rank 1 projectiom

Question

A projection $p$ is called the rank 1 projection on a Hilbert space $H$ if $p=x\otimes x$ for some unit vector $x$

Suppose $q$ is any non-trivial projection on $H$,does there exsit a relationship between $q$ and rank 1 projections? Can $q$ be experessed by linear combinations of rank 1 projections?

Answer 1

Let $M$ be the range of $q$ an d let $(e_n)$ be an orthonormal basis for $M$. Then any vector $x$ can be written as $x =\sum a_ne_n+y$ with $ y \in M^{\perp}$ and we have $qx=\sum a_n e_n$. If $q_i$ is the rank $1$ projection with range $span \{e_i\}$ the we can write $qx=\sum q_nx$. it is not possible to write $q$ in terms of a finite number rank $1$ projections.