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 . In fact for 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  proved that for T ∈ B(H).
So it is clear that if T2 = 0; then (1)
Also it is known that w(.) is weakly unitarily invariant, that is (2) 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  that
Moreover, in the case that T1, T2 = T2 T1,
However, the sharp inequality still has not been reached. A useful result in this direction, which can be found in , says that for T1, T2 ∈ B(H),
If T1, T2 ∈ B(H), and T1 is positive operator, Kittaneh in  showed that
Recently, the authors of  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)
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.
Theorem 2.2.: Let Ai ∈ 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 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 □
Theorem 2.3.: Let Ai ∈ B(H), (i = 1, 2, ..., n) where n ≥ 3 and 1, α, α2,..., αn − 1 are the roots of unity and . Then
Proof.: Let and Then it is easy to show that U1, U2 and U3 are unitary operators so by (2) we have , which completes the proof. □As a direct consequence of Theorem 2.3 we obtain the following corollary
Corollary 2.4.: Let X1, X2, X3 ∈ B(H) and 1; α, α2 be the roots of x3 = 1. Then
Proof.: For , we have
Another relation for the numerical radius of the operator is as follows.
Theorem 2.5.: Let Xi ∈ B(H), (i = 1, 2, ..., n) where n ≥ 3, and Then An application of Theorem 2.5 yields
Corollary 2.6.: Let X1, X2, X3 ∈ B(H). Then Using a straightforward technique we derive the following lemma.
Lemma 2.7: Let Ai, Xi ∈ 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.
Theorem 2.8.: Let Ai, Bi, Xi, Yi ∈ B(H), i = 1, 2, ..., n. Then where
Proof.: Assume that and Applying Lemma 2.1 and Lemma 2.7 we get Now, if we replace Ai and Bi by t Ai and , t > 0, respectively, then we have Also, since 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: If Ti ∈ B(H), i = 1, 2,..., n, then
Proof.: Let Xi = Yi = Ti and Ai = Bi = I. □
Theorem 2.10: Let Aij ∈ B(H) where 1 ≤ i, j ≤ n. Then
Proof.: For 1 ≤ x, y ≤ n where x ≠ y. Define the operator matrix Txy = [tij] where Then so □
Corollary 2.11: Let A, 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 (5)
Theorem 3.1: If z is any zero of p(z), then
Proof.: Partition C(p) as where and 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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Published Online: 2016-03-01
Published in Print: 2016-01-01
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