Some refinements of numerical radius inequalities
UDC 517.5 In this paper, we give some refinements for the second inequality in $\dfrac{1}{2}\|A\| \leq w(A) \leq \|A\|,$  where $A\in B(H).$  In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot),...
Saved in:
| Date: | 2020 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/6027 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512231311015936 |
|---|---|
| author | Heydarbeygi, Z. Amyari, M. Khanehgir, M. Heydarbeygi, Z. Amyari, M. Khanehgir, M. A. |
| author_facet | Heydarbeygi, Z. Amyari, M. Khanehgir, M. Heydarbeygi, Z. Amyari, M. Khanehgir, M. A. |
| author_sort | Heydarbeygi, Z. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:43Z |
| description | UDC 517.5
In this paper, we give some refinements for the second inequality in $\dfrac{1}{2}\|A\| \leq w(A) \leq \|A\|,$  where $A\in B(H).$  In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot),$  we show that $w(A)\leq \dfrac{1}{\displaystyle {2\inf\nolimits_{\| x \|=1}}\zeta(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|,$  where $\zeta(x)=K\left(\dfrac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2\right)^{r},$ $r=\min\{\lambda,1-\lambda\}$ and $0\leq \lambda \leq 1$ . We also give a reverse for the classical numerical radius power inequality $w(A^{n})\leq w^{n}(A)$ for any operator $A \in B(H)$ in the case when $n=2.$ 
|
| doi_str_mv | 10.37863/umzh.v72i10.6027 |
| first_indexed | 2026-03-24T03:25:30Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i10.6027
UDC 517.5
Z. Heydarbeygi, M. Amyari, M. Khanehgir (Mashhad Branch, Islamic Azad Univ., Iran)
SOME REFINEMENTS OF NUMERICAL RADIUS INEQUALITIES
ДЕЯКI УТОЧНЕННЯ НЕРIВНОСТЕЙ ДЛЯ ЧИСЛОВИХ РАДIУСIВ
In this paper, we give some refinements for the second inequality in
1
2
\| A\| \leq w(A) \leq \| A\| , where A \in B(H). In
particular, if A is hyponormal by refining the Young inequality with the Kantorovich constant K(\cdot , \cdot ), we show that
w(A) \leq 1
2 \mathrm{i}\mathrm{n}\mathrm{f}\| x\| =1\zeta (x)
\| | A| + | A\ast | \| \leq 1
2
\| | A| + | A\ast | \| , where \zeta (x) = K
\biggl(
\langle | A| x, x\rangle
\langle | A\ast | x, x\rangle , 2
\biggr) r
, r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \lambda , 1 - \lambda \} and
0 \leq \lambda \leq 1 . We also give a reverse for the classical numerical radius power inequality w(An) \leq wn(A) for any operator
A \in B(H) in the case when n = 2.
Запропоновано деякi уточнення другої нерiвностi у
1
2
\| A\| \leq w(A) \leq \| A\| , де A \in B(H). Зокрема, якщо A
є гiпонормальним, то за допомогою нерiвностi Юнга з константою Канторовича K(\cdot , \cdot ) доведено, що w(A) \leq
\leq 1
2 \mathrm{i}\mathrm{n}\mathrm{f}\| x\| =1\zeta (x)
\| | A| + | A\ast | \| \leq 1
2
\| | A| + | A\ast | \| , де \zeta (x) = K
\biggl(
\langle | A| x, x\rangle
\langle | A\ast | x, x\rangle , 2
\biggr) r
, r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \lambda , 1 - \lambda \} i 0 \leq \lambda \leq 1.
Також доведено нерiвнiсть для числових радiусiв, що є оберненою до класичної степеневої нерiвностi w(An) \leq
\leq wn(A) для будь-якого оператора A \in B(H) у випадку n = 2.
1. Introduction. Suppose that (H, \langle \cdot , \cdot \rangle ) is a complex Hilbert space and B(H) denotes the C\ast -
algebra of all bounded linear operators on H. For A \in B(H), let w(A) and \| A\| denote the
numerical radius and the usual operator norm of A, respectively. It is well-known that w(\cdot ) defines
a norm on B(H), which is equivalent to the usual operator norm \| \cdot \| . In fact, for every A \in B(H),
1
2
\| A\| \leq w(A) \leq \| A\| . (1.1)
An important inequality for w(A) is the power inequality stating that
w(An) \leq wn(A) (1.2)
for each n \in \BbbN . Many authors have investigated several inequalities involving numerical radius
inequalities (see, e.g., [1, 5, 6, 8, 13, 14]). If x, y \in H are arbitrary, then the angle between x and
y is defined by
\mathrm{c}\mathrm{o}\mathrm{s}\phi x,y =
\mathrm{R}\mathrm{e}\langle x, y\rangle
\| x\| \| y\|
or by
\mathrm{c}\mathrm{o}\mathrm{s}\psi x,y =
| \langle x, y\rangle |
\| x\| \| y\|
.
The following inequality for angles between two vectors was obtained by Krein [11]
\phi x,z \leq \phi x,y + \phi y,z (1.3)
for any nonzero elements x, y, z \in H. By using the representation
c\bigcirc Z. HEYDARBEYGI, M. AMYARI, M. KHANEHGIR, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10 1443
1444 Z. HEYDARBEYGI, M. AMYARI, M. KHANEHGIR
\psi x,y = \mathrm{i}\mathrm{n}\mathrm{f}
\lambda ,\mu \in \BbbC - \{ 0\}
\phi \lambda x,\mu y = \mathrm{i}\mathrm{n}\mathrm{f}
\lambda \in \BbbC - \{ 0\}
\phi \lambda x,y = \mathrm{i}\mathrm{n}\mathrm{f}
\mu \in \BbbC - \{ 0\}
\phi x,\mu y
and inequality (1.3), he showed that the following triangle inequality is valid:
\psi x,y \leq \psi x,z + \psi y,z (1.4)
for any nonzero elements x, y, z \in H.
In Section 2, we first introduce some new refinements of numerical radius inequality (1.1) by
applying the Krein – Lin triangle inequality (1.3) and obtain a reverse of inequality (1.2) in the case
when n = 2. In Section 3, we obtain some refinements of inequality (1.1) by applying a refinement
of the Young inequality.
2. Some refinements of inequality (1.1) by Krein – Lin triangle inequality. In order to achieve
our goals, we need the following lemmas. The first lemma is a simple consequence of the classical
Jensen and Young inequalities.
Lemma 2.1 ([12], Lemma 2.1). Let a, b \geq 0 and 0 \leq \lambda \leq 1. Then
a\lambda b1 - \lambda \leq \lambda a+ (1 - \lambda )b \leq [\lambda ar + (1 - \lambda )br]
1
r
for any r \geq 1.
The second lemma is a simple consequence of the classical Jensen inequality for convex function
f(t) = tr, where r \geq 1.
Lemma 2.2. If a and b are nonnegative real numbers, then
(a+ b)r \leq 2r - 1(ar + br)
for any r \geq 1.
Lemma 2.3 ([4], Lemma 2.4). Suppose that x, y \in H with \| y\| = 1. Then
\| x\| 2 - | \langle x, y\rangle | 2 = \mathrm{i}\mathrm{n}\mathrm{f}
\lambda \in \BbbC
\| x - \lambda y\| 2.
The following lemma is known as a generalized mixed Schwarz inequality.
Lemma 2.4 ([12], Lemma 2.3). Let A \in B(H) and x, y \in H be two vectors.
(i) If 0 \leq \lambda \leq 1, then
| \langle Ax, y\rangle | 2 \leq \langle | A| 2\lambda x, x\rangle \langle | A\ast | 2(1 - \lambda )y, y\rangle .
(ii) If f and g are nonnegative continuous functions on [0,\infty ) satisfying f(t)g(t) = t, then
| \langle Ax, y\rangle | \leq \| f(| A| )x\| \| g(| A\ast | )y\| .
In the next result, we use some ideas of [3].
Theorem 2.1. Let A \in B(H) and f, g be nonnegative continuous functions on [0,\infty ) satisfying
f(t)g(t) = t. Then, for r \geq 1,
w2r(A) \leq 1
2r
\biggl(
\| f2(| A2| ) + g2(| (A2)\ast | )\| r + 2r \mathrm{i}\mathrm{n}\mathrm{f}
\lambda \in \BbbC
\| A - \lambda I\| 2r
\biggr)
. (2.1)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
SOME REFINEMENTS OF NUMERICAL RADIUS INEQUALITIES 1445
Proof. By (1.4), we get the inequality (9) of [2] as follows:
| \langle x, z\rangle |
\| x\| \| z\|
| \langle y, z\rangle |
\| y\| \| z\|
\leq | \langle x, y\rangle |
\| x\| \| y\|
+
\sqrt{}
1 - | \langle x, z\rangle | 2
\| x\| 2\| z\| 2
\sqrt{}
1 - | \langle y, z\rangle | 2
\| y\| 2\| z\| 2
(2.2)
for any x, y, z \in H \setminus \{ 0\} .
If we multiply (2.2) by \| x\| \| y\| \| z\| 2, then we deduce
| \langle x, z\rangle | | \langle y, z\rangle | \leq | \langle x, y\rangle | \| z\| 2 +
\sqrt{}
\| x\| 2\| z\| 2 - | \langle x, z\rangle | 2
\sqrt{}
\| y\| 2\| z\| 2 - | \langle y, z\rangle | 2. (2.3)
Applying Lemma 2.3 for any x, y, z \in H with \| z\| = 1, we obtain
| \langle x, z\rangle | | \langle y, z\rangle | \leq | \langle x, y\rangle | + \mathrm{i}\mathrm{n}\mathrm{f}
\lambda \in \BbbC
\| x - \lambda z\| \mathrm{i}\mathrm{n}\mathrm{f}
\mu \in \BbbC
\| y - \mu z\| . (2.4)
Put x = Az, y = A\ast z in (2.4) to get
| \langle Az, z\rangle | 2 \leq | \langle A2z, z\rangle | + \mathrm{i}\mathrm{n}\mathrm{f}
\lambda \in \BbbC
\| Az - \lambda z\| \mathrm{i}\mathrm{n}\mathrm{f}
\mu \in \BbbC
\| A\ast z - \mu z\| \leq
\leq | \langle A2z, z\rangle | + \| Az - \lambda z\| \| A\ast z - \mu z\| (2.5)
for any z \in H with \| z\| = 1 and \lambda , \mu \in \BbbC .
On the other hand, by applying Lemma 2.4 and the AM-GM inequality, we have
| \langle A2z, z\rangle | \leq \| f(| A2| )z\| \| g(| (A2)\ast | )z\| =
=
\sqrt{}
\langle f2(| A2| )z, z\rangle \langle g2(| (A2)\ast | )z, z\rangle \leq
\leq 1
2
\langle (f2(| A2| ) + g2(| (A2)\ast | ))z, z\rangle . (2.6)
Applying again the AM-GM inequality, we get
\| Az - \lambda z\| \| A\ast z - \mu z\| \leq \| Az - \lambda z\| 2 + \| A\ast z - \mu z\| 2
2
. (2.7)
By combining inequalities (2.5), (2.6) and (2.7), we obtain
| \langle Az, z\rangle | 2 \leq 1
2
\bigl(
\langle (f2(| A2| ) + g2(| (A2)\ast | ))z, z\rangle + \| Az - \lambda z\| 2 + \| A\ast z - \mu z\| 2
\bigr)
\leq
\leq 1
2
\bigl(
\langle (f2(| A2| ) + g2(| (A2)\ast | ))z, z\rangle r + (\| Az - \lambda z\| 2 + \| A\ast z - \mu z\| 2)r
\bigr) 1
r (\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 2.1) \leq
\leq 1
2
\bigl(
\langle (f2(| A2| ) + g2(| (A2)\ast | ))z, z\rangle r + 2r - 1(\| Az - \lambda z\| 2r + \| A\ast z - \mu z\| 2r)
\bigr) 1
r (\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 2.2).
Hence
| \langle Az, z\rangle | 2r \leq 1
2r
\bigl(
\langle (f2(| A2| ) + g2(| (A2)\ast | ))z, z\rangle r + 2r - 1(\| Az - \lambda z\| 2r + \| A\ast z - \mu z\| 2r)
\bigr)
.
By taking the supremum over z \in H with \| z\| = 1, we deduce
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1446 Z. HEYDARBEYGI, M. AMYARI, M. KHANEHGIR
w2r(A) \leq 1
2r
\bigl(
\| f2(| A2| ) + g2(| (A2)\ast | )\| r + 2r - 1(\| A - \lambda I\| 2r + \| A\ast - \mu I\| 2r)
\bigr)
for any \lambda , \mu \in \BbbC .
Finally, taking the infimum over \lambda , \mu \in \BbbC in the inequality above and utilizing
\mathrm{i}\mathrm{n}\mathrm{f}
\mu \in \BbbC
\| A\ast - \mu I\| = \mathrm{i}\mathrm{n}\mathrm{f}
\mu \in \BbbC
\| A - \mu I\| = \mathrm{i}\mathrm{n}\mathrm{f}
\lambda \in \BbbC
\| A - \lambda I\|
we obtain the result (2.1).
Theorem 2.1 is proved.
Remark 2.1. In Theorem 2.1 if we choose r = 1, f(t) = g(t) =
\surd
t, we get
w2(A) \leq 1
2
\biggl(
\| | A2| + | (A2)\ast | \| + 2 \mathrm{i}\mathrm{n}\mathrm{f}
\lambda \in \BbbC
\| A - \lambda I\| 2
\biggr)
.
Now, suppose that s > 0 such that s \leq
\sqrt{}
\| A\| 2 - 1
2
\| | A2| + | (A2)\ast | \| , if there is \lambda 0 \in \BbbC in
which \| A - \lambda 0I\| \leq s, then w(A) \leq
\sqrt{}
1
2
\| | A2| + | (A2)\ast | \| + s2 \leq \| A\| , that is an improvement of
inequality (1.1) for nonnormal operators.
Recall that if A \in M2(\BbbR ), then \| A\| = \mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n \sigma i, where \sigma \prime is are the square root of eigen-
values of A\ast A, which are called the singular values of A, and w(A) for matrix of the form
A =
\biggl[
a1 b
0 a2
\biggr]
or A =
\biggl[
a1 0
b a2
\biggr]
is defined by
w(A) =
1
2
| a1 + a2| +
1
2
\sqrt{}
| a1 - a2| 2 + | b| 2,
where a1, a2, b \in \BbbR .
Example 2.1. By taking A =
\Biggl[
1
1
2
0 1
\Biggr]
and \lambda 0 =
1
2
in Remark 2.1, we have w2(A) \simeq 1.5625,
\| A\| 2 \simeq 3.2822, \| A - \lambda 0I\| \simeq 0.5201 and
1
2
\| | A2| + | (A2)\ast | \| \simeq 1.5652. If s2 \leq \| A\| 2 - 1
2
\| | A2| +
+| (A\ast )2| \| \simeq 1.7170, then s \leq 1.3103. Hence, inequality w(A) \leq
\sqrt{}
1
2
\| | A2| + | (A2)\ast | \| + s2 \leq \| A\|
provides an improvement of inequality (1.1).
Remark 2.2. If there exists \lambda 0 \in \BbbC in which \| A - \lambda 0I\| \leq s, then by putting \lambda = \mu = \lambda 0 and
by taking supremum over z \in H with \| z\| = 1 in (2.5), we deduce
w2(A) - w(A2) \leq \| A - \lambda 0I\| \| A\ast - \lambda 0I\| .
Therefore
w2(A) - w(A2) \leq s2.
Now, if \| A - \lambda 0I\| \leq s \leq
\sqrt{}
\| A\| 2 - w(A2), we have w(A) \leq
\sqrt{}
w(A2) + s2 \leq \| A\| , that is an
improvement of inequality (1.1).
Example 2.2. By taking A =
\biggl[
2 - 1
0 3
\biggr]
and \lambda 0 = 2.5 in Remark 2.2, we have w(A2) \simeq
\simeq 6.4142, \| A\| 2 \simeq 10.6054, \| A - \lambda 0I\| \simeq 0.955 for s \leq
\sqrt{}
\| A\| 2 - w(A2) \simeq 2.0472. Hence,
inequality w(A) \leq
\sqrt{}
w(A2) + s2 \leq \| A\| provides an improvement of inequality (1.1).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
SOME REFINEMENTS OF NUMERICAL RADIUS INEQUALITIES 1447
Recall that the vector x \in H is orthogonal to y \in H (denote by x \bot y), if \langle x, y\rangle = 0. Now, an
argument similar to the proof of Theorem 2.1 with the aid of Lemmas 2.1 and 2.3 gives the following
proposition.
Proposition 2.1. Let x, y, z \in H with \| z\| = 1 and \lambda , \mu \in \BbbC , a, b > 0, and r \geq 1 such that
\| x - \lambda z\| \leq a, \| y - \mu z\| \leq b.
Then
(| \langle x, z\rangle | | \langle y, z\rangle | - | \langle x, y\rangle | )r \leq a2r + b2r
2
. (2.8)
In particular, if x \bot y, then
(| \langle x, z\rangle | | \langle y, z\rangle | )r \leq a2r + b2r
2
(2.9)
for any r \geq 1.
Proof. Since z is a unit vector, from (2.3) we have
| \langle x, z\rangle | | \langle y, z\rangle | - | \langle x, y\rangle | \leq
\sqrt{}
\| x\| 2 - | \langle x, z\rangle | 2
\sqrt{}
\| y\| 2 - | \langle y, z\rangle | 2 \leq
\leq 1
2
(\| x\| 2 - | \langle x, z\rangle | 2 + \| y\| 2 - | \langle y, z\rangle | 2) (\mathrm{b}\mathrm{y} \mathrm{A}\mathrm{M}-\mathrm{G}\mathrm{M} \mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}) =
=
1
2
\biggl(
\mathrm{i}\mathrm{n}\mathrm{f}
\lambda \in \BbbC
\| x - \lambda z\| 2 + \mathrm{i}\mathrm{n}\mathrm{f}
\mu \in \BbbC
\| y - \mu z\| 2
\biggr)
(\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 2.3) \leq
\leq 1
2
\bigl(
\| x - \lambda z\| 2 + \| y - \mu z\| 2
\bigr)
\leq
\leq a2 + b2
2
\leq
\biggl(
a2r + b2r
2
\biggr) 1
r
(\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 2.1).
Hence
(| \langle x, z\rangle | | \langle y, z\rangle | - | \langle x, y\rangle | )r \leq a2r + b2r
2
.
Proposition 2.1 is proved.
Corollary 2.1. Let A \in B(H) and B be a nonzero self-adjoin element in B(H), under assump-
tions of Proposition 2.1, if we choose x = Az and y = Bz with \| z\| = 1 in (2.9) yields
(| \langle Az, z\rangle | | \langle Bz, z\rangle | )r \leq a2r + b2r
2
.
By taking supremum over z \in H with \| z\| = 1, we get
wr(A) \leq a2r + b2r
2
\| B\| - r
provided \| A - \lambda I\| \leq a, \| B - \mu I\| \leq b and for any r \geq 1 and a, b > 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1448 Z. HEYDARBEYGI, M. AMYARI, M. KHANEHGIR
Proposition 2.1 induces several inequalities as special cases, but here we only focus on the case
r = 1, i.e.,
| \langle x, z\rangle | | \langle y, z\rangle | \leq a2 + b2
2
+ | \langle x, y\rangle | , (2.10)
whenever \| x - \lambda z\| \leq a, \| y - \mu z\| \leq b with \| z\| = 1 and \lambda , \mu \in \BbbC .
Remark 2.3. Suppose that the assumptions of Proposition 2.1 are still valid.
As an application of inequality (2.10) the following reverse of inequality (1.2) for n = 2, i.e., an
upper bound for w2(A) - w(A2) can be obtained. In fact, by choosing x = Az and y = A\ast z with
\| z\| = 1 and taking supremum over z \in H with \| z\| = 1, we get
w2(A) - w(A2) \leq a2 + b2
2
provided \| A - \lambda I\| \leq a, \| A\ast - \mu I\| \leq b.
By choosing x = Az and y = A - 1z with \| z\| = 1, in inequality (2.10) and taking supremum
over z \in H with \| z\| = 1, we have
K(A; z) - 1 \leq a2 + b2
2
provided \| A - \lambda I\| \leq a, \| A - 1 - \mu I\| \leq b, where K(A; z) = \langle Az, z\rangle \langle A - 1z, z\rangle is the Kantorovich
functional.
3. Some refinements of inequality (1.1) by using Young’s inequality. In this section, we
obtain some refinements of inequality (1.1) by applying refinements of the Young inequality. The
next lemma is an additive refinement of the scalar Young inequality.
Lemma 3.1 ([9], Theorem 2.1). If a, b \geq 0 and 0 \leq \lambda \leq 1, then
a\lambda b1 - \lambda + r(
\surd
a -
\surd
b)2 \leq \lambda a+ (1 - \lambda )b,
where r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \lambda , 1 - \lambda \} .
The main result of this section reads as follows.
Theorem 3.1. If A \in B(H), r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \lambda , 1 - \lambda \} , where 0 \leq \lambda \leq 1, then
w(A) \leq 1 - 2r
2
\| | A| + | A\ast | \| + 2r\| A\| .
Proof. Let x \in H be a unit vector. Then we have
| \langle Ax, x\rangle | \leq
\sqrt{}
\langle | A| x, x\rangle \langle | A\ast | x, x\rangle (\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 2.4) =
=
\Bigl(
\langle | A| x, x\rangle 1 - \lambda \langle | A\ast | x, x\rangle \lambda
\Bigr) 1
2
\Bigl(
\langle | A\ast | x, x\rangle 1 - \lambda \langle | A| x, x\rangle \lambda
\Bigr) 1
2 \leq
\leq 1
2
\Bigl(
\langle | A| x, x\rangle 1 - \lambda \langle | A\ast | x, x\rangle \lambda + \langle | A\ast | x, x\rangle 1 - \lambda \langle | A| x, x\rangle \lambda
\Bigr)
(\mathrm{b}\mathrm{y} \mathrm{A}\mathrm{M}-\mathrm{G}\mathrm{M} \mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}) \leq
\leq 1
2
\Bigl(
(1 - \lambda )\langle | A| x, x\rangle + \lambda \langle | A\ast | x, x\rangle - r(
\sqrt{}
\langle | A| x, x\rangle -
\sqrt{}
\langle | A\ast | x, x\rangle )2+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
SOME REFINEMENTS OF NUMERICAL RADIUS INEQUALITIES 1449
+(1 - \lambda )\langle | A\ast | x, x\rangle + \lambda \langle | A| x, x\rangle - r(
\sqrt{}
\langle | A| x, x\rangle -
\sqrt{}
\langle | A\ast | x, x\rangle )2
\Bigr)
(\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 3.1) =
=
1
2
\Bigl(
\langle (| A| + | A\ast | )x, x\rangle - 2r\langle (| A| + | A\ast | )x, x\rangle ) + 4r
\sqrt{}
\langle | A| x, x\rangle \langle | A\ast | x, x\rangle
\Bigr)
,
so
| \langle Ax, x\rangle | + r\langle (| A| + | A\ast | )x, x\rangle \leq 1
2
(\langle (| A| + | A\ast | )x, x\rangle + 4r
\sqrt{}
\langle | A| x, x\rangle \langle | A\ast | x, x\rangle ).
By taking supremum over x \in H with \| x\| = 1, we deduce
w(A) \leq 1 - 2r
2
\| | A| + | A\ast | \| + 2r\| A\| ,
which is an improvement of inequality (1.1).
Theorem 3.1 is proved.
Example 3.1. Let A =
\biggl[
1 1
0 2
\biggr]
be as in Theorem 3.1 and r = 0.1. Then by straightforward
computation, we get w(A) \simeq 2.2071, \| A\| \simeq 2.2882 and
1
2
\| | A| + | A\ast | \| \simeq 2.2518. Hence
w(A) \leq 1 - 2r
2
\| | A| + | A\ast | \| + 2r\| A\| \leq \| A\| ,
provides an improvement of inequality (1.1). In fact, 2.2071 \leq 2.2590 \leq 2.2882.
The following lemma is a multiplicative refinement of the Young inequality with the Kantorovich
constant.
Lemma 3.2 ([7], Corollary 3). Let a, b > 0. Then
(1 - \lambda )a+ \lambda b \geq k(h, 2)ra1 - \lambda b\lambda ,
where 0 \leq \lambda \leq 1, r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \lambda , 1 - \lambda \} , h =
b
a
such that K(h, 2) =
(h+ 1)2
4h
for h > 0, which has
properties K(h, 2) = K
\Bigl( 1
h
, 2
\Bigr)
\geq 1 (h > 0) and K(h, 2) is increasing on [1,\infty ) and is decreasing
on (0, 1).
In [10], Kittaneh obtained the inequality
w(A) \leq 1
2
\| | A| + | A\ast | \| . (3.1)
In the following theorem, we improve inequality (3.1) for hyponormal operators. Before pro-
ceeding recall that the operator A \in B(H) is said to be hyponormal if A\ast A - AA\ast \geq 0.
Theorem 3.2. If A \in B(H) is hyponormal, r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \lambda , 1 - \lambda \} , where 0 \leq \lambda \leq 1, then
w(A) \leq 1
\mathrm{i}\mathrm{n}\mathrm{f}\| x\| =1 \zeta (x)
\| | A| + | A\ast | \|
2
,
where \zeta (x) = K
\biggl(
\langle | A| x, x\rangle
\langle | A\ast | x, x\rangle
, 2
\biggr) r
is a refinement of inequality (1.1).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1450 Z. HEYDARBEYGI, M. AMYARI, M. KHANEHGIR
Proof. Let x \in H be a unit vector.
| \langle Ax, x\rangle | \leq
\sqrt{}
\langle | A| x, x\rangle \langle | A\ast | x, x\rangle (\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 2.4) =
=
\Bigl(
\langle | A\ast | x, x\rangle 1 - \lambda \langle | A| x, x\rangle \lambda
\Bigr) 1
2
\Bigl(
\langle | A| x, x\rangle 1 - \lambda \langle | A\ast | x, x\rangle \lambda
\Bigr) 1
2 \leq
\leq 1
2
\Bigl( \bigl(
\langle | A\ast | x, x\rangle 1 - \lambda \langle | A| x, x\rangle \lambda
\bigr)
+
\bigl(
\langle | A| x, x\rangle 1 - \lambda \langle | A\ast | x, x\rangle \lambda
\bigr) \Bigr)
\leq (\mathrm{b}\mathrm{y} \mathrm{A}\mathrm{M}-\mathrm{G}\mathrm{M} \mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}) \leq
\leq 1
2
\left( 1
K
\biggl(
\langle | A| x, x\rangle
\langle | A\ast | x, x\rangle
, 2
\biggr) r
\bigl(
(1 - \lambda )\langle | A\ast | x, x\rangle + \lambda \langle | A| x, x\rangle
\bigr)
+
+
1
K
\biggl(
\langle | A| x, x\rangle
\langle | A\ast | x, x\rangle
, 2
\biggr) r
\bigl(
(1 - \lambda )\langle | A| x, x\rangle + \lambda \langle | A\ast | x, x\rangle
\bigr) \right) (\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 3.2) =
=
1
2
\left( 1
K
\biggl(
\langle | A| x, x\rangle
\langle | A\ast | x, x\rangle
, 2
\biggr) r
\bigl(
\langle | A\ast | x, x\rangle + \langle | A| x, x\rangle
\bigr) \right) .
Taking supremum over x \in H with \| x\| = 1, we have
w(A) \leq 1
\mathrm{i}\mathrm{n}\mathrm{f}
\| x\| =1
\zeta (x)
\| | A| + | A\ast | \|
2
,
where \zeta (x) = K
\biggl(
\langle | A| x, x\rangle
\langle | A\ast | x, x\rangle
, 2
\biggr) r
.
Note that 2\langle | A| x, x\rangle \langle | A\ast | x, x\rangle \leq \langle | A| x, x\rangle 2 + \langle | A\ast | x, x\rangle 2, so\bigl(
\langle | A| x, x\rangle + \langle | A\ast | x, x\rangle
\bigr) 2 \geq 4\langle | A| x, x\rangle \langle | A\ast | x, x\rangle .
Hence \bigl(
\langle | A| x, x\rangle + \langle | A\ast | x, x\rangle
\bigr) 2
4\langle | A| x, x\rangle \langle | A\ast | x, x\rangle
\geq 1.
Therefore, K
\biggl(
\langle | A| x, x\rangle
\langle | A\ast | x, x\rangle
, 2
\biggr)
\geq 1.
Theorem 3.2 is proved.
References
1. M. Boumazgour, A. H. Nabwey, A note concerning the numerical range of a basic elementary operator, Ann. Funct.
Anal., 7, № 3, 434 – 441 (2016).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
SOME REFINEMENTS OF NUMERICAL RADIUS INEQUALITIES 1451
2. S. S. Dragomir, A note on numerical radius and the Krein – Lin inequality, RGMIA Res. Rep. Collect., 18, Article
113 (2015).
3. S. S. Dragomir, A note on new refinements and reverses of Young’s inequality, Transylv. J. Math. and Mech., 8, № 1,
46 – 49 (2016).
4. S. S. Dragomir, Some Gr\"uss type inequalities in inner product spaces, J. Inequal. Pure and Appl. Math., 4, № 2,
Article 42 (2003), 10 p.
5. S. S. Dragomir, Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces,
Tamkang J. Math., 39, № 1, 1 – 7 (2008).
6. R. Golla, On the numerical radius of a quaternionic normal operator, Adv. Oper. Theory, 2, № 1, 78 – 86 (2017).
7. M. Fuji, H. Zuo, G. Shi, Refined Young inequality with Kantorovich constant, J. Math. Inequal., 4, № 4, 551 – 556
(2011).
8. F. Kittaneh, Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. and Appl., 361, № 1,
262 – 269 (2010).
9. F. Kittaneh, Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear and Multilinear Algebra, 59,
1031 – 1037 (2011).
10. F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion
matrix, Stud. Math., 158, № 1, 11 – 17 (2003).
11. M. G. Krein, The angular localization of the spectrum of a multiplicative integral in Hilbert space (in Russian),
Funkcional. Anal. i Prilozhen., 3, 89 – 90 (1969).
12. M. Satari, M. S. Moslehian, T. Yamazaki, Some generalized numerical radius inequalities for Hilbert space operators,
Linear Algebra and Appl., 470, 216 – 227 (2015).
13. A. Sheikhhosseini, M. S. Moslehian, K. Shebrawi, Inequalities for generalized Euclidean operator radius via Young’s
inequality, J. Math. Anal. and Appl., 445, № 2, 1516 – 1529 (2017).
14. A. Zamani, Some lower bounds for the numerical radius of Hilbert space operators, Adv. Oper. Theory, 2, № 2,
98 – 107 (2017).
Received 15.09.17,
after revision — 16.11.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
|
| id | umjimathkievua-article-6027 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:30Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9d/3bec738e90b7ef591c96efadb779ad9d.pdf |
| spelling | umjimathkievua-article-60272025-03-31T08:49:43Z Some refinements of numerical radius inequalities Some refinements of numerical radius inequalities Heydarbeygi, Z. Amyari, M. Khanehgir, M. Heydarbeygi, Z. Amyari, M. Khanehgir, M. A. UDC 517.5 In this paper, we give some refinements for the second inequality in $\dfrac{1}{2}\|A\| \leq w(A) \leq \|A\|,$  where $A\in B(H).$  In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot),$  we show that $w(A)\leq \dfrac{1}{\displaystyle {2\inf\nolimits_{\| x \|=1}}\zeta(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|,$  where $\zeta(x)=K\left(\dfrac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2\right)^{r},$ $r=\min\{\lambda,1-\lambda\}$ and $0\leq \lambda \leq 1$ . We also give a reverse for the classical numerical radius power inequality $w(A^{n})\leq w^{n}(A)$ for any operator $A \in B(H)$ in the case when $n=2.$  УДК 517.5 Деякі уточнення нерівностей для числових радіусів Запропоновано деякі уточнення другої нерівності у $\dfrac{1}{2}\|A\| \leq w(A) \leq \|A\|,$ де $A\in B(H).$ Зокрема, якщо $A$ є гіпонормальним, то за допомогою нерівності Юнга з константою Канторовича $K(\cdot, \cdot)$ доведено, що $w(A) \leq \dfrac{1}{\displaystyle {2\inf\nolimits_{\| x \|=1}}\zeta(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|,$ де $\zeta(x)=K\left(\dfrac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2\right)^{r},$ $r=\min\{\lambda,1-\lambda\}$ i $0\leq \lambda \leq 1.$Також доведено нерівність для числових радіусів, що є оберненою до класичної степеневої нерівності $w(A^{n})\leq w^{n}(A)$ для будь-якого оператора $A \in B(H)$ у випадку $n=2.$ Institute of Mathematics, NAS of Ukraine 2020-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6027 10.37863/umzh.v72i10.6027 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 10 (2020); 1443 - 1451 Український математичний журнал; Том 72 № 10 (2020); 1443 - 1451 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6027/8767 |
| spellingShingle | Heydarbeygi, Z. Amyari, M. Khanehgir, M. Heydarbeygi, Z. Amyari, M. Khanehgir, M. A. Some refinements of numerical radius inequalities |
| title | Some refinements of numerical radius inequalities |
| title_alt | Some refinements of numerical radius inequalities |
| title_full | Some refinements of numerical radius inequalities |
| title_fullStr | Some refinements of numerical radius inequalities |
| title_full_unstemmed | Some refinements of numerical radius inequalities |
| title_short | Some refinements of numerical radius inequalities |
| title_sort | some refinements of numerical radius inequalities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6027 |
| work_keys_str_mv | AT heydarbeygiz somerefinementsofnumericalradiusinequalities AT amyarim somerefinementsofnumericalradiusinequalities AT khanehgirm somerefinementsofnumericalradiusinequalities AT heydarbeygiz somerefinementsofnumericalradiusinequalities AT amyarim somerefinementsofnumericalradiusinequalities AT khanehgirm somerefinementsofnumericalradiusinequalities AT a somerefinementsofnumericalradiusinequalities |