Abstract
Let Ai ∈ B(H), (i = 1, 2, ..., n), and
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
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 T2 = 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 T1, T2 ∈ B(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, T2 ∈ B(H),
If T1, T2 ∈ B(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, T4 ∈ 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]). LetX1, X2, ..., Xn ∈ B(H). Then
Let us use this lemma to generalize the inequality (4).
Let Ai ∈ B(H), i = 1, 2,..., n and
Then if n is even,
Let
Then if n is an even number we have
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.
Let Ai ∈ B(H), (i = 1, 2, ..., n) where n ≥ 3 and 1, α, α2,..., αn − 1are the roots of unity and
Let
Then it is easy to show that U1, U2 and U3 are unitary operators so by (2) we have
As a direct consequence of Theorem 2.3 we obtain the following corollary
LetX1, X2, X3 ∈ B(H) and 1; α, α2be the roots of x3 = 1. Then
For
Another relation for the numerical radius of the operator
LetXi ∈ B(H), (i = 1, 2, ..., n) where n ≥ 3,
An application of Theorem 2.5 yields
LetX1, X2, X3 ∈ B(H). Then
Using a straightforward technique we derive the following lemma.
LetAi, Xi ∈ B(H), i = 1, 2, ..., n. Then
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 Ai, Bi, Xi, Yi ∈ B(H), i = 1, 2, ..., n. Then
where
Assume that
Applying Lemma 2.1 and Lemma 2.7 we get
Now, if we replace Ai and Bi by t Ai and
Also, since
Finally, replace Yi by −Yi to get
As an application of Theorem 2.8, we obtain the following result.
IfTi ∈ B(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.
LetAij ∈ B(H) where 1 ≤ i, j ≤ n. Then
For 1 ≤ x, y ≤ n where x ≠ y. Define the operator matrix Txy = [tij] where
Then
□
An application of Theorem 2.10 and Corollary 2.9 yields
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.
If z is any zero ofp(z), then
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.
References
1 Bhatia R., Matrix Analysis, Berlin. Springer-Verlage, 1997.210.1007/978-1-4612-0653-8Search in Google Scholar
2 Dragomir S., A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces, Banach J. Math. Anal., 2007, 2(1), 154– 175.10.15352/bjma/1240336213Search in Google Scholar
3 Dragomir S., Inequalities for the norm and the numerical radius of linear operator in Hilbert spaces, Demonstatio Math., 2007, 2(40), 411– 417.10.1515/dema-2007-0213Search in Google Scholar
4 Dragomir S., Norm and numerical radius inequalities for sums of bounded linear operators in Hilbert spaces, Ser. Math. Inform., 2007, 1(22), 61– 75. General numerical radius inequalities for matrices of operators 11710.15352/bjma/1240336213Search in Google Scholar
5 Kittaneh F., A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math., 2003, 158(1), 11– 17.10.4064/sm158-1-2Search in Google Scholar
6 Omidvar E., Moslehian M., Niknam A., Some numerical radius inequalities for Hilbert space operators. Involve, a journal of mathematics, 2009, 4(2), 471– 478.10.2140/involve.2009.2.471Search in Google Scholar
7 Kittaneh F., Numerical radius inequalities for Hillbert space operators, Studia Math., 2005, 168(1), 73– 80.10.4064/sm168-1-5Search in Google Scholar
8 Fong C., Holbrook J., Unitarily invariant operator norms, Can. J. Math., 1983, 135, 274– 299.10.4153/CJM-1983-015-3Search in Google Scholar
9 Kittaneh F., Notes on some inequalities for Hilbert space operators, Pub1. Res. Inst. Math. Sci., 1988, 24, 283– 293.10.2977/prims/1195175202Search in Google Scholar
10 Hirzallah O., Kittaneh F., Shebrawi K., Numerical radius inequalities for commutators of Hilbert space operators, Num. Func. Anal. and Opti., 2011, 7(32), 739– 749.10.1080/01630563.2011.580875Search in Google Scholar
11 Hirzallah O., Kittaneh F., Shebrawi K., Numerical radius inequalities for certian 2 × 2 operator matrices, Integr. Equ. Oper. Theory, 2011, 71, 129– 147.10.1007/s00020-011-1893-0Search in Google Scholar
12 Kittaneh F., Moslehian M. Takeaki Y., decommpostion and numerical radius inequalities. Linear Algebra 2015, 471, 46-53.10.1016/j.laa.2014.12.016Search in Google Scholar
13 Horn R., Johnson C., Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985.10.1017/CBO9780511810817Search in Google Scholar
14 Amer A.,Kittaneh F., Estimates for the numerical radius and the spectral radius of the Frobenius companion matrix and bound for the zeros of polynomials. Ann. funct. Anal. 5 2014, 1, 56-62.10.15352/afa/1391614569Search in Google Scholar
© 2016 Al-Dolat et al., published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.