Rigidity of the structured singular value and applications

The structured singular value $\mu_E$ for a linear subspace $E$ of $M_n(\mathbb C)$ is defined by $$\mu_E(A)=1 / \inf\{\|X\| \ : \ X \in E, \ \det(I_n-AX)=0 \} \quad (A \in M_n(\mathbb{C})),$$ and $\mu_E(A)=0$ if there is no $X \in E$ with $\det(I_n-AX)=0$. It is well-known that $\mu_E(A)$ coincides with the spectral radius $r(A)$ when $E=\{cI_n: c \in \mathbb C \}$ and $\mu_E(A)=\|A\|$ when $E=M_n(\mathbb C)$, for all $A\in M_n(\mathbb C)$. Also, for any linear subspace $E$ satisfying $\{cI_n: c \in \mathbb C \} \subseteq E \subseteq M_n(\mathbb C)$, we have $r(A)\leq \mu_E(A) \leq \|A\|$. We prove that if $E=\{cI_n: c \in \mathbb C \}$ and $F$ is any linear subspace of $M_n(\mathbb C)$ containing $E$, then $\mu_E=\mu_F$ if and only if $E=F$. We prove the exact same rigidity theorem for the linear subspace consisting of the diagonal matrices of order $n$. On the contrary, when $E=M_n(\mathbb C)$, we show that there is a proper subspace $F$ of $M_n(\mathbb C)$, viz. the space of symmetric matrices such that $\mu_E=\mu_F=$ operator norm. Further, we characterize all linear subspaces $F\subseteq M_n(\mathbb C)$ such that $\mu_F$ coincides with the operator norm. Next, we show that in general there is no subspace $E$ of $M_n(\mathbb C)$ such that $\mu_E=$ the numerical radius, not even for $M_2(\mathbb C)$. We establish the rigidity of the structured singular value for each of the subspaces $E$ of $M_2(\mathbb C)$ such that the corresponding $\mu_E$-unit ball induces the domains -- symmetrized bidisc, tetrablock, pentablock, hexablock.