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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!

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In this case I think the question is too broad. Can you not at least try to pin down a focused list of what you want to know about algebraic operators? Otherwise you are almost asking for an encyclopaedia entry –  user16299 Jul 2 '12 at 2:32
    
@YemonChoi Yes. I understand that. But it's quite difficult to pin something down due to my limited knowledge about algebraic operators. But I am thinking about something concerning finite rank algebraic operators of the form $PT-TP$, where $P$ is a projection. I want to know some relation between this operator and $P$, maybe there invariant subspaces. –  Hui Yu Jul 2 '12 at 4:16
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Well, since this question never got an answer I will share some references I found on this topic.

The most useful reference is Halmos' Ten Problems in Hilbert Space, where he concluded $T\in B(\mathcal{H})$ iff $T$ is 'block upper-triangular' and all the diagonal blocks are scalars.

Also, Radjavi and Rosenthal's Invariant Subspace book contains good information on the connection between algebraic operators and invariant subspace problem. In particular, they showed that $T$ is algebraic iff there is a natural number $D$ such that $\mathcal{H}=\oplus_{1}^{\infty}\mathcal{H}_j$, where $dim(\mathcal{H}_j)\le D$ for all $j$.

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