Inequalities for accretive-dissipative matrices

ABSTRACT

For $i,j=1,2,\dots ,n,$ let $T_{\mathit{ij}}$ be square matrices of the same size such that the block matrix $T=\left(\begin{array}{cccccc}{T}_{11}& T_{12}& .& .& .& T_{1n}\\ T_{21}& T_{22}& .& .& .& T_{2n}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ T_{n1}& T_{n2}& .& .& .& T_{\mathit{nn}}\end{array}\right)$ is accretive-dissipative. It is shown that $$\sum _{i\ne j}^{}{∥T_{\mathit{ij}}∥}_p^p\le (n-1)2^{p-2}\sum _{i=1}^n{∥T_{\mathit{ii}}∥}_p^p\text{for}p\ge 2$$

and $$\sum _{i\ne j}{∥T_{\mathit{ij}}∥}_p^p\le (n-1)2^{2-p}\sum _{i=1}^n{∥T_{\mathit{ii}}∥}_p^p\text{for}0<p\le 2.$$

It is also proved that for any two $n\times n$ accretive-dissipative matrices S and T, we have $$\sqrt{2}{\left|det(S+T)\right|}^{\frac{1}{n}}\ge {\left|detS\right|}^{\frac{1}{n}}+{\left|detT\right|}^{\frac{1}{n}}$$

and $${\left|detS\right|}^{\alpha }{\left|detT\right|}^{1-\alpha }\le 2^{\frac{n}{2}}\left|det(\alpha S+(1-\alpha )T)\right|\text{for}0<\alpha <1.$$

KEYWORDS:

• Accretive-dissipative matrix
• Schatten p-norm
• determinant
• inequality

AMS SUBJECT CLASSIFICATIONS:

• Primary: 15A60
• Secondary: 15A15
• 15A45
• 47A30

Notes

No potential conflict of interest was reported by the authors.