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.


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.

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