I have the following problem, and I don't know how to approach it. Let $\mathcal{H}$ be a finite-dimensional Hilbert space and fix $d\in\mathbb{N}$ such that $d>1$. Also, let $\omega=e^{\frac{2\pi}{d}}$ be a root of unity. Consider two spaces of operators:
- $\mathcal{U}(\mathcal{H})_d$ to be the subspace of the unitary matrices $A$ such that $A^d=1$.
- $P^{(d)}_{\mathcal{H}}$ to be the space of $d$-tuples of projective operators in $\mathcal{H}$.
We consider the following map: $$ sp\colon \mathcal{U}(\mathcal{H})_d\to P^{(d)}_{\mathcal{H}} $$ defined by $$ sp(A)=(\Pi_{\omega^0},\Pi_{\omega^1},\ldots,\Pi_{\omega^{d-1}}) $$ where $\\Pi_{\omega^i}$ is the projection on the eigenspace of $A$ corresponding to the eigenvalue of $\omega^i$. Note that by the condition $A^d=1$ we get that eigenvalues of $A$ are always roots of unity.
Both the domain and the codomain here have a topology - the first one from being a subspace of $\mathcal{U}(\mathcal{H})$ and the second one from being a subspace of $L(\mathcal{H})^d$. The question is whether map $sp$ is continuous?
So far I know that if two operators in $\mathcal{U}(\mathcal{H})_d$ are sufficiently close to each other, than they have the same eigenvalues. However, I don't know how to approach eigenspaces. My problem here is that even if two operators have the same eigenvalues, their eigenspaces might have (potentially) different dimensions and be completely unrelated. However, if these operators are sufficiently close w.r.t. operator norm, then such situation is excluded.
A remark here is in place. I am not fluent in the field I am asking about here. So there might be possibly other norms in which this problem will be simpler. However, since we are over finite dimensional space, all norms are equivalent. I am only interested in the continuity of this map.
Thanks!
Perhaps the simplest way to do this is to note that the spectral projections are polynomials in the operator. That is, for any $j$ the spectral projection $\Pi_{\omega^j}(A)$ of $A \in \cal U(\cal H)_d$ on eigenvalue $\omega^j$ is $f_j(A)$, where $f_j(z)$ is a polynomial that takes value $1$ at $\omega^j$ and $0$ at all the other $d$-th roots of unity.