Strong-principal Bimodules of Nest Algebras

We shall show that B(H) can be represented by the strong closure of the linear span of the compounds of a fixed operator in B(H) and the rank one operators, composed only by the vectors of a certain orthonormal basis of H, in a nest algebra, even, under some assumption, in the radical of a nest algebra. 0. Notations, basic relationships, and introduction Throughout, the nonzero Hilbert space H under consideration is complex and separable. Subspace means "closed subspace of H " and operator means "bounded linear operator from H into itself." ç is used for "is contained in," while c is reserved for "is properly contained in." We write BÍH) for the set of all operators and KÍH) for the set of all compact operators. The symbol x ® y denotes the rank-1 operator (•, x)y for x, y c H. If S ç H and S" c BÍH), we write RXÍS) for the set of all rank-1 operators composed by vectors in S and R\5^ and RX3P for RxiS)n3' and RxiH)n3" respectively. If M is a subspace, Pm stands for the projection from H onto M. If 3* ç BÍH), [3*]^ ([«5^"') denotes the strong closure (norm closure) of the linear span of 3". If 3* ,$f ç BÍH), and if there exists G in BÍH) such that 3> = [stfGstf^s\3' = [j/Gs/]W), then we call 3* a strong-principal (normprincipal) bimodule of s/ with respect to G. A nest J" is a family of subspaces totally ordered by inclusion. Jf is said to be complete if (i) it contains {0} and H and (ii) given any subfamily Jó of Jf, the subspaces A{^: L e -^6} and \J{L: L c Jo} are both members of jr. If ¿V € JT, we define AL = \J{L: L C N, L c ¿V} and N+ = /\{L: N c L, Le JT}. Obviously, if yf is complete, then AL , N+ eJ^ for all N cjr. Throughout, a nest is complete. The set {T: T à BÍH), TN ç N, N eJT} is called the nest algebra (associated with Jf ) and is denoted in this paper by AJ^. The space M e N, with M, N c J? and M D N, is called an yT-interval. A finite yf-partition is a finite set {Ex, ... , En} of mutually orthogonal yT-intervals with Ex®--®En = H. The Jacobson radical [5, p. 69] of AJf is denoted by RAJf . We can regard the following theorem as another Received by the editors December 5, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 47A65, 47C65, 47D15.