Inequalities for partial determinants of accretive block matrices

Abstract

Abstract Let A = [ A i , j ] i , j = 1 m ∈ M m ( M n ) $A=[A_{i,j}]^{m}_{i,j=1}\in \mathbf{M}_{m}(\mathbf{M}_{n})$ be an accretive block matrix. We write det1 and det2 for the first and second partial determinants, respectively. In this paper, we show that ∥ det 1 ( Re A ) ∥ ≤ ∥ ( tr ( | A | ) m ) m I n ∥ $$\begin{aligned} \bigl\Vert \operatorname{det}_{1} (\operatorname{Re}A) \bigr\Vert \leq \biggl\Vert \biggl(\frac{\operatorname{tr}( \vert A \vert )}{m} \biggr)^{m}I_{n} \biggr\Vert \end{aligned}$$ and ∥ det 2 ( Re A ) ∥ ≤ ∥ ( tr ( | A | ) n ) n I m ∥ $$\begin{aligned} \bigl\Vert \operatorname{det}_{2} (\operatorname{Re}A) \bigr\Vert \leq \biggl\Vert \biggl( \frac {\operatorname{tr}( \vert A \vert )}{n} \biggr)^{n}I_{m} \biggr\Vert \end{aligned}$$ hold for any unitarily invariant norm ∥ ⋅ ∥ $\|\cdot \|$ . The two inequalities generalize some known results related to partial determinants of positive-semidefinite block matrices.

Keywords

• Partial determinants
• Accretive block matrices
• Partial traces
• Unitarily invariant norms