Properties of nest algebras

Let H be a Hilbert space and B(H) be the set of all bounded linear operators on H. A nest ${\cal N}$ in H is a totally ordered set of orthogonal projections. The corresponding nest algebra $Alg{\cal N}$ is $$Alg{\cal N} \equiv \{T \in B(H) \mid PTP = TP, \forall P \in {\cal N}\}.$$A nest is said to be finite if it contains only finitely many orthogonal projections.

In this dissertation, we show that ${\cal N}$ is finite if and only if $Alg{\cal N} + (Alg{\cal N})\sp* = B(H)$ if and only if $Alg{\cal N}$ is complemented in $B(H)$ as a subspace of $B(H),$ which is also equivalent to one of the following: (UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{&(1)\quad Alg{\cal N} + (Alg{\cal N})\sp*\ \rm is\ an\ algebra.\cr&(2)\quad Alg{\cal N} + (Alg{\cal N})\sp*\ {\rm is\ norm\ closed\ in}\ B(H).\cr&(3)\quad Alg{\cal N} + (Alg{\cal N})\sp*\ {\rm is\ norm\ dense\ in}\ B(H).\cr}$$(TABLE/EQUATION ENDS)

We also investigate the homotopy of invertibles in nest algebras.