DOI: 10.1515/gmj-2026-3020 ISSN: 1072-947X
New inequalities in semi-Hilbert spaces with joint numerical radius applications
Salma Aljawi, Silvestru Sever Dragomir, Kais Feki Abstract
Let
A
be a nonzero positive operator on a complex Hilbert space
(
ℋ
,
〈
⋅
,
⋅
〉
)
{(\mathcal{H},\langle\,\cdot\,,\cdot\,\rangle)}
, inducing the semi-inner product
〈
x
,
y
〉
A
:=
〈
A
x
,
y
〉
{\langle x,y\rangle_{A}:=\langle Ax,y\rangle}
for
x
,
y
∈
ℋ
{x,y\in\mathcal{H}}
. The space
(
ℋ
,
∥
⋅
∥
A
)
{(\mathcal{H},\|\cdot\|_{A})}
, where
∥
x
∥
A
=
〈
x
,
x
〉
A
{\|x\|_{A}=\sqrt{\langle x,x\rangle_{A}}}
, is a semi-Hilbert space equipped with a seminorm induced by
〈
⋅
,
⋅
〉
A
{\langle\,\cdot\,,\cdot\,\rangle_{A}}
.
In this paper, we establish the inequality
|
∑
i
=
1
n
c
i
〈
x
,
y
i
〉
A
|
2
≤
∥
x
∥
A
2
(
∑
i
=
1
n
|
c
i
|
p
(
∑
j
=
1
n
|
〈
y
i
,
y
j
〉
A
|
)
)
1
p
(
∑
i
=
1
n
|
c
i
|
q
(
∑
j
=
1
n
|
〈
y
i
,
y
j
〉
A
|
)
)
1
q
,
see text
\Bigg{|}\sum_{i=1}^{n}c_{i}\langle x,y_{i}\rangle_{A}\Bigg{|}^{2}\leq\|x\|_{A}%
^{2}\Bigg{(}\sum_{i=1}^{n}|c_{i}|^{p}\Bigg{(}\sum_{j=1}^{n}|\langle y_{i},y_{j%
}\rangle_{A}|\Bigg{)}\Bigg{)}^{\frac{1}{p}}\Bigg{(}\sum_{i=1}^{n}|c_{i}|^{q}%
\Bigg{(}\sum_{j=1}^{n}|\langle y_{i},y_{j}\rangle_{A}|\Bigg{)}\Bigg{)}^{\frac{%
1}{q}},
where
x
,
y
1
,
…
,
y
n
∈
ℋ
{x,y_{1},\dots,y_{n}\in\mathcal{H}}
,
c
1
,
…
,
c
n
{c_{1},\dots,c_{n}}
are complex
numbers, and
p
,
q
>
1
{p,q>1}
satisfy
1
p
+
1
q
=
1
{\frac{1}{p}+\frac{1}{q}=1}
. We derive
analogues of the Bombieri, Selberg, and Heilbronn inequalities in the
context of semi-Hilbert spaces. Applications to
A
-seminorms associated
with
n
-tuples of operators in semi-Hilbert spaces, including the joint
A
-numerical radius and the Euclidean
A
-seminorm, are also presented.