We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$

Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \langle \bullet,e_n \rangle e_n.$

Take any $n\in\mathbb{N}_0$. My question is this: Does there exist a bounded linear operator $T$ on $\ell^2(\mathbb{N}_0)$ that commutes with all $P_m$ for $m \neq n$ but not with $P_n$?