It is Shown that a nest in a Hilbert space H is the lattice of closed invariant subspaces of a band algebra in B(H) (i.e. an algebra generated by a semigroup of idempotent operators) if and only if all finite-dimensional atoms of the nest have dimension 1. A canonical operator matrix form for operator bands, developed by the authors, is used to demonstrate that the set of algebraic operators in B(H) coincides with the union of all band subalgebras of B(H). Several sufficient conditions for an Show moreIt is Shown that a nest in a Hilbert space H is the lattice of closed invariant subspaces of a band algebra in B(H) (i.e. an algebra generated by a semigroup of idempotent operators) if and only if all finite-dimensional atoms of the nest have dimension 1. A canonical operator matrix form for operator bands, developed by the authors, is used to demonstrate that the set of algebraic operators in B(H) coincides with the union of all band subalgebras of B(H). Several sufficient conditions for an operator band to be reducible and triangularizable are presented, and a new proof is given for a theorem on algebraic triangularizability of arbitrary operator bands. Show less