Lower and upper bounds for the $ p $-(A-M)-norm of two operators in Hilbert spaces with applications

Najla Altwaijry; Silvestru Sever Dragomir; Najla Altwaijry; Silvestru Sever Dragomir

doi:10.3934/math.2026454

AIMS Mathematics

2026, Volume 11, Issue 4: 11050-11071. doi: 10.3934/math.2026454

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Lower and upper bounds for the $ p $-(A-M)-norm of two operators in Hilbert spaces with applications

Najla Altwaijry ^{1
,
,},
Silvestru Sever Dragomir ²

1.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Department of Mathematical and Geospatial Sciences, School of Science, RMIT University, GPO Box 2476V, Melbourne, Victoria 3001, Australia

Received: 21 August 2025 Revised: 11 December 2025 Accepted: 25 December 2025 Published: 21 April 2026
MSC : 46C05, 47A63, 47A99

For $ \nu \in \lbrack 0, 1] $, $ p\geq 1 $ and $ A, B\in \mathcal{B}\left(H\right) $, we define the $ p $-arithmetic-mean (A-M)-norm for the pair of operators $ \left(A, B\right) $ by
$ \begin{equation*} \left\Vert \left( A,B\right) \right\Vert _{p,\nu }: = \sup\limits_{\left\Vert x\right\Vert = 1}\left( \left( 1-\nu \right) \left\Vert Ax\right\Vert ^{p}+\nu \left\Vert Bx\right\Vert ^{p}\right) ^{1/p}. \end{equation*} $
In this paper, we obtain several lower and upper bounds for this norm. Some inequalities for the numerical radius of the off-diagonal operator matrix are given. In the case when $ \left(A, B\right) = \left(T, T^{\ast }\right) $ and $ \left(A, B\right) = \left(\mathrm{Re}T, \mathrm{Im}T\right) $, where $ \mathrm{Re}T: = \frac{T+T^{\ast }}{2} $ is the real part of $ T $ and $ \mathrm{Im}T: = \frac{T-T^{\ast }}{2\text{i}} $ is the imaginary part of $ T $, respectively, some inequalities for one operator are also provided.
- inner product spaces,
- operator norm,
- numerical radius,
- off-diagonal operator matrix,
- norms for pairs of operators,
- A-G inequality
Citation: Najla Altwaijry, Silvestru Sever Dragomir. Lower and upper bounds for the $ p $-(A-M)-norm of two operators in Hilbert spaces with applications[J]. AIMS Mathematics, 2026, 11(4): 11050-11071. doi: 10.3934/math.2026454

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Abstract

For $ \nu \in \lbrack 0, 1] $, $ p\geq 1 $ and $ A, B\in \mathcal{B}\left(H\right) $, we define the $ p $-arithmetic-mean (A-M)-norm for the pair of operators $ \left(A, B\right) $ by

$ \begin{equation*} \left\Vert \left( A,B\right) \right\Vert _{p,\nu }: = \sup\limits_{\left\Vert x\right\Vert = 1}\left( \left( 1-\nu \right) \left\Vert Ax\right\Vert ^{p}+\nu \left\Vert Bx\right\Vert ^{p}\right) ^{1/p}. \end{equation*} $

In this paper, we obtain several lower and upper bounds for this norm. Some inequalities for the numerical radius of the off-diagonal operator matrix are given. In the case when $ \left(A, B\right) = \left(T, T^{\ast }\right) $ and $ \left(A, B\right) = \left(\mathrm{Re}T, \mathrm{Im}T\right) $, where $ \mathrm{Re}T: = \frac{T+T^{\ast }}{2} $ is the real part of $ T $ and $ \mathrm{Im}T: = \frac{T-T^{\ast }}{2\text{i}} $ is the imaginary part of $ T $, respectively, some inequalities for one operator are also provided.

References

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