Power inequalities for weighted numerical radius and norm of operators

Najla Altwaijry; Silvestru Sever Dragomir; Kais Feki

doi:10.1515/gmj-2025-2052

Artikel

Power inequalities for weighted numerical radius and norm of operators

Najla Altwaijry , Silvestru Sever Dragomir und Kais Feki

Veröffentlicht/Copyright: 25. Juni 2025

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Aus der Zeitschrift Georgian Mathematical Journal

Abstract

For a bounded linear operator T on a complex Hilbert space ℋ , we define the weighted real and imaginary parts of the operator T as ℜ t ⁢ ( T ) := ( 1 - t ) ⁢ T ∗ + t ⁢ T and ℑ t ⁢ ( T ) := ( 1 - t ) ⁢ T - t ⁢ T ∗ i for t ∈ [ 0 , 1 ] , where T ∗ denotes the adjoint operator of T. By setting T t := ℜ t ⁢ ( T ) + i ⁢ ℑ t ⁢ ( T ) , we define the weighted norm ∥ T ∥ t := ∥ T t ∥ . Another weighted numerical radius is given by w ~ t ⁢ ( T ) = sup ⁡ { ∥ ℜ t ⁢ ( e i ⁢ φ ⁢ T ) ∥ : φ ∈ ℝ } for t ∈ [ 0 , 1 ] . In this paper, we present new power inequalities for the weighted operator norm ∥ ⋅ ∥ t and the weighted numerical radius w ~ t ⁢ ( ⋅ ) . Additionally, we discuss the special cases of the classical numerical radius and norm of Hilbert space operators.

Keywords: Weighted numerical radius; weighted operator norm; power inequalities; Hilbert space; bounded linear operator

MSC 2020: 47A12; 47A30; 47A63; 47A99

Funding statement: Supported by Distinguished Scientist Fellowship Program under Researchers Supporting Project number RSP2025R187, King Saud University, Riyadh, Saudi Arabia.

Acknowledgements

The authors sincerely appreciate the reviewer for his/her valuable comments and suggestions, which have greatly improved this paper. Additionally, the first author wishes to express her heartfelt gratitude for the support received from the Distinguished Scientist Fellowship Program under the Researchers Supporting Project number (RSP2025R187), King Saud University, Riyadh, Saudi Arabia.

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Received: 2025-02-13

Revised: 2025-04-21

Accepted: 2025-04-30

Published Online: 2025-06-25

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https://doi.org/10.1515/gmj-2025-2052

Schlagwörter für diesen Artikel

Weighted numerical radius; weighted operator norm; power inequalities; Hilbert space; bounded linear operator