A Geometry for the Set of Split Operators

ESTEBAN ANDRUCHOW, GUSTAVO CORACH, MOSTAFA MBEKHTA

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 Hilbert space H : X = fT 2 B(H ) : N(T)\R(T) = f0g and N(T)+R(T) = H g: Inside X, we consider the set Y : Y = fT 2 X : N(T) ? R(T)g: Several characterizations of these sets are given. For instance T 2 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 Y . Two natural maps are considered: q : X !Q := f oblique projections in H g; q(T) = PR(T)==N(T) and p : Y !P := f orthogonal projections in H g; 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.