If the range of an operator $T$ is one-dimensional, then it is said to have $\newcommand{\rank}{\operatorname{rank}}\rank 1$ as stated in N.Young's book An Introduction to Hilbert Space, pg.84. Also, if $T$ is a bounded operator of $\rank 1$ on a Hilbert Space $H$, then $Tx = \langle x, \phi \rangle \psi$ for all $x\in H$ where $\psi$ is a non-zero vector in range of $T$ and $\phi$ is a fixed unique element of $H$.
So, $\psi = Ty$ for some $y\in H$, but then $Tx= \langle x, \phi \rangle Ty$. And this goes on forever, $Tx= \langle x, \phi \rangle \langle y, \phi \rangle Tz$... So, $T$ becomes an infinite product. What do I miss? What is the exact definition of $\rank 1$ operator? Thanks in advance.
Let $Tx=\langle x,\phi\rangle\psi$. If $y$ is such that $Ty=\psi$ then $\langle y,\phi\rangle=1$. Then $$ Tx = \langle x,\phi\rangle Ty = \langle x,\phi\rangle \langle y,\phi\rangle Ty = \dots $$ so all additional factors are $1$ and these extra points $Tz$ are all equal to $Ty$, $Tz=Ty$.