A geometry for split operators

Abstract

We study the set X of split operators acting in the Hilbert space H: X = {T ∈ B(H) : N(T) ∩ R(T) = {0} and N(T) + R(T) = H}. Inside X , we consider the set Y: Y = {T ∈ X : N(T) ⊥ R(T)}. Several characterizations of these sets are given. For instance T ∈ X if and only if there exists an oblique projection Q whose range is N(T) such that T + Q is invertible, if and only if T posseses a commuting (necessarilly unique) pseudo-inverse S (i.e. T S = ST,TST = T and STS = S). Analogous characterizations are given for Y. Two natural maps are considered: q : X → Q := {oblique projections in H}, q(T) = PR(T )//N(T ) and p : Y → P := {orthogonal projections in H}, p(T) = PR(T ), where PR(T )//N(T ) denotes the projection onto R(T) with nullspace N(T), and PR(T ) denotes the orthogonal projection onto R(T). These maps are in general non continuous, subsets of continuity are studied. For the map q these are: similarity orbits, and the subsets Xck ⊂ X of operators with rank k < ∞, and XFk ⊂ X of Fredholm operators with nullity k < ∞. For the map p there are analogous results. We show that the interior of X is XF0 ∪ XF1 , and that Xck and XFk are arc-wise connected differentiable manifolds.