抄録
By definition the cosine of the angle between the two subspaces M and N is { \left| {u, v} \
ight|:u \in M, v \in N, \left// u \
ight// = 1 = \left// v \
ight//} . For operators A and B with closed range in Hilbert spaces, AB has closed range if and only if the angle between ker A and B({(\ker AB)^ ⊥ }) is positive. Moreover, if we denote by {A^Ψ } the pseudo-inverse of A, then {(AB)^Ψ } = {B^Ψ }{A^Ψ } if and only if B({(\ker AB)^ ⊥ }) \subset {(\ker A)^ ⊥ } and {A^ * }({(\ker {B^ * }{A^ * })^ ⊥ }) \subset {(\ker {B^ * })^ ⊥ }.
