Inter national J our nal of Electrical and Computer Engineering (IJECE) V ol. 15, No. 5, October 2025, pp. 4732 ∼ 4739 ISSN: 2088-8708, DOI: 10.11591/ijece.v15i5.pp4732-4739 ❒ 4732 New appr oximations f or the numerical radius of an n × n operator matrix Amer Hasan Darweesh 1 , Adel Almalki 2 , Kamel Al-Khaled 1 1 Department of Mathematics and Statistics, Jordan Uni v ersity of Science and T echnology , Irbid, Jordan 2 Department of Mathematics, Al-Gunfudah Uni v ersity Colle ge, Umm Al-Qura Uni v ersity , Mecca, Saudi Arabia Article Inf o Article history: Recei v ed Sep 6, 2024 Re vised Mar 25, 2025 Accepted Jul 3, 2025 K eyw ords: Block operators Inequalities Numerical radius Operator matrix Spectral radius ABSTRA CT Man y mathematicians ha v e been interested in establishing more stringent bounds on the numerical radius of operators on a Hilbert space. Studying the numerical radii of operator matrices has pro vided v aluable insights using operator matrices. In this paper , we present ne w , sharper bounds for the numerical radius 1 4 | A | 2 + | A ∗ | 2 ≤ w 2 ( A ) ≤ 1 2 | A | 2 + | A ∗ | 2 , that found by Kittaneh. Specically , we de v elop a ne w bound for the numerical radius w ( T ) of block operators. Moreo v er , we sho w that these bounds not only impro v e upon b ut also generalize some of the current lo wer and upper bounds. The concept of nding and understanding these bounds in matrices and linear operators is re visited throughout this research. Furthermore, the study emphasizes the importance of these bounds in mathematics and their potential applications in v arious mathematical elds. This is an open access article under the CC BY -SA license . Corresponding A uthor: Amer Hasan Darweesh Department of Mathematics and Statistics, Jordan Uni v ersity of Science and T echnology P .O.Box 3030, Irbid 22110, Jordan Email: ahdarweesh@just.edu.jo 1. INTR ODUCTION Consider a comple x Hilbert space H . The C ∗ − algebra of all bounded linear operators on H is denoted as B ( H ) . Throughout this paper , we dene the real part Re( . ) , the imaginary part Im( . ) , the absolute v alue | . | , the numerical radius w ( . ) , and the standard operator norm || . || for A ∈ B ( H ) as follo ws: Re( A ) = A + A ∗ 2 , Im( A ) = A − A ∗ 2 i , | A | = ( A ∗ A ) 1 2 , w ( A ) = sup {|⟨ Ax, x ⟩| : x ∈ H , ∥ x ∥ = 1 } , ∥ A ∥ = sup n p ⟨ Ax, Ax ⟩ : x ∈ H , ∥ x ∥ = 1 o . It is kno wn that the norms w ( . ) and || . || are equi v alent and satisfy 1 2 ∥ A ∥ ≤ w ( A ) ≤ ∥ A ∥ , (1) for e v ery A ∈ B ( H ) . W ith a fe w e xceptions, it is typically challenging to determine the precise v alue of the numerical radius w ( A ) for a n y matrix A ∈ B ( H ) . As a result, researchers ha v e w ork ed on determining tighter upper and lo wer bounds of w ( A ) for general matrices A , better than those presented in (1). Readers can consult [1]–[5] and the references therein for the lates t de v elopments on the upper and lo wer bounds of the numerical J ournal homepage: http://ijece .iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 4733 radius. The authors in v estig ated numerical radius inequalities for sectorial matrices, a particular type of ma- trix, in [6]–[8]. The y created se v eral more precise upper and lo wer bounds for the numerical radius of these matrices. The study in [9] e xplains the basics of Schur complements in the conte xt of a particular numerical radius problem. Ne w inequalities for a generalized numerical radius of block operators are presented in [10]. An impro v ed bound for the numerical radius of n × n operator matrices is de v eloped in [11]. Additionally , the y gi v e numerical radius bounds for the product of tw o operators and the commutator of operators. Ne w upper and lo wer bounds for the numerical radii of some operator matrices are found by the authors in [12]. In [13], a ne w bound for polynomial zeros is deri v ed. Understanding norm-related and numerical radius-related inequalities is crucial when performing mathematical analysis. This kno wledge of fers important insights into operator beha vior and approximation accurac y . A comprehensi v e re vie w of se v eral inequalities concerning the Euclidean operator radius is gi v en in [14]. It co v ers addi tion and multiplication for groups of n -tuple opera- tors. The need for more stringent and general bounds on the numerical radius of operators on Hilbert spaces is discussed in this paper . The ef fecti v eness of e xisti ng upper and lo wer bounds in a v ariety of applications is limited because the y are frequently sub-optimal, especially for block operator matrices. Our ndings pro vide a more thorough understanding of numerical radi us beha vior , particularly for block operators, by both impro ving and generalizing current estimates. The enhanced bounds complement theoretical adv ancements in functional analysis and operator theory . W ith possible uses in mathematical ph ysics, quantum mechanics, and numerical analysis, the y of fer impro v ed analytical tools for researching operator matrices. Our method ensures wider ap- plicability by methodically generalizing e xisting upper and lo wer bounds, in contrast to earlier studies that only pro vide isolated bounds. Adv anced inequalities ar guments are used to construct theoretical proofs, guarantee- ing the generality and rob ustness of our ndings. In particular , we impro v e estimates for block operators by deri ving ne w , sharper bounds for the numerical radius of operators. These bounds of fer a more comprehensi v e frame w ork for numerical radius analysis by both impro ving and generalizing pre vious ndings. The enhanced bounds may resul t in more accurate numerical techniques for linear algebra and functional analysis, which w ould be adv antageous for computational mathematics. Applications for the ndings could be found in control theory stability analysis, quantum mechanics, and other domains where operator . This w ork opens t he door for further e xploration of numerical radius bounds in more comple x operator classes, potentially inspiring future research on non-normal operators and unbounded operators in Hilbert spaces. Numerous mathematicians ha v e aimed to impro v e the inequalities in (1). F or e xample, see [15]–[20]. In [16], Kittaneh rened the inequality in (1) by pro ving that if A ∈ B ( H ) , then 1 4 | A | 2 + | A ∗ | 2 ≤ w 2 ( A ) ≤ 1 2 | A | 2 + | A ∗ | 2 . (2) Let H 1 , H 2 ...., H n be comple x Hilbert spaces, and let B ( H j , H i ) be the space of all bounded linear operators from H j into H i . Based on this structure, an y operator T ∈ B n ⊕ i =1 H i (where n ⊕ i =1 H i is the direct sum of H i , i = 1 , 2 , ..., n ) can be represented by an n × n operator matrix T = [ T ij ] , where T ij ∈ B ( H j , H i ) . T o disco v er more important results related to the numerical radius of operator mat rices, see [21]–[26]. The results gi v en in [15] moti v ated us to de v elop ne w l o wer and upper bounds for n × n operator matrices. In particular , we sho w that if T = 0 Λ 1 Λ 2 . . . Λ n 0 , where { Λ i } n i =1 ⊆ B ( H ) , and Λ 1 , Λ 2 , ..., Λ n ∈ H , then w ( T ) ≥ 1 4 max 1 ≤ i ≤ n | Λ i | 2 + | Λ ∗ n − i +1 | 2 + 1 8 max 1 ≤ i ≤ n || Λ i + Λ ∗ n − i +1 || − || Λ i − Λ ∗ n − i +1 || and w ( T ) ≤ 1 4 max 1 ≤ i ≤ n | Λ ∗ i | 2 + | Λ n − i +1 | 2 + 1 2 max 1 ≤ i ≤ n w | Λ ∗ n − i +1 | | Λ i | . As special cases of these bounds, we will rene the inequalities in (1) and (2). W e will also pro vide some concrete e xamples sho wing ho w these ne w bounds impro v e upon those in (1) and (2). Ne w appr oximations for the numerical r adius of an n × n oper ator matrix (Amer Hasan Darweesh) Evaluation Warning : The document was created with Spire.PDF for Python.
4734 ❒ ISSN: 2088-8708 2. B A CKGR OUND PRELIMIN ARIES In this section, some k e y results about the numerical radius and the operator norm on a comple x Hilbert space are re vie wed. These results are essential for pro ving our main ndings. The follo wing lemma describes the numerical radius of an operator in terms of the numerical radius of its blocks, as seen in [23]. Lemma 1. Let Λ 1 , Λ 2 be tw o bounded linear operators on H . Then (a) w  Λ 1 0 0 Λ 2  = max { w (Λ 1 ) , w (Λ 2 ) } ; (b) w  Λ 1 Λ 2 Λ 2 Λ 1  = max { w (Λ 1 + Λ 2 ) , w (Λ 1 − Λ 2 ) } . In particular , w  0 Λ 2 Λ 2 0  = w (Λ 2 ) . A special case of the mix ed Schw arz inequality , which is found in [18], is the lemma that follo ws. Lemma 2. Let Λ be a bounded linear operator on H . Then, for an y x, y ∈ H , we ha v e |⟨ Λ x, y ⟩| 2 ≤ ⟨| Λ | x, x ⟩ ⟨| Λ ∗ | y , y ⟩ . The follo wing lemma is one of the most important results about the numerical radius that we will use in our proofs. This lemma can be found in [19]. Lemma 3. Let Λ be a bounded linear operator on H . Then, w (Λ) = max θ ∈ R Re e iθ Λ = max θ ∈ R Im e iθ Λ . The ne xt lemma is the Buzano inequality (see [27]). Lemma 4. Let a, b, e ∈ H with ∥ e ∥ = 1 . Then |⟨ a, e ⟩⟨ e, b ⟩| ≤ 1 2 ( ∥ a ∥ ∥ b ∥ + |⟨ a, b ⟩| ) . As stated in [29], Theorem 7 and Theorem 12 can be pro v ed using the Kittaneh result, which is the subject of the follo wing lemma. Lemma 5. Let A, B be positi v e bounded linear operators on H . Then ∥ A + B ∥ ≤ max {|| A || , || B ||} + A 1 2 B 1 2 . F or A ∈ B ( H ) , the spectral radius is dened as r ( A ) = sup {| λ | : λ ∈ σ ( A ) } . Before concluding this section, we introduce the follo wing lemma, which pro vides tw o important properties for r ( A ) and is k e y to the proof of our rst result. Lemma 6. Let A, B be bounded linear operators on H . Then (a) r ( A ) ≤ w ( A ) ≤ || A || (the equality holds when A is normal), (b) r ( AB ) = r ( B A ) . 3. THE MAIN RESUL TS A ne w upper bound for the numerical radius of a n × n operator matrix is introduced by the follo wing theorem, which we use to start our results. Theorem 7. Let A 1 , A 2 , ..., A n be bounded operators on a comple x Hilbert space H , and let M = [ m ij ] n × n where m ij = A i , i + j = n + 1 0 , otherwise . Then w ( M ) ≤ 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 2 max 1 ≤ i ≤ n | A n − i +1 | 1 2 | A ∗ i | 1 2 = 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 2 max 1 ≤ i ≤ n r 1 2 ( | A n − i +1 | | A ∗ i | ) . W e start by noting that | M | = [ p ij ] n × n and | M ∗ | = [ q ij ] n × n where p ij = | A i | , i = j 0 , otherwise , and q ij = | A ∗ i | , i = j 0 , otherwise . Int J Elec & Comp Eng, V ol. 15, No. 5, October 2025: 4732-4739 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 4735 No w , for an y unit v ector y ∈ H ⊕ H ⊕ · · · ⊕ H , we ha v e |⟨ M y , y ⟩| ≤ ⟨| M | y , y ⟩ 1 2 ⟨| M ∗ | y , y ⟩ 1 2 ≤ 1 2 ⟨ ( | M | + | M ∗ | ) y , y ⟩ = 1 2 D [ p ij + q ij ] n × n y , y E ≤ 1 2 w [ p ij + q ij ] n × n = 1 2 max 1 ≤ i ≤ n ∥| A n − i +1 | + | A ∗ i |∥ = 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 2 max 1 ≤ i ≤ n | A n − i +1 | 1 2 | A ∗ i | 1 2 . (by Lemma 5) Therefore, w ( M ) = sup ∥ y ∥ =1 |⟨ M y , y ⟩| ≤ 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 2 max 1 ≤ i ≤ n | A n − i +1 | 1 2 | A ∗ i | 1 2 = 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 2 max 1 ≤ i ≤ n | A n − i +1 | 1 2 | A ∗ i | 1 2 | A n − i +1 | 1 2 | A ∗ i | 1 2 ∗ 1 2 = 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 2 max 1 ≤ i ≤ n | A n − i +1 | 1 2 | A ∗ i || A n − i +1 | 1 2 1 2 = 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 2 max 1 ≤ i ≤ n r 1 2 | A n − i +1 | 1 2 | A ∗ i || A n − i +1 | 1 2 (by Lemma 6(a)) = 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 2 max 1 ≤ i ≤ n r 1 2 ( | A n − i +1 || A ∗ i | ) . (by Lemma 6(b)) This completes the proof of the theorem. It is w orth mentioning that Theorem 7 generalizes the result found in [15] for the case n = 2 . Addi- tionally , when A 1 = A 2 = ... = A n = A, we get w ( A ) ≤ 1 2 ∥ A ∥ + 1 2 r 1 2 ( | A || A ∗ | ) . (3) It is e vident that the upper bound in (3) is tighter than the upper bound in (1). F or e xample, if we consider A = 0 1 0 0 , then then the upper bounds of (1) and (2) are 1 and 1 √ 2 , respecti v ely . While the upper bound of (3) is 1 2 , which emphasize our claim . Ne xt, we pro vide four lo wer bounds for w ( M ) , where M is an n × n operator matrix. Theorem 8. Let { A i } n i =1 ⊆ B ( H ) and let M be as in Theorem 7. Then w ( M ) ≥ 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 4 max 1 ≤ i ≤ n A i + A ∗ n − i +1 − A i − A ∗ n − i +1 . By Lemma 3, we ha v e w ( M ) ≥ ∥ Re( M ) ∥ = 1 2 max 1 ≤ i ≤ n A i + A ∗ n − i +1 , w ( M ) ≥ ∥ Im( M ) ∥ = 1 2 max 1 ≤ i ≤ n A i − A ∗ n − i +1 . Thus, for each k ∈ { 1 , 2 , ..., n } we ha v e that w ( M ) ≥ 1 2 max A k + A ∗ n − k +1 , A k − A ∗ n − k +1 = 1 4 A k + A ∗ n − k +1 + A k − A ∗ n − k +1 + 1 4 A k + A ∗ n − k +1 − A k − A ∗ n − k +1 ≥ 1 4 A k + A ∗ n − k +1 ∓ A k − A ∗ n − k +1 + 1 4 A k + A ∗ n − k +1 − A k − A ∗ n − k +1 . This implies that w ( M ) ≥ 1 2 max 1 ≤ i ≤ n ∥ A i ∥ + 1 4 max 1 ≤ i ≤ n A i + A ∗ n − i +1 − A i − A ∗ n − i +1 . The follo wing renement of inequality (1) is a direct result of Theorem 8. Ne w appr oximations for the numerical r adius of an n × n oper ator matrix (Amer Hasan Darweesh) Evaluation Warning : The document was created with Spire.PDF for Python.
4736 ❒ ISSN: 2088-8708 Corollary 9. Let A ∈ B ( H ) . Then w ( A ) ≥ 1 2 ∥ A ∥ + 1 4 |∥ A + A ∗ ∥ − ∥ A − A ∗ ∥| ≥ 1 2 ∥ A ∥ , where 1 2 ∥ A ∥ is the lo wer bound of (1). Theorem 10. Let { A i } n i =1 ⊆ B ( H ) and let M be as in Theorem 7. Then w 2 ( M ) ≥ 1 4 max 1 ≤ i ≤ n | A i | 2 + | A ∗ n − i +1 | 2 + 1 8 max 1 ≤ i ≤ n A i + A ∗ n − i +1 2 − A i − A ∗ n − i +1 2 . F or each k ∈ { 1 , 2 , ..., n } , we ha v e that w ( M ) ≥ 1 2 max A k + A ∗ n − k +1 , A k − A ∗ n − k +1 . Therefore, w 2 ( M ) ≥ 1 4 max n A k + A ∗ n − k +1 2 , A k − A ∗ n − k +1 2 o = 1 8 A k + A ∗ n − k +1 2 + A k − A ∗ n − k +1 2 + 1 8 A k + A ∗ n − k +1 2 − A k − A ∗ n − k +1 2 . This implies that w 2 ( M ) ≥ 1 2 ∥ Re( M ) ∥ 2 + ∥ Im( M ) ∥ 2 + 1 8 max 1 ≤ i ≤ n A i + A ∗ n − i +1 2 − A i − A ∗ n − i +1 2 = 1 2 Re 2 ( M ) + Im 2 ( M ) + 1 8 max 1 ≤ i ≤ n A i + A ∗ n − i +1 2 − A i − A ∗ n − i +1 2 ≥ 1 2 Re 2 ( M ) + Im 2 ( M ) + 1 8 max 1 ≤ i ≤ n A i + A ∗ n − i +1 2 − A i − A ∗ n − i +1 2 = 1 4 max 1 ≤ i ≤ n | A i | 2 + | A ∗ n − i +1 | 2 + 1 8 max 1 ≤ i ≤ n A i + A ∗ n − i +1 2 − A i − A ∗ n − i +1 2 . Corollary 11. If A ∈ B ( H ) , then w 2 ( A ) ≥ 1 4 | A | 2 + | A ∗ | 2 + 1 8 || A + A ∗ || 2 − || A − A ∗ || 2 ≥ 1 4 | A | 2 + | A ∗ | 2 , where 1 4 | A | 2 + | A ∗ | 2 is the lo wer bound of (2). Theorem 12. Let { Λ i } n i =1 ⊆ B ( H ) and let T be as in Theorem 7. Then, w 2 ( T ) ≥ 1 8 max max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 2 , max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 2 + 1 8 max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 ≥ 1 4 max 1 ≤ i ≤ n | Λ i | 2 + | Λ ∗ n − i +1 | 2 . F or the rst inequality , w 2 ( T ) = 1 2 w 2 ( T ) + 1 2 w 2 ( T ) ≥ 1 2 max n ∥ Re( T ) ∥ 2 , ∥ Im( T ) ∥ 2 o + 1 2 ∥ Re( T ) ∥ ∥ Im( T ) ∥ = 1 8 max max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 2 , max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 2 + 1 8 max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 . Int J Elec & Comp Eng, V ol. 15, No. 5, October 2025: 4732-4739 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 4737 F or the second inequality , 1 4 max 1 ≤ i ≤ n | Λ i | 2 + | Λ ∗ n − i +1 | 2 = 1 2 Re 2 ( T ) + Im 2 ( T ) ≤ 1 2 max Re 2 ( T ) , Im 2 ( T ) + 1 2 ∥ Re( T ) ∥ ∥ Im( T ) ∥ = 1 8 max max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 2 , max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 2 + 1 8 max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 . By Theorem 12 and Lemma 1(b), we ha v e the follo wing corollary . Corollary 13. Let Λ ∈ B ( H ) . Then w 2 (Λ) ≥ 1 8 max {∥ Λ + Λ ∗ ∥ , ∥ Λ − Λ ∗ ∥} + 1 8 ∥ Λ + Λ ∗ ∥ ∥ Λ − Λ ∗ ∥ ≥ 1 4 | Λ | 2 + | Λ ∗ | 2 , where 1 4 | Λ | 2 + | Λ ∗ | 2 is the lo wer bound of (2). T o pro v e our ne xt result, we need the follo wing lemma, which can be found in [28]. Lemma 14. Let A, B ∈ B ( H ) . Then ∥ A + B ∥ 2 ≤ 2 max ||| A | 2 + | B | 2 || , ||| A ∗ | 2 + | B ∗ | 2 || . Theorem 15. Let { Λ i } n i =1 ⊆ B ( H ) and let T be as in Theorem 7. Then w 4 ( T ) ≥ 1 32 max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 4 + 1 32 max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 4 ≥ 1 16 max 1 ≤ i ≤ n | Λ ∗ i | 2 + | Λ n − i +1 | 2 . F or the rst inequality , we ha v e w 4 ( T ) ≥ max n ∥ Re( T ) ∥ 4 , ∥ Im( T ) ∥ 4 o ≥ 1 2 ∥ Re( T ) ∥ 4 + ∥ Im( T ) ∥ 4 = 1 32 max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 4 + 1 32 max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 4 . No w , for the second inequality , we ha v e 1 16 max 1 ≤ i ≤ n | Λ i ∗ | 2 + | Λ n − i +1 | 2 2 = 1 4 Re 2 ( T ) + Im 2 ( T ) 2 ≤ 1 2 Re 4 ( T ) + Im 4 ( T ) (by Lemma 14) ≤ 1 2 ∥ Re( T ) ∥ 4 + ∥ Im( T ) ∥ 4 = 1 32 max 1 ≤ i ≤ n Λ i + Λ ∗ n − i +1 4 + 1 32 max 1 ≤ i ≤ n Λ i − Λ ∗ n − i +1 4 . Remark 1. Let Λ 1 = Λ 2 = ... = Λ n = Λ . Then by Theorem 15, we ha v e w 2 (Λ) ≥ 1 4 √ 2 q ∥ Λ + Λ ∗ ∥ 4 + ∥ Λ − Λ ∗ ∥ 4 ≥ 1 4 | Λ | 2 + | Λ ∗ | 2 , where 1 4 | Λ | 2 + | Λ ∗ | 2 is the lo wer bound of (2). At the end of this paper , we remark that all the inequalities in our results become equalities if Λ 1 = Λ , where Λ is a bounded linear operator , and Λ 2 = Λ 3 = ... = Λ n = 0 . Ne w appr oximations for the numerical r adius of an n × n oper ator matrix (Amer Hasan Darweesh) Evaluation Warning : The document was created with Spire.PDF for Python.
4738 ❒ ISSN: 2088-8708 4. CONCLUSION In this paper , we ha v e introduced se v eral ne w inequalities t hat help limit the Euclidean numer ical radius and its arithmetic operations. These results also pro vide useful tools for establishing inequalities for the numerical radius w ( T ) of block operators. The study of fers a ne w inequality that pro vides more accurate bounds for the numerical radius by carefully analyzing inequalities for numerical radii in n × n operator matrices that include block operators. Additionally , we sho w that the bounds we obtained here not only enhance b ut also generalize some of the e xisting lo wer and upper bounds. This analysis emphasizes the signicance of understanding bounds in matrices and linear operators, and it highlights the k e y rol e that symmetry plays in mathematics across v arious disciplines. REFERENCES [1] P . Bhunia and K. P aul, “Furtherance of numerical radius inequalities of Hilbert space operators, ” Ar c hiv der Mathe- matik , v ol. 117, no. 5, pp. 537–546, 2021, doi: 10.1007/s00013-021-01641-w . [2] P . Bhunia and K. P aul, “Renements of norm and numerical radius inequalities, ” Roc k y Mountain J ournal of Mathe- matics , v ol. 51, no. 6, pp. 1953–1965, 2021, doi: 10.1216/rmj.2021.51.1953. [3] P . Bhunia and K. P aul, “Proper impro v ement of well-kno wn numerical radius inequalities and their applications, ” Results in Mathematics , v ol. 76, no. 4, P aper No. 177, 2021, doi: 10.1007/s00025-021-01478-3. [4] P . Bhunia, K. P aul, and R. K. Nayak, “Sha rp inequalities for the numerical radius of Hilbert space operators and operator matrices, ” arXiv pr eprint arXiv:1908.04499 , v ol. 24, no. 1, pp. 167–183, 2021. [5] P . Bhunia and K. P aul, “Ne w upper bounds for the numerical radius of Hilbert space operators, ” Bulletin des Sciences Math ´ ematiques , v ol. 167, P aper No. 102959, 2021, doi: 10.1016/j.b ulsci.2021.102959. [6] S. Ab u Sammour , F . Kittaneh, and M. Sababheh, “ A geometric approach to numerical radius inequalities, ” Linear Alg ebr a and its Applications , v ol. 652, pp. 1–17, 2022, doi: 10.1016/j.laa.2022.07.001. [7] N. Altw aijry , C. Conde, S. S. Dragomir , and K. Feki, “Norm and numerical radi us inequalities related to the Selber g operator , ” Symmetry , v ol. 15, no. 10, p. 1860, 2023, doi: 10.3390/sym15101860. [8] Y . Bedrani, F . Kit taneh, and M. Sababheh, “Numerical radii of accreti v e matri ces, ” Linear and Multilinear Alg ebr a , v ol. 69, no. 5, pp. 957–970, 2020, doi: 10.1080/03081087.2020.1813679. [9] R. Bhatia and T . Jain, “The numerical radius and positi vity of block matrices, ” Linear Alg ebr a and its Applications , v ol. 656, pp. 463–482, 2023, doi: 10.1016/j.laa.2022.10.009. [10] S. Sahoo and M. Sababheh, “Hilbert-Schmidt numerical radius of block operators, ” F ilomat , v ol. 35, no. 8, pp. 2663– 2678, 2021, doi: 10.2298/FIL2108663S. [11] P . Bhunia, “Sharper bounds for the numerical radius of n × n operator matrices, ” Ar c hiv der Mathematik , v ol. 123, pp. 173–183, 2024, doi: 10.1007/s00013-024-02017-6. [12] M. Al-Dolat and F . Kittaneh, “Numerical radius inequalities for certain operator matrices, ” The J ournal of Analy sis , 2024, doi: 10.1007/s41478-024-00782-9. [13] A. Frakis, F . Kittaneh, and S. Soltani, “Ne w numerical radius inequaliti es for operator matrices and a bound for the zeros of polynomials, ” Advances in Oper ator Theory , v ol. 8, no. 6, 2023, doi: 10.1007/s43036-022-00232-y . [14] M. Rashid and W . Salameh, “Inequalities for the Euclidean operator radius of n -tuple operators and operator matrices in Hilbert C ∗ -modules, ” Symmetry , v ol. 16, p. 647, 2024, doi: 10.3390/sym16060647. [15] A. Darweesh, M. Hayajneh, S. Hayajneh, and F . Kittaneh, “Norm inequalities for positi v e denite matrices re- lated to a question of Bourin, ” Linear and Multilinear Alg ebr a , v ol. 71, no. 12, pp. 1933–1947, 2023, doi: 10.1080/03081087.2022.2091507. [16] F . Kittaneh, “Numerical radius inequalities for Hilbert space operators, ” Studia Mathematica , v ol. 168, pp. 73–80, 2005. [17] A. Abo-Omar and F . Kittaneh, “Upper and lo wer bounds for the numerical radius with an application to in v olution operators, ” Roc k y Mountain J ournal of Mathematics , v ol. 45, pp. 1055–1065, 2015. [18] F . Kittaneh, “Notes on some inequalities for Hilbert space operators, ” Publications of the Resear c h Institut e for Mathematical Sciences , v ol. 24, pp. 283–293, 1988, doi: 10.2977/prims/1195175202. [19] T . Y amazaki, “On upper and lo wer bounds of the numerical radius and an equality condition, ” Studia Mathematica , v ol. 178, pp. 83–89, 2007, doi: 10.4064/sm178-1-5. [20] N. Altw aijry and K. Feki, “Further inequal ities for the weighted numerical radius of operators, ” Mathematics , v ol. 10, p. 3576, 2022, doi: 10.3390/math10193576. [21] W . Bani-Domi and F . Kittaneh, “Norm equalities and i nequalities for operator matrices, ” Linear Alg ebr a and its Applications , v ol. 429, pp. 57–67, 2008, doi: 10.1016/j.laa.2008.02.004. [22] W . Bani-Domi and F . Kittaneh, “Numerical radius inequalities for operator matrices, ” Linear and Multilinear Alg ebr a , v ol. 57, pp. 421–427, 2009, doi: 10.15352/aot.1804-1359. [23] O. Hirzallah, F . Kittaneh, and K. Shebra wi, “Numerical radius inequalities for certain 2 × 2 operator matrices, ” Int J Elec & Comp Eng, V ol. 15, No. 5, October 2025: 4732-4739 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 4739 Inte gr al Equations and Oper ator Theory , v ol. 71, pp. 129–147, 2011. [24] W . Bani-Domi, “Some general numerical radius inequalities for the of f-diagonal parts of 2 × 2 operator matrices, ” Italian J ournal of Pur e and Applied Mathematics , v ol. 35, pp. 433–442, 2015. [25] M. Al-Dolat, K. Al-Zoubi, M. Ali, and F . Bani-Ahamed, “General numerical radius inequalities for matrices of operators, ” Open Mathematics , v ol. 4, pp. 1–9, 2016. [26] A. Ab u-Omar and F . Kittaneh, “Numerical radius inequalities for n × n operator matrices, ” Linear Alg ebr a and its Applications , v ol. 468, pp. 18–26, 2015, doi: 10.1016/j.laa.2013.09.049. [27] M. L. Buzano, “Generalizzazione della dise guaglianza di Cauch y-Schw arz (in Italia), ” Rendiconti del Seminario Matematico della Univer sit ` a e P olitecnico di T orino , v ol. 31, pp. 405–409, 1974. [28] P . Bhunia, S. Bag, and K. P aul, “Bounds for zeros of polynomials using numerical radius of Hilbert space operators, ” Annals of Functional Analysis , v ol. 12, no. 2, 2021, doi: 10.1007/s43034-020-00107-4. [29] F . Kittaneh, “Norm inequalities for certain operator sums, ” J ournal of Functional Analysis , v ol. 143, pp. 337–348, 1997, doi: 10.1006/jf an.1996.2957. BIOGRAPHIES OF A UTHORS Amer Hasan Darweesh is an ass ociate professor at the Department of Mathematics and Statistics, F aculty of Science and Arts, Jordan Uni v ersity of Science and T echnology , Irbid 22110, Jordan. He can be contacted at ahdarweesh@just.edu.jo. Adel Almalki is an assistant profes sor at the Department of Mathematics, Al-Gunfudah Uni v ersity Colle ge, Umm Al-Qura Uni v ersity , Mecca, Saudi Arabia. He can be contacted at aaa- malki@uqu.edu.sa. Kamel Al-Khaled is a professor at the Department of Mathematics and Statistics, F aculty of Science and Arts, Jordan Uni v ersity of Science and T echnology , Irbid 22110, Jordan. He can be contacted at kamel@just.edu.jo. Ne w appr oximations for the numerical r adius of an n × n oper ator matrix (Amer Hasan Darweesh) Evaluation Warning : The document was created with Spire.PDF for Python.