Find all the subspaces of $l_2$ that reduces the right shift operator en $l_2$ Consider the Hilbert Space $l_2$.
Let the right and left shift operator $R$ and $L$ of $l_2$ to $l_2$ $$R(a_0, a_1, a_2,...)=(0, a_0, a_1, a_2,...).$$ $$L(a_0, a_1, a_2,...)=(a_1, a_2, a_3,...).$$
I need to find all the subspaces $W$ of $l_2$ that reduces to $R$ and $L$.
A subspace $W$ reduces to operator $T$ if $T(W)\subseteq W$ and $T(W^\perp)\subseteq W^\perp$.
Since that $R^\ast =L$ and $L^\ast=R$ and
$W$ reduces $T$ $\Leftrightarrow$ $T(W)\subseteq W$ and $T^\ast(W)\subseteq W$.
Then $W$ reduces $R$ if and only if $W$ reduces $L$.
Then I need to find the subspaces $W$ of $l_2$ so that $R(W)\subseteq W$ and $L(W)\subseteq W$ but i don´t have idea how to find this subspaces.
Thank you for your help.
 A: Lemma: The algebra of operators on $\ell^2$ generated by $\{I, L, R\}$ contains all operators whose matrix has
finitely many nonzero entries.
Proof: Denoting the canonical basis of $\ell^2$ by $\{e_n\}_{n=0}^\infty $,  notice that
$$
  p:= I-RL
  $$
is the projection onto the space spanned by $e_0$.  Moreover $R^mpL^n$ is the operator which sends all basis vectors
to zero, except for $e_n$, which is sent to $e_m$.  So the matrix of
$R^mpL^n$  coincides with $e_{m, n}$,  namely the matrix having all entries zero except for the $(m,n)$.th entry
which is 1.
This said, any operator whose matrix $A=(a_{i, j})_{i, j}$  has finitely many nonzero entries can be written as
$$
  \sum_{i,j=0}^N a_{i,j} e_{i,j} =  \sum_{i,j=0}^N a_{i,j} R^ipL^j.
  $$
QED.
Now, if $W$ is a subspace of $\ell^2$ reducing $L$, it is clear that it is invariant under any operator in the above
algebra, and hence also under any finite matrix by the Lemma.
In case $W$ is closed,  then clearly $W$ can only be $\{0\}$ or the whole $\ell^2$.
On the other hand, I suppose there must be a huge number of non-closed spaces invariant under both $R$ and $L$,   and the characterization of these might be
a difficult problem.
