Let A_{i} ∈ B(H), (i = 1, 2, ..., n), and
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 T^{2} = 0; then
Also it is known that w(.) is weakly unitarily invariant, that is
A fundamental inequality for the numerical radius is the power inequality, which says that for T ∈ B(H), we have
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
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],
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.
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]). LetX_{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,
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, 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
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
LetX_{1}, X_{2}, X_{3} ∈ B(H) and 1; α, α^{2}be the roots of x^{3} = 1. Then
For
Another relation for the numerical radius of the operator
LetX_{i} ∈ B(H), (i = 1, 2, ..., n) where n ≥ 3,
An application of Theorem 2.5 yields
LetX_{1}, X_{2}, X_{3} ∈ B(H). Then
Using a straightforward technique we derive the following lemma.
LetA_{i}, X_{i} ∈ 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 A_{i}, B_{i}, X_{i}, Y_{i} ∈ B(H), i = 1, 2, ..., n. Then
where
Assume that
Applying Lemma 2.1 and Lemma 2.7 we get
Now, if we replace A_{i} and B_{i} by t A_{i} and
Also, since
Finally, replace Y_{i} by −Y_{i} to get
As an application of Theorem 2.8, we obtain the following result.
IfT_{i} ∈ B(H), i = 1, 2,..., n, then
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.
LetA_{ij} ∈ B(H) where 1 ≤ i, j ≤ n. Then
For 1 ≤ x, y ≤ n where x ≠ y. Define the operator matrix T_{xy} = [t_{ij}] where
Then
An application of Theorem 2.10 and Corollary 2.9 yields
LetA, B, C, D = ∈ B(H). Then
Let p(z) = z^{n} + a_{n}z^{n-1} + ... + a_{2}z + a_{1} be a monic polynomial of degree n ≥ 3 with complex coefficients a_{1}, a_{2}, ..., a_{n}. 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.
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