Let $\mathcal{H}$ be a Hilbert space and $d$ is an inner derivation on $\mathcal{L}(\mathcal{H})$. An operator $T\in\mathcal{L}(\mathcal{H})$ is algebraic if $p(T)=0$ for some polynomial $p$.
In their 1996 paper, Bresar and Semrl show that if $d(T)$ has finite spectrum, then $d(T)$ is algebraic (Thm3.2). This provides a link with the problem of almost invariant spaces through the derivation:\begin{equation} d(T)=PT-TP, \end{equation}
where $P$ is a fixed projection.
To be more specific, a subspace $Y$ is almost reducing for $T$ if $d(T)$ is of finite rank where $d$ is the derivation defined above and $P$ is the orthogonal projection onto $Y$. So in particular, $PT-TP$ will be a finite rank algebraic operator.
Some particularly interesting question maybe:
1) Can we get some information concerning the invariant/ reducing subspaces from this operator?
2) If we further assume $d(T)$ is finite rank for all $T$ in an algebra of operators, can we know something about the invariant subspaces, or almost invariant subspaces of the algebra?
3) On the other direction, if for a fixed operator $T$, $PT-TP$ is of finite rank for many $P$, (eg. all $P$ in a masa) what can we say about this $T$?
However, to gain more information, some more properties about algebraic operators are needed. Therefore I wonder whether there is some good references talking about algebraic operators. The problem is quite open and any suggestion is welcome.
Thanks!