Split partial isometries

Abstract

A partial isometry V is said to be a split partial isometry if H = R(V) + N(V), with R(V)∩ N(V ) = {0} (R(V) = range of V, N(V ) = null-space of V).We study the topological properties of the set I0 of such partial isometries. Denote by I the set of all partial isometries of B(H), and by IN the set of normal partial isometries. Then IN ⊂ I0 ⊂ I, and the inclusions are proper. It is known that I is a C∞-submanifold of B(H). It is shown here that I0 is open in I, therefore is has also C∞-local structure. We characterize the set I0, in terms of metric properties, existence of special pseudoinverses, and a property of the spectrum and the resolvent of V. The connected components of I0 are characterized: V0, V1 ∈ I0 lie in the same connected component if and only if dim R(V0) = dim R(V1) and dim R(V0)⊥ = dim R(V1)⊥.