Norm inequalities involving a special class of functions for sector matrices
Abstract In this paper, we present some unitarily invariant norm inequalities for sector matrices involving a special class of functions. In particular, if Z = ( Z 11 Z 12 Z 21 Z 22 ) is a 2 n × 2 n $2n\times 2n$ matrix such that numerical range of Z is contained in a sector region S α $S_{\alpha }...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-05-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-020-02383-z |
| _version_ | 1818241200637870080 |
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| author | Davood Afraz Rahmatollah Lashkaripour Mojtaba Bakherad |
| author_facet | Davood Afraz Rahmatollah Lashkaripour Mojtaba Bakherad |
| author_sort | Davood Afraz |
| collection | DOAJ |
| description | Abstract In this paper, we present some unitarily invariant norm inequalities for sector matrices involving a special class of functions. In particular, if Z = ( Z 11 Z 12 Z 21 Z 22 ) is a 2 n × 2 n $2n\times 2n$ matrix such that numerical range of Z is contained in a sector region S α $S_{\alpha } $ for some α ∈ [ 0 , π 2 ) $\alpha \in [0,\frac{\pi }{2} ) $ , then, for a submultiplicative function h of the class C $\mathcal{C} $ and every unitarily invariant norm, we have ∥ h ( | Z i j | 2 ) ∥ ≤ ∥ h r ( sec ( α ) | Z 11 | ) ∥ 1 r ∥ h s ( sec ( α ) | Z 22 | ) ∥ 1 s , $$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{ \frac{1}{s} }, \end{aligned}$$ where r and s are positive real numbers with 1 r + 1 s = 1 $\frac{1}{r}+\frac{1}{s}=1 $ and i , j = 1 , 2 $i,j=1,2$ . We also extend some unitarily invariant norm inequalities for sector matrices. |
| first_indexed | 2024-12-12T13:25:34Z |
| format | Article |
| id | doaj.art-f6ff8fe829a6479bb04224507ad21a11 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-12T13:25:34Z |
| publishDate | 2020-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-f6ff8fe829a6479bb04224507ad21a112022-12-22T00:23:12ZengSpringerOpenJournal of Inequalities and Applications1029-242X2020-05-012020111110.1186/s13660-020-02383-zNorm inequalities involving a special class of functions for sector matricesDavood Afraz0Rahmatollah Lashkaripour1Mojtaba Bakherad2Department of Mathematics, Faculty of Mathematics, University of Sistan and BaluchestanDepartment of Mathematics, Faculty of Mathematics, University of Sistan and BaluchestanDepartment of Mathematics, Faculty of Mathematics, University of Sistan and BaluchestanAbstract In this paper, we present some unitarily invariant norm inequalities for sector matrices involving a special class of functions. In particular, if Z = ( Z 11 Z 12 Z 21 Z 22 ) is a 2 n × 2 n $2n\times 2n$ matrix such that numerical range of Z is contained in a sector region S α $S_{\alpha } $ for some α ∈ [ 0 , π 2 ) $\alpha \in [0,\frac{\pi }{2} ) $ , then, for a submultiplicative function h of the class C $\mathcal{C} $ and every unitarily invariant norm, we have ∥ h ( | Z i j | 2 ) ∥ ≤ ∥ h r ( sec ( α ) | Z 11 | ) ∥ 1 r ∥ h s ( sec ( α ) | Z 22 | ) ∥ 1 s , $$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{ \frac{1}{s} }, \end{aligned}$$ where r and s are positive real numbers with 1 r + 1 s = 1 $\frac{1}{r}+\frac{1}{s}=1 $ and i , j = 1 , 2 $i,j=1,2$ . We also extend some unitarily invariant norm inequalities for sector matrices.http://link.springer.com/article/10.1186/s13660-020-02383-zUnitarily invariant normAccrative–dissipative matrixNumerical rangeSector matrix |
| spellingShingle | Davood Afraz Rahmatollah Lashkaripour Mojtaba Bakherad Norm inequalities involving a special class of functions for sector matrices Journal of Inequalities and Applications Unitarily invariant norm Accrative–dissipative matrix Numerical range Sector matrix |
| title | Norm inequalities involving a special class of functions for sector matrices |
| title_full | Norm inequalities involving a special class of functions for sector matrices |
| title_fullStr | Norm inequalities involving a special class of functions for sector matrices |
| title_full_unstemmed | Norm inequalities involving a special class of functions for sector matrices |
| title_short | Norm inequalities involving a special class of functions for sector matrices |
| title_sort | norm inequalities involving a special class of functions for sector matrices |
| topic | Unitarily invariant norm Accrative–dissipative matrix Numerical range Sector matrix |
| url | http://link.springer.com/article/10.1186/s13660-020-02383-z |
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