Norm inequalities involving a special class of functions for sector matrices

Abstract

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.

Keywords

• Unitarily invariant norm
• Accrative–dissipative matrix
• Numerical range
• Sector matrix