Let $A\in\mathbb{C}^{n\times n}$ be a complex valued square matrix which can be written as $A=PUP$ in which $P$ is a projector and $U$ is a unitary matrix. The interesting case is $P$ and $U$ do not commute.
The matrix $A$ is not normal, and I would like to upper bound the 2-norm of $A^{n-1}$. It is the case that $\|A\|=1$, so an trivial upper bound is $\|A^{n-1}\|\leq\|A\|^{n-1}=1$. Is it possible to find a bound which is strictly less than 1?
I tried to read through some articles, e.g., this thesis, in particular, Chapter 3, but did not find an obvious approach.
Taking $$ P = \pmatrix{1&0\\0&0}, \quad U = \pmatrix{\sqrt{1 - \epsilon^2} & -\epsilon \\ \epsilon & \sqrt{1 - \epsilon^2}} $$ is enough for us to see that $\|(PUP)^{n-1}\|$ can be made arbitrarily close to $1$ in the case of $n = 2$. A similar example works for arbitrary $n$. That is, it's enough to take $$ \pmatrix{1&0\\0&0_{(n-1)\times(n-1)}}, \quad \pmatrix{U&0\\0&I_{n-2}} $$ for sufficiently small $\epsilon > 0$.