TITLE: What is the spectral radius of $PBD$ with $P$ projection, $\|B\|_\infty=1$, and $D$ diagonal with $\|D\|_2<1$?
QUESTION [6 upvotes]: Assume we have the matrix product:
$$A=PBD$$
where $P$ is a projection matrix (i.e., $P=P^2$, $P=P^\top$, and $\|P\|_2=1$), $B$ is a matrix whose infinite norm is equal to one ($\|B\|_\infty=1$), and $D$ is a diagonal matrix whose $\ell_2$-norm is less than one ($\|D\|_2<1$). 
Is it correct to say that the spectral radius of $A$ is less than 1 ($\rho(A)< 1$). If yes, how to prove it?

REPLY [5 votes]: Edit: the old answer is wrong. Here is a correction.
No. Counterexample: let $0<c<1$ and
$$
P=\pmatrix{\frac{16}{25}&-\frac{12}{25}\\ -\frac{12}{25}&\frac{9}{25}},\ B=\pmatrix{1&0\\ -1&0},\ D=cI,
\ A=PBD=\pmatrix{\frac{28}{25}r&0\\ -\frac{21}{25}r&0}.
$$
Then $\rho(A)=\frac{28}{25}c>1$ when $c$ is close to $1$.