Some characterizations of minimal matrices with operator norm

This paper studies matrices A in \(M_n(\mathbb C)\) $M_n(\mathbb{C})$ satisfying \(\begin{aligned} \Vert A\Vert =\min \{\Vert A+B\Vert :B\in {\mathcal {B}}\}, \end{aligned}\) $$\begin{array}{c}\hfill \Vert A\Vert =min\{\Vert A+B\Vert :B\in \mathcal{B}\},\end{array}$$ where \({\mathcal {B}}\) $\mathcal{B}$ is a C*-subalgebra of \(M_n(\mathbb C)\) $M_n(\mathbb{C})$ and \(\Vert \cdot \Vert \) $\Vert ·\Vert$ denotes the operator norm. Such an A is called \({\mathcal {B}}\) $\mathcal{B}$ -minimal. The necessary and sufficient conditions for A to be \({\mathcal {B}}\) $\mathcal{B}$ -minimal are characterized, and a constructive method to obtain \({\mathcal {B}}\) $\mathcal{B}$ -minimal normal matrices is provided. Moreover, \(\bigoplus _{i=1}^k{\mathcal {B}}\) ${⨁}_{i=1}^k\mathcal{B}$ -minimal normal matrices with anti-diagonal block form are studied.