Basically the title says it all. Every construction I've seen for QSP and QSVT involves finding 1 and 2-dimensional invariant subspaces of the iterate, $O \equiv U_A \exp(i \pi/2 Z_\Pi)$, where $U_A$ is an $(\alpha = 1, m, \epsilon=0)$ block encoding of $A$ and $Z_\Pi$ is a reflection operator formed from the projection $\Pi$ as $Z_\Pi = 2I_2-\Pi$. Without loss of generality we can always dilate (at the cost of an extra ancilla) $$A\mapsto\pmatrix{0 & A^\dagger\\A & 0}$$ and therefore we may assume that $U_A^\dagger=U_A$ and that $U_A$ is a Hermitian block-encoding. For each eigenvalue $\lambda_i \neq \pm 1$ of $A$ there exists a (2D) invariant subspace spanned by $H_i = \text{span}(\{|0^m\rangle|v_i\rangle,|\perp_i\rangle\})$, where $|v_i\rangle$ is an eigenvector of the block encoded matrix $A$. In the case where $\lambda_i = \pm 1$, we get a 1D invariant subspace $H_i = \text{span}(|0^m\rangle|v_i\rangle)$. The whole QSP procedure becomes interleaving a controlled phase difference between these invariant subspaces, the phase angles determining the matrix polynomial implemented by the QSP.
Do we need to know the spectrum of $A$ to find these invariant subspaces? If so, why even go through this process to begin with? If not, how does this algorithm work without knowing the spectrum of $A$ $\textit{a priori}$?
