanda1  0. 15

The second open problem is possible to complement the upper bound (9) by giving a lower bound estimate for the zeros of p .To see this,observe that the zeros of the polynomial

qz  _azn

1 p 1z, 16

are the reciprocals of those of p . Thus ,applying the upper bound (9) to the zeros of q yields the desired lower bound estimate for the zeros of p . this enables us to present a new annulus containing the zeros of p .

References

[1] F.Kittaneh (2003), A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math 158, 11 -- 17.

[2] F.Kittaneh (2005), Numerical radius inequalities for Hilbert Space operators. Studia Math 168, 73-80.

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University press , Cambridge.

[4] K.E.Gustafson , and K.M. Rao, (1997), Numerical Range. Springer--Verlag, New York.

[5] M. Al-Hawari, (2009), New estimate for the numerical radius of a given matrix and new bounds for zeros of polynomials, studia math(Romania)., .

[6] M. Al-Hawari, (2009), Sharper Inequalities for Numerical Radius of Hilbert Space Operators, Int. J. of Math.analysis,21(3), 1021-1025.

[7] M. El-Haddad and F. Kittaneh, (2007), Numerical radius inequalities for Hilbert space operators ii, Studia Math., 182(2),133-140.

[8] R Bhatia, F.Kittaneh, (2008), The matrix arithmetic-geometric mean inequalities. Linear Algebra Appl.428, 2177-2191.

[9] S.S.Dragomir, (2006), Some inequalities for the Eculidean operator radius of two operators in Hilbert spaces, Linear Algebra Appl. 419, 256-264.