General numerical radius inequalities for matrices of operators

Mohammed Al-Dolat; Khaldoun Al-Zoubi; Mohammed Ali; Feras Bani-Ahmad

doi:10.1515/math-2016-0011

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

https://doi.org/10.1515/math-2016-0011

Abstract

Let A_i ∈ B(H), (i = 1, 2, ..., n), and T=[0⋯0A1⋮⋰A200⋰⋰⋮An0⋯0]. 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.

Key Words: Numerical radius; Operator norm; Cartesian decomposition

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 T ∈ B(H); denoted by W(T), is the subset of complex numbers given by

W (T) = { 〈T x,x〉 : x ∈ H, ||x|| = 1}.

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

w (T) = { 〈T x,x〉 : x ∈ H, ||x|| = 1}.

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 T ∈ B(H), we have

12 || T || ≤ w(T) ≤ || T ||.

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).

w( T ) ≤ 12 ( ||T|| + ||T2||12).

So it is clear that if T² = 0; then

(1)w( T ) = 12 || T ||.

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

(2)w ( U T U*) = w( T ),

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

w ( Tn ) ≤ wn (T ),

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, T₂ ∈ B(H), it is known [1] that

w (T1 T2 ) ≤ 4w (T1) w (T2).

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

w (T1 T2 ) ≤ 2w (T1) w (T2).

However, the sharp inequality

w (T1 T2 ) ≤ w (T1) w (T2)

still has not been reached. A useful result in this direction, which can be found in [8], says that for T₁, T₂ ∈ B(H),

w ( T1 T2 ± T2 T1* ) ≤ 2|| T1 || w ( T2).

If T₁, T₂ ∈ B(H), and T₁ is positive operator, Kittaneh in [9] showed that

w(T1T2−T2T1)≤12‖T1‖(‖T2‖+‖T22‖12).

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, T₄ ∈ B(H),

(3)w ( T1 T3 T2* ± T2 T4 T1* ) ≤ 2 || T1 || || T2 || w ([0T3T40]) .

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

(4)w([0XY0]) ≤ ||X|| + ||Y||2,

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.

Lemma 2.1

([3]). LetX₁, X₂, ..., X_n ∈ B(H). Then

w ([X10X2⋱0Xn]) = max { w ( X1 ), w ( X2 ),..., w ( Xn ) }

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

Theorem 2.2.

Let A_i ∈ B(H), i = 1, 2,..., n andT=[0A1A2⋰An0]

Then if n is even,

w ( T ) ≤ 12 ∑i=1n || Ai ||

and if n is odd,

w( T ) ≤ w (An + 12) + 12 ∑i= 1n|| Ai ||.

Proof.

Let X1 = [0A10⋰00],X2 = [00A20⋰00],...,Xn = [00⋰0An0].

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

w ( T ) = w (∑i = 1nXi) ≤ ∑i=1nw (Xi) = 12∑i= 1n|| Xi ||.

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

w ( T ) = w (∑i = 1nXi) ≤ w(Xn +12)+∑i≠n+12nw (Xi) =w(An +12)+ 12∑i≠n+12n|| Ai ||.

□

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 A_i ∈ B(H), (i = 1, 2, ..., n) where n ≥ 3 and 1, α, α²,..., α^{n − 1}are the roots of unity andT=[0A1A2⋰An0]. Then

w(T) = w([0αn−1Anα2n−3An−1α3n−5An−2⋰α(n−1)2A10])=w([0αAnα3An−1α5An−2⋰α2n−1A10])=w([0⋯0α3An−100⋮⋰α5An−20⋰⋰⋰⋮α2n−3A2⋮000αn−1A10⋯0αn+1An0])

Proof.

Let U1 = [0α2n−2Iα2n−3I⋰αn−1I0], U2 = [0IαIα2I⋰αn−1I0] and [0⋯0I0αI⋮⋰α2I0⋰⋰⋮αn−2I00⋯0αn−1I]

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, X₃ ∈ B(H) and 1; α, α²be the roots of x³ = 1. Then

w([X2α2X1αX3αX3X2α2X1α2X1αX3X3]) ≤ 3w([00X30X20X100])

Proof.

For T = [X2α2X1αX3αX3X2α2X1α2X1αX3X2] , we have

w(T) ≤ ([00αX30X20α2X100]) + w([X20000α2X10αX30])+w([0α2X10αX30000X2])=2w ([00X30X20X100]) +w ([0α2X10αX30000X2]) =2w ([00X30X20X100]) +w ([00α2X10X20αX300]) =2w ([00X30X20X100]) +w ([00α2X3*0X2*0αX1*00]) =2w ([00X30X20X100]) +w ([00X30X20X100]*) =3w ([00X30X20X100]) .

Another relation for the numerical radius of the operator [0A1A2⋰An0] is as follows.

Theorem 2.5.

LetX_i ∈ B(H), (i = 1, 2, ..., n) where n ≥ 3, T = [0A1A2⋰An0],T1 = [0A2A3⋰An−10]andT1t = [0An−1An−2⋰A20]Then

w(T) = w([00A10T10An00])=w([00An0T1t0A100]) =w([T10000An0A10]) = w([T1t0000A10An0])=w([0An0A10000T1]) =w([0A10An0000T1t]).

An application of Theorem 2.5 yields

Corollary 2.6.

LetX₁, X₂, X₃ ∈ B(H). Then

w([X1X2X3X3X1X2X2X3X1]) ≤ 3w ([00X30X10X200]) .

Using a straightforward technique we derive the following lemma.

Lemma 2.7

LetA_i, X_i ∈ B(H), i = 1, 2, ..., n. Then

w(∑i=1nAiXiAi*) ≤ ∑i=1n||Ai||2w(Xi)

Proof.

Assume that x ∈ H is a unit vector. Then

|〈∑i=1nAiXiAi*x,x〉| = | ∑i=1n〈XiAi*x,Ai*x〉| ≤ ∑i=1n|〈XiAi*x,Ai*x〉| ≤ ∑i=1n||Ai*x||2w(Xi) ≤ ∑i=1n||Ai||2w(Xi).

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

Our next result can be stated as follows.

Theorem 2.8.

Let A_i, B_i, X_i, Y_i ∈ B(H), i = 1, 2, ..., n. Then

w(∑i=1nAiXiBn−i+1* ∓ BiYiAn−i+1*) ≤ 2 ((∑i=1n||Ai||2)(∑i=1n||Bi||2))w(T),

whereT = [0X1⋰XnY1⋰Yn0]

Proof.

Assume that C = [A1A2⋯AnB1B2⋯Bn0⋯0] and Z = [0X1⋰XnY1⋰Yn0]

Applying Lemma 2.1 and Lemma 2.7 we get

w(∑i=1nAiXiBn−i+1*+BiYiAn−i+1*) = w([∑i=1nAiXiBn−i+1*+BiYiAn−i+1*000])= w (CZC*)≤ ||C||2w(Z)=||∑i=1nAiAi*+BiBi*||w(Z)≤(∑i=1n||AiAi*+BiBi*||)w(Z)≤(∑i=1n||Ai||2 + ||Bi||2)w(Z).

Now, if we replace A_i and B_i by t A_i and 1t Bi, t > 0, respectively, then we have

w (∑i=1nAiXiBn−i+1*+BiYiAn−i+1*) ≤ w(Z)∑i=1n(t4||Ai||2+||Bi||2t2).

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

w (∑i=1nAiXiBn−i+1*+BiYiAn−i+1*) ≤ 2w(Z)(∑i=1n||Ai||2∑i=1n||Bi||2).

Finally, replace Y_i by −Y_i to get

w (∑i=1nAiXiBn−i+1*+BiYiAn−i+1*) ≤ 2w(Z)(∑i=1n||Ai||2∑i=1n||Bi||2).

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

Corollary 2.9

IfT_i ∈ B(H), i = 1, 2,..., n, then

w([0T1T2⋰Tn0]) ≥ 1nw(∑i=1nTi)

Proof.

LetX_i = Y_i = T_i and 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.

Theorem 2.10

LetA_ij ∈ B(H) where 1 ≤ i, j ≤ n. Then

w([A11A12⋯A1nA21A22⋯A2n⋮⋮⋮⋮An1An2⋯Ann]) ≤ max {w(Aii) : 1 ≤ i ≤ n} + 12 ∑i,j=1,i≠jn||Aij||.

Proof.

For 1 ≤ x, y ≤ n where x ≠ y. Define the operator matrix T_xy = [t_ij] where

tij = {0 otherwise.Aij if i = x, j = y

Then Txy2 =0 so

w([A11A12⋯A1nA21A22⋯A2n⋮⋮⋮⋮An1An2⋯Ann]) =w([A110A22⋱0Ann] + ∑x,y=1,x≠ynTxy) ≤w([A110A22⋱0Ann]) + ∑x,y=1,x≠ynw(Txy)=max{w(Aii): 1 ≤ i ≤ n}+12∑i,j=1,i≠jn||Aij||

□

An application of Theorem 2.10 and Corollary 2.9 yields

Corollary 2.11

LetA, B, C, D = ∈ B(H). Then

max{max{w(A),w(D)}, 12w(A + D)} ≤ w ([ABCD]) ≤ max{w(A), w(D)}+||B||+||C||2

3 A bound for the zeros of polynomials

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

C(p) = [−an−an−1…−a2−a110⋯⋯001⋱⋮⋮⋱⋱⋱⋮0⋯010].

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

(5)|z| ≤ w (C(p))

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

|z| ≤ max{w(A),cos(πn+1)}+12(1+(∑i=1n−1|ai|2)12).

Proof.

Partition C(p) as

C(p) = [ABCD],

where A = [−an−an−110],B = [−an−2⋯−a2−a10⋯⋯0], C = [0100⋮⋮00] and D = [00⋯⋯010⋯⋱⋮01⋱⋱⋮⋮⋱⋱⋱00⋯010]

Then we have

| z | ≤ ω (C(p)) = ω ([ABCD]) ≤max{ ω(A), cos (πn + 1) } + 12(1+ (∑ | ai |2i = 1n − 1)12).

□

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

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

General numerical radius inequalities for matrices of operators

Abstract

1 Introduction

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

3 A bound for the zeros of polynomials

References

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