A Geometry for the Set of Split Operators

Abstract

We study the set ${\mathcal{X}}$ of split operators acting in the Hilbert space ${\mathcal{H}}$ :

$$\mathcal{X}=\{T\in \mathcal{B}(\mathcal{H}): N(T)\cap R(T)=\{0\} \ {\rm and} \ N(T)+R(T)=\mathcal{H}\}.$$

Inside ${\mathcal{X}}$, we consider the set ${\mathcal{Y}}$ :

$$\mathcal{Y}=\{T\in\mathcal{X}: N(T)\perp R(T)\}.$$

Several characterizations of these sets are given. For instance ${T\in\mathcal{X}}$ 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. TS = ST, TST = T and STS = S). Analogous characterizations are given for ${\mathcal{Y}}$. Two natural maps are considered:

$${\bf q}:\mathcal{X} \to \mathbb{Q}:=\{{\rm oblique \ projections \ in} \, \mathcal{H} \}, \ {\bf q}(T)=P_{R(T)//N(T)}$$

and

$${\bf p}:\mathcal{Y} \to \mathbb{P}:=\{{\rm orthogonal \ projections \ in} \ \mathcal{H} \}, \ {\bf p}(T)=P_{R(T)}, $$

where ${P_{R(T)//N(T)}}$ denotes the projection onto R(T) with nullspace N(T), and P _R(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 ${\mathcal{X}_{c_k}\subset \mathcal{X}}$ of operators with rank ${k<\infty}$, and ${\mathcal{X}_{F_k}\subset\mathcal{X}}$ of Fredholm operators with nullity ${k<\infty}$. For the map p there are analogous results. We show that the interior of ${\mathcal{X}}$ is ${\mathcal{X}_{F_0}\cup\mathcal{X}_{F_1}}$, and that ${\mathcal{X}_{c_k}}$ and ${\mathcal{X}_{F_k}}$ are arc-wise connected differentiable manifolds.