1. Introduction
Let be the Banach algebra of all bounded linear operators defined on a complex Hilbert space with the identity operator in . When , we identify with the algebra of n-by-n complex matrices. The cone of n-by-n positive semidefinite matrices is then . This is adopted for all matrices, whether self-adjoint (symmetric) or not.
The numerical range
of a bounded linear operator
T on a Hilbert space
is the image of the unit sphere of
associated with the operator under the quadratic form
. More precisely, we have
Furthermore, the numerical radius is
The spectral radius of an operator
T is indicated as
We recall that the usual operator norm of an operator
T is defined as
and
It is well-known that the numerical radius is not submultiplicative, but it satisfies
for all
. In particular, if
T and
S commute, then
Moreover, if T and S are normal, then is submultiplicative .
The absolute value of the operator T is denoted by . Then we have It is convenient to mention that the numerical radius norm is weakly unitarily invariant, i.e., for all unitary U. Furthermore, let us not miss the chance to mention the important properties that and for every .
A popular problem is the following: does the numerical radius of the product of operators commute, i.e., for any operators ?
This problem has been given serious attention by many authors and in several resources (see [
1], for example). Fortunately, it has been shown recently that for any bounded linear operators
,
and
always have the same numerical radius for all rank one
if and only if
is a multiple of a unitary operator for some
. This fact was proved by Chien et al. in [
2]. For other related problems involving numerical ranges and radiuses, see [
2,
3] as well as the elegant work of Li [
4] and the references therein. For more classical and recent properties of numerical range and radiuses, see [
2,
3,
4] and the comprehensive books [
5,
6,
7].
On the other hand,
is well-known to define an operator norm on
, which is equivalent to the operator norm
. Moreover, we have
for any
. The inequality is sharp.
In 2003, Kittaneh [
8] refined the right-hand side of (
1), where he proved that
for any
.
After that, in 2005, the same author in [
9] proved that
These inequalities were also reformulated and generalized in [
10] but in terms of Cartesian decomposition. Both of them have been generalized recently in [
11,
12], respectively.
In 2007, Yamazaki [
13] improved (
1) by proving that
where
with unitary
U.
In 2008, Dragomir [
14] used the Buzano inequality to improve (
1), where he proved that
This result was also recently generalized by Sattari et al. in [
15]. This result was also recently generalized by Sattari et al. in [
15] and Alomari in [
16,
17,
18,
19]. For more recent results about the numerical radius, see the recent monograph study in [
14,
20,
21,
22].
According to the Schwarz inequality for positive operators, for any positive operator
A in
, we have
for any vectors
.
In 1951, Reid [
23] proved an inequality, which in some senses considered a variant of the Schwarz inequality. In fact, he proved that for all operators
such that
A is positive and
is self-adjoint, then
for all
. In [
24], Halmos presented his stronger version of the Reid inequality (
7) by substituting
for
.
In 1952, Kato [
25] introduced a companion inequality of (
6), called the mixed Schwarz inequality, which asserts
for every operators
and any vectors
, where
.
In 1988, Kittaneh [
26] proved a very interesting extension combining both the Halmos–Reid Inequality (
2) and the mixed Schwarz Inequality (
3). His result says that
for any vectors
, where
such that
and
are nonnegative continuous functions defined on
satisfying that
. Clearly, if we choose
and
with
, then we may refer to (
8). Moreover, choosing
, some manipulations refer to the Halmos version of the Reid inequality. The cartesian decomposition form of (
9) was recently proved by Alomari in [
16].
In 1994, Furuta [
27] proved another attractive generalization of Kato’s inequality (
3), as follows:
for any
and
with
.
The inequality (
5) was generalized for any
with
by Dragomir in [
22]. Indeed, as noted by Dragomir, the condition
was assumed by Furuta to fit with the Heinz–Kato inequality, which reads:
for any
and
, where
A and
B are positive operators such that
and
for any
.
In the same work [
22], Dragomir provides a useful extension of Furuta’s inequality, as follows:
for any
and any vectors
. The equality in (
11) holds iff the vectors
and
are linearly dependent in
.
Indeed, since
and
, the Inequality (
11) can be rewritten as
If one setting
(
U is unitary),
,
and
such that
, then we recapture (
10).
Based on the most recent Dragomir extension of Furuta’s inequality, various numerical radius inequalities are derived in this paper. Additionally, several specific examples are given.
The rest of the paper is composed of the following sections:
Section 2 presents some crucial lemmas. Numerical radius inequalites are determined and proved in
Section 3. The conclusion is made in
Section 4.
3. Numerical Radius Inequalities
In this section, we provide some numerical radius inequalities. Let us begin with the following key result.
Theorem 1. Let . Let f be a positive, increasing and convex function on . If f is twice differentiable such that , thenwhere . Proof. Let
in (
17), then we obtain
Taking the supremum over all unit vectors , we obtain the required result. □
Corollary 4. Let . Then we have Proof. Take in Theorem 1, in such a way that the required would be ‘2’. □
Corollary 5. Let . Let f be a positive, increasing and convex function on . If f is twice differentiable such that , then we havewhere , for all such that . Proof. Let
in (
18), we obtain
Taking the supremum over all unit vectors , we obtain the required result. □
Corollary 6. Let . Let f be a positive, increasing and convex function on . If f is twice differentiable such that , then we havewhere . Proof. Setting
and
in (
25), we establish the stated result. □
Corollary 7. Let . Let f be a positive, increasing and convex function on . If f is twice differentiable such that , then we havewhere . Proof. Setting
and
in (
25), we obtain the desired result. □
Corollary 8. Let . Let f be a positive, increasing and convex function on . If f is twice differentiable such that , then we havewhere . Proof. Setting
in (
25), the desired result follows. □
Theorem 2. Let . Let f be a positive, increasing, convex and supermultiplicative function on , i.e., for all . Then, we have For all such that and all such that , where Proof. Let
in (
19), we obtain
Taking the supremum over all unit vectors , we obtain the required result. □
Corollary 9. Let . Let f be a positive, increasing, convex and supermultiplicative function on , i.e., for all . Then we have For all such that and all such that , where Proof. Let
in (
20), and then taking the supremum over all unit vectors
, we obtain the required result. □
Corollary 10. Let . Then we havefor all such that , wherefor all such that . Proof. Applying Corollary 9 for the convex increasing function , , we obtain the stated result. □
Remark 1. In (29), let , we obtainfor all such that , where In particular, for , we havefor all . Example 1. Let . Applying (31) with , simple calculations yield that , and . Thus, we havewhich means that (31) is a non-trivial improvement of the right-hand side of (10). Theorem 3. Let . Let f be a positive, increasing, convex and supermultiplicative function on , i.e., for all . Then we havewhere As a special case, we have Proof. Let
in (
21), we obtain
Taking the supremum over all unit vectors
, we obtain the required result. The particular case follows by setting
in (
22) and then taking the supremum over all unit vectors
. □
Corollary 11. Let . Then, we havewhere In this particular case, we have Proof. Applying Theorem 3 for , we obtain the required result. □
Corollary 12. Let . Let f be a positive, increasing, convex and supermultiplicative function on , i.e., for all . Then we havewhere Proof. The proof follows by considering
,
,
and
such that
in (
32). □
Corollary 13. Let . Then we havefor all such that , where Proof. Setting in Corollary 12, we obtain the required result. □
Remark 2. By choosing in (37), we obtainfor all . Furthermore, for in (38), we obtainfor all . In general, for in (38), we havefor all . In particular, for , we havewhich refines the right-hand side of (10), where we have used the fact thatfor every non-negative convex function f and all positive operators (see [32]), in the second inequality above. Example 2. Let . Applying (39), simple calculations yield that , , , and . Thus, we havewhich means that (39) is a non-trivial improvement of the right-hand side of the celebrated Kittaneh Inequality (10). The numerical radius inequality of special type of Hilbert space operators for commutators can be established as follows:
Lemma 9. Let . Then, for all , the following inequality:holds for all vectors . Proof. Employing the triangle inequality and the Inequality (
6), we have
for all vectors
, which proves the result. □
Corollary 14. Let . Then, the following inequality:holds for all . Proof. Let
in (
40) and then taking the supremum over all unit vectors
, we obtain the mentioned result. □
Corollary 15. Let . Then we havefor all vectors . Proof. Let
in (
40) and consider
. In the proof of (
42), combining the inner products, then taking the supremum over all unit vectors
, we obtain the required result. □
In special cases, a particular choice of in the Corollaries 14 and 15 would give the following result:
Corollary 16. Let , such that and . Then we havefor all . Proof. Let
,
,
and
such that
in (
42), then we have
also, we have
and
. □
Corollary 17. Let , such that and . Then we have Proof. It is enough to consider
,
,
and
such that
in (
42). □
Remark 3. Setting in (44), we obtain In particular, take , we obtain Example 3. Let . Applying (45), simple calculations yield that , , and . Thus, we havewhich means that (45) is a non-trivial improvement of the celebrated Kittaneh Inequality (2). Remark 4. Setting in (45), we obtain In particular, take , we obtain