Regarding upper semicontinuity of a function

Question

Let $E$ be a linear subspace of $\mathbb{C}^{n\times n}$. Define the function $\mu_E:\mathbb{C}^{n\times n}\longrightarrow \mathbb{R}_+$ as $$ \mu_E(A)=\frac{1}{\inf\{\|X\|:X\in E\text{ and }\det(I_n-AX)=0\}},\quad A\in \mathbb{C}^{n\times n}. $$ Where $\|.\|$ denotes the operator norm. Can anyone help me in showing that $\mu_E$ is upper semicontinuous? I know that if $\|X\|<1$, then $I_n-X$ is invertible. But this involves invertibility of $I_n-AX$.

Also does $\mu_E$ satisfy the triangular inequality?

Note: I know that $\mu_E(A)=\|A\|$ if $E$ is the whole space $\mathbb{C}^{n\times n}$ and $\mu_E(A)$ is the spectral radius of $A$ if $E$ is the subspace of the scalar multiples of identity.

Answer 1

Fix $A$ and take a sequence $A_k\to A$, denote $\mu_E(A_k)=a_k$. We have to prove that $\mu_E(A)\geqslant \limsup a_k$. Assume the contrary. Then $a:=\limsup a_k>\mu_E(A)\geqslant 0$. Passing to a subsequence, we may suppose that $a_k>0$ for all $k$ and the limit $a=\lim a_k>\mu_E(A)$ exists (maybe infinite).

Choose $X_k\in E$ such that $\|X_k\|=a_k^{-1}$ and $\det(I_n-A_kX_k)=0$ (such minimizing matrices $X_k$ exist by compactness). Again by compactness, we may pass to a subsequence and suppose that the matrices $X_k$ converge (their norms $a_k^{-1}$ have finite limit $a^{-1}$) to certain matrix $X_0\in E$, $\|X_0\|=a^{-1}$. Then $\det(I_n-AX_0)=\lim \det(I_n-A_kX_k)=0$, therefore $\mu_E(A)\geqslant 1/\|X_0\|=a$, a contradiction.