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An operator is unicellular if its lattice of invariant subspaces is totally ordered by inclusion. The list of nests which are known to be the set of invariant subspaces of a unicellular operator is surprisingly short. We construct unicellular operators on $ {l^p},1 \leqslant p < \infty $ with lattices isomorphic to $ \alpha + X + {\beta ^{\ast}}$ and $ \beta $ is in this short list. Certain other nests are attained as well.

In this note, we show that if Lis a commutative subspace lattice generated by a completely distributive lattice and finitely many commuting chains, then Lis compact in the strong operator topology if and only if Lis completely distributive. The key device used is a new characterization of complete distributivity which we call upper (or lower) semicontinuity.

Every power partial isometry (p.p.i.) in the Calkin algebra lifts to a p.p.i. in .

A short proof is given of an important theorem of N. Andersen: All continuous nests are approximately unitarily equivalent.

For every absolutely continuous contraction T with spectrum on the unit circle, we exhibit an $ {H^\infty }$h and a sequence of operators $ {T_n}$T and differ from T by a sequence of compact operators converging to 0 in norm such that $ h({T_n})$. When T is diagonal, it can also be arranged that $ T - {T_n}$ commutes with T. Pour toute contraction absolument continue T dont le spectre rencontre le cercle unité, il existe une fonction h de $ {H^\infty }$ d'opérateurs unitairement equivalents à T telle que $ T - {T_n}$ $ h({T_n}) - h(T)$n. Dans le cas où T est diagonal, la suite $ {T_n}$ est un opérateur à trace, et $ {T_n}$T.

Extrapolating from Ringrose's characterization of the Jacobson radical of a nest algebra, Hopenwasser conjectured that the radical of a CSL algebra coincides with the Ringrose ideal (the closure of the union of zero diagonal elements with respect to finite sublattices). A general interpolation theorem is proved that reduces this conjecture for completely distributive lattices to a strictly combinatorial problem. This problem is solved for all width two lattices (with no restriction of complete distributivity), verifying the conjecture in this case.

We define the notion of essentially spectral operator as an operator similar to an operator of the form N+Qwhere Nis essentially normal, Qis quasinilpotent, and NQ–QNis compact. We investigate the problem of perturbing such operators by a compact operator to obtain a spectral operator. We show that in addition to certain index obstructions related to the Brown‐Douglas‐Fillmore theory, there are also some other obstructions depending on the difference between biquasitriangularity and quasidiagonality.