Unicellular Operators

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 lp, 1 ^ p < oo, and on c0 with lattices isomorphic to a + X + ß* where a and ß are countable (finite or zero) ordinals, and X is in this short list. Certain other nests are attained as well. An operator F is unicellular if its lattice of invariant subspaces is totally ordered by inclusion (i.e. Lat F is a nest). The purpose of this paper is to construct unicellular operators with prescribed lattices. These lattices include nests of order type a + 1 + ß* for any countable ordinals a and ß. Halmos (see [13]) first asked for a description of those lattices which may be the invariant subspace lattice of a single operator acting on a separable Hilbert space. Which nests are attainable is an important special case, and it has been investigated by many authors. The best known unicellular operator is the Volterra integral operator on Lp(0,1), which has a lattice order isomorphic to the unit interval [4]. Donoghue [6] showed that certain weighted unilateral shifts have only the obvious invariant spaces, a nest of order type 1 + w*. The weighted shifts have been extensively studied, especially by Nikol'skii [10-12], as well as [7,15]. Construction of unicellular operators for more complicated nests has resisted attack, and only a few have been found. Notably, a unicellular bilateral weighted shift (with lattice order isomorphic to 1 + Z + 1) has only recently been constructed by Domar [5]. More general unicellular Volterra operators have been found in [2,14]. Harrison and Longstaff [8] constructed a unicellular operator with lattice of order type w + w + 1 by putting two weighted shifts together. Their computation was tricky, and became very complex in a generalization by the first author [1] to the case com + 1 + (to«)*. In this paper, we simplify and generalize the method of [8]. This will form the major building block in our constructions. Let B be an operator such that ||F"|| = 0((n\)~J) for some d > 0 which has a cyclic vector and a cocyclic vector. We show (Corollary 4.6) that a + Lat B + ß* is attainable for any countable ordinals a and ß (they may be finite or zero, also). Since unicellular operators always have cyclic vectors [13, Theorem 4.4], these Received by the editors April 20, 1983. 1980 Mathematics Subject Classification. Primary 47A15, 47B37.