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 $P\_n$?



Operator that commutes with projections
Maybe a quicker way to see this is, if $TP\_m = P\_mT$ for all $m \neq n$ then $TP = PT$ where $P =\sum\_{m\neq n} P\_m = I - P\_n$. Since $T$ commutes with $I$, it must therefore commute with $P\_n$.


