The geometric rank and isometries of radical-type ideals in nest algebras

Abstract

Let T( N) be a nest algebra. A left (right) ideal J of T( N) is said to be radical-type if a compact operator K belongs to J if and only if K belongs to the Jacobson radical of T( N) . In this paper, the geometric rank of finite rank operators in radical-type left ideals and isometries of these ideals are studied. Let J be a radical-type left ideal of T( N) . It is shown that any finite rank operator in J has finite geometric rank if and only if the condition is satisfied: if N∈ N with 0 +< N< I such that dim( N− N −)=∞ then both dim( N −) and dim( N ⊥) are infinite. It is also shown that if both dim(0 +) and dim( I + ⊥) are either zero or infinite then the geometric rank of a rank n operator in J is not more than n 2 and not less than (1/2) n( n+1). Using these results, we prove that if dim(0 +)≠1 and dim( I + ⊥)≠1 then linear surjective isometries between the radical-type left ideals of T( N) are of the form A→ UAV or A→UJA ∗JV , where U and V are suitable unitary operators and J is a fixed involution on a Hilbert space.