1. Introduction
Let be a complex Hilbert space, where the norm induced by the inner product is denoted by . The -algebra of bounded linear operators on H is denoted by . For any operator , its adjoint is represented as , and we define , which is the positive square root of . The real and imaginary parts of A are given by and , respectively.
The operator norm and numerical radius of
A are denoted by
and
, respectively. The operator norm is defined as
while the numerical radius is defined as
It is well known that the numerical radius
defines a norm on
, which is equivalent to the operator norm
. Specifically, the following inequalities hold
For further insights into inequalities involving norms and numerical radii, readers may refer to [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10] and the references therein.
For a pair of bounded linear operators
on
H, the spherical operator radius is given by:
As noted in [
11],
defines a norm, and the following inequalities hold:
for
, where the constants
and 1 are optimal in (
1). Additional results on the spherical operator radius can be found in [
12,
13,
14,
15] and the references therein.
Following [
16] (see also [
17]), the Davis–Wielandt radius of an operator
, denoted by
, is defined as
It is evident that , and if and only if . For any scalar , the following relations hold: when , when , and when .
Moreover, the triangle inequality
does not generally hold for all
. However, it is valid when
, as shown in Corollary 2.2 in [
18]. Additionally, it is straightforward to verify that:
Zamani and Shebrawi in Theorem 2.1 [
19] demonstrated that:
Furthermore, in Theorems 2.13, 2.14, and 2.17 [
19], the authors established the following bounds:
and
for
.
Recently, Bhunia et al. [
18] [Theorem 2.4] established the following upper bound:
for
. Additionally, in [
20], the authors obtained inequalities for the sum of operators
for
.
It is worth noting that for
and
, we have
Substituting
and
into (
1), we obtain:
which provides both the upper bound from (
3) and a corresponding lower bound.
In [
21], for
, the concept of a generalized numerical radius for a pair of operators
, known as the
p-spherical numerical radius, was introduced as follows:
This concept generalizes two important notions: the Euclidean numerical radius for
and the so-called diamond numerical radius for
. Specifically, the diamond numerical radius is defined as
In other words, the p-spherical numerical radius we tested in our manuscript interpolates Euclidean numerical radius and diamond numerical radius as special cases, so that we could give the generalized results, including the existing ones. The Davis–Wielandt radius can be derived from the Euclidean numerical radius by choosing . This connection highlights that the spherical radius for naturally extends the Davis–Wielandt radius to values beyond .
Furthermore, in the section dedicated to applications for a single operator, the spherical radius yields several meaningful special cases. These cases provide upper bounds for the power p of the numerical radius , where . This demonstrates the versatility and utility of the spherical radius in unifying and extending various classical concepts in operator theory.
In 2017, Moslehian et al. [
22] established several fundamental inequalities for the
p-spherical numerical radius of a pair of operators
, as follows:
and
The authors of [
22] also applied their findings to the Cartesian decomposition of an operator, expressed as
.
Inspired by (
4), we naturally define the Davis–Wielandt
p-radius for
as
For
, we recover the standard Davis–Wielandt radius (
2).
Building on the results mentioned above, this paper introduces new lower and upper bounds for the generalized p-spherical numerical radius and the spherical numerical radius of a pair of operators . We demonstrate that some of these bounds refine recent findings by other researchers. Additionally, we derive new inequalities for the Davis–Wielandt p-radius and the Davis–Wielandt radius as natural consequences. Applications for specific cases where and are also included.
Among other results, we establish the following lower bounds:
and
where
,
,
, and
.
We also establish the following upper bounds:
for
, and for
and
, we have
where
.
2. Main Results
In this paper, we build upon the results obtained in [
22] by providing several new lower bounds for
in terms of various families of parameters, as well as upper bounds that generalize some results from [
1,
2,
12]. Moreover, we establish new inequalities involving families of parameters for the Davis–Wielandt radius, which are related to recent results obtained in [
18,
20].
In this section, we present our primary findings. We assume that throughout this discussion. Our first result is stated as follows.
Theorem 1. Let and with and such that Thenand Proof. Using Hölder’s inequality, we determine for
and
with
that
for all
If we take the power 2 and the supremum over
, then we obtain
Similarly, we deduce that
Since
and
then by (
7) and (
8), we obtain the desired inequality (
5).
Now, if we replace
B by
and
C by
in (
5), then we obtain
Observe that
and
which gives that
and from (
9) we derive (
6). □
Remark 1. If we set, for example, and in Theorem 1, then we obtainand Similar lower bounds can be derived for other choices of the parameters , where two of them are 0 and the others are either or . These cases are left to the interested reader.
To illustrate our result in Theorem 1, we give a numerical example which may be helpful for the readers to understand our result. Take Thus, the inequalitybecomes The above inequality holds, since is decreasing and if .
Thus, the inequalitybecomes The above inequality is also true.
If we take a simpler example such as Then, the inequality (10) becomes which is true. We can confirm that the equality holds when or . Also, the inequality (10) becomes For simplicity, we consider the case with . For the case , the above inequality is true since is decreasing for and, using this, the inequality becomes . For the case , the inequality similarly becomes , which is true for .
The spherical case is of interest for applications.
Corollary 1. Let with Then,and The inequality (
12) follows by (
5) for
while (
13) is obtained from (
6) for
and observing that
Remark 2. Let and with and such that If we take and in Theorem 1, then we haveand If we take in (14), then we obtain If then by Corollary 1and If we set and in Theorem 1, then for , we deduce thatand Remark 3. If then by Corollary 1, we haveand Let and with such that If we set and in Theorem 1, thenand If then by Corollary 1, we deduce thatand We have the following trigonometric inequalities as well:
and
where
.
If we take
in (
22) and (
23), then
and
These follow by (
15) and (
16), while from (
18) and (
19), we infer that
and
From (
20) and (
21), we also obtain
and
If we take
in (
24) and (
25), then we obtain
and
If we use (
26) for
instead of
then we also obtain
while from (
27),
Remark 4. To address the sharpness question in our lower bounds above, consider, for instance, the inequalitywhich holds for all . If we set , then we obtain the inequalitywhich is sharp. Indeed, if is a self-adjoint operator, both sides of the inequality yield the same quantity . Let us emphasize that we have shown that for self-adjoint operators , equality is achieved in the bound Thus, any numerical example such aswhere are real numbers which will satisfy the equality, since this matrix is symmetric. To establish some upper bounds for
, we first need to recall Kato’s inequality [
23], which is given by
for
,
, and any
.
Theorem 2. For any and , we determine for thatand Proof. We determine by (
30) that
for any
,
and
.
By the Cauchy–Schwarz inequality, we obtain
for any
,
and
By McCarthy’s inequality for
We also have
and
for
with
Therefore, by the geometric mean-arithmetic mean inequality
for
with
By (
33)–(
35), we then obtain a sequence of inequalities as follows
for
with
By using (
36), we deduce the desired results (
31) and (
32). □
We also have the following theorem.
Theorem 3. For any , , we determine for and that Proof. For
, we also determine for
with
that
Observe that for
and, similarly
By (
38), we then obtain
for
with
From (
33), (
34) and (
39), we obtain
for
with
By taking the supremum over
with
in the last inequality, we obtain (
37). □
Also, recall the following result for operator matrices, established by F. Kittaneh in [
24].
Lemma 1. Let with Then, the operator matrixis positive, if and only iffor all , Using this lemma, we can also state the following result.
Theorem 4. Let with , , andare positive. Then, for and Proof. From Lemma 1, we have
and
for all
Therefore, we obtain
for all
We have
which proves the first inequality in (
40).
By McCarthy’s inequality, we also have for
that
and
giving that
which proves the second part of (
40).
We also observe from the above that for
which, by taking the supremum over
, gives (
41). □
Remark 5. Under the assumptions of Theorem 4, we observe that for , we obtainand For , we obtain the following inequalities for the spherical numerical radius:and Corollary 2. For and , we haveand Proof. Observe that the operator matrices
and
are positive. The proof follows now by Theorem 4 by taking
and
□
Remark 6. Under the assumptions of Corollary 2, we observe that for , we obtainand From a different perspective, we also have the following theorem:
Theorem 5. Let with andare positive. Then,where and In particular, the following inequality for the spherical numerical radius holds Proof. We also determine for
and
that
Observe that for
and similarly,
for
,
By (
43), we obtain
for
,
By taking the supremum over
we obtain
which proves (
42). □
We can also state the following corollary.
Corollary 3. For and , we havewhere and In particular, the following inequality for the spherical numerical radius holds: Proof. The proof now follows by Theorem 5 by taking and □
Now, let us move on to some applications for the pair
. We note that for
, with
and
, we have
By (
5), we then obtain for
and
with
and such that
that
If
with
then applying Corollary 1 with
and
, we obtain
and
Let
and
with
and such that
By (
14), we obtain for
,
that
while by (
17),
If
and
, we obtain from (
18) and (
19) that
and
Also, if
then by (
20) and (
21),
and
From (
27), we obtain
while from (
29)
since the operators
,
,
and
are normal operators.
For any
,
, and
, we have by Theorem 2 with
and
that
and
The case
gives for
that
and
By Theorem 3, for any
,
,
, and
, we have
In particular, for
,
From Corollary 3 for
and
, we obtain
and
Now, if we take
and
, we obtain
From Corollary 3 we obtain for
and
that
where
and
Now, if we take
and
then
where
and
Our focus now shifts to presenting some applications for the Davis–Wielandt p-radius.
Let
and
with
and such that
Then by (
5) for
, we obtain
Let
and
with
and such that
Then, by (
14), we have
and, in particular,
Let
with
Then, by Corollary 1 for
, we obtain the following lower bounds for the Davis–Wielandt radius:
and
If
then by (
15) and (
16), we obtain for
that
and
From (
26), we derive that
while from (
28), we obtain
Further, we can provide the following upper bounds.
From Theorem 2 for
, we obtain for any
,
and
that
and
For
, we obtain
and
for
In particular, for
, we derive
and
For
, we have the following upper bounds for the Davis–Wielandt radius
and
for
In particular, for
, we obtain
and
From Theorem 3, for any
and
, we have, for
and
, that
which for
gives
For
, we obtain
and in particular,
We conclude this paper with some applications for the pair .
Let
and
. For the pair
, we consider the functional
Observe that and if and only if .
Also, by the properties of the
r-spherical numerical radius, we have for
that
and for
, we have
which shows that
is a norm on
For
, we observe that
Let
and
with
and such that
Then, by (
5) for
, we have
Let
and
with
and such that
By (
14), we obtain
and, in particular for
If
, then by (
17), we obtain
Let
with
Then, by Corollary 1 with
, we obtain
and
If
then by (
15) and (
16), we also have
and
Also, by (
18) and (
19), we obtain
and
From (
26), we obtain, by taking
, that
while, from (
28), we have
We determine for
and
from Theorem 2 that
and
For
, we obtain
and
for
If we take
then we have
For
, we obtain
and
for
If we take
then we obtain
From Theorem 3 we obtain for
,
and
that
For
, we derive that
for
which gives for
that
For
, we obtain
for
which gives for
that
3. Conclusions
As we have seen, the authors have shown the lower and upper bounds for the generalized
r–numerical radius
for a pair of the bounded linear operators
B and
C. It is natural to consider the bounds of the following
r–numerical radius
introduced in [
22] for
n–tuples of the bounded linear operators
:
Then we can obtain the following bounds of by the usual numerical radius by using a proof similar to that of Theorems 1 and 2.
Proposition 1. Let with and such that for any . For any and , Proof. For any
, by using the same method as in the proof of Theorem 1,
Taking the supremum over with , we obtain the result. □
Proposition 2. Let and . For any , Proof. We calculate by the same method as that used in the proof of Theorem 2,
Taking the power on both sides and the supremum over with , we obtain the result. □
We consider the case
. To this end, we review that the following inequalities hold (e.g., [
25] [Theorem 2.11]):
for every unit vector
and a positive operator
A such that
for positive scalars
, where
is often called the generalized Kantorovich constant and is given by
Then, we can obtain the following result.
Proposition 3. Let and . For any such that and for some scalar and all . Then, Proof. We calculate by a method similar to that used in the proof of Proposition 2,
In the third inequality above, we used the first inequality in (
44). Taking the power
on both sides and the supremum over
with
, we obtain the result. □
From Proposition 2, we see that
In [
22], the following upper bounds have been established for
:
and
It may be of interest to compare (
45) with (
46) and (
47). By the convexity of the function
for
on
, we have
From this, the inequality (
46) gives a better upper bound than our inequality (
45).
Also, we easily observe that our inequality (
45) coincides with the inequality (
47) when
. In addition, there is no general ordering between
and
for
and
. Of course, our inequality (
45) and the inequality (
47) have different conditions regarding
p, so they should not be directly compared. However, we believe that there is no superiority or inferiority between the upper bounds in our inequality (
45) and the inequality (
47) for
.
However, we would like to emphasize that the important point in the main results is that we precisely derive upper and lower bounds for the pair of operators.