A geometry for split operators

ANDRUCHOW E, CORACH G, MBEKHTA M

INTEGRAL EQUATIONS AND OPERATOR THEORY

BIRKHAUSER VERLAG AG

Lugar: BASEL; Año: 2013 vol. 77 p. 559 - 579

0378-620X

We study the set X of split operators acting in the Hilbertspace 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 ifand 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 andSTS = S). Analogous characterizations are given for Y. Two naturalmaps are considered:q :X →Q := {oblique projections in H}, q(T) = PR(T)//N(T)andp :Y →P := {orthogonal projections in H}, p(T) = PR(T),where PR(T)//N(T) denotes the projection onto R(T) with nullspaceN(T), and PR(T) denotes the orthogonal projection onto R(T). Thesemaps are in general non continuous, subsets of continuity are studied.For the map q these are: similarity orbits, and the subsets Xck⊂ X ofoperators with rank k < ∞, and XFk⊂ X of Fredholm operators withnullity k < ∞. For the map p there are analogous results. We showthat the interior of X is XF0∪XF1, and that Xck and XFk are arc-wiseconnected differentiable manifolds.