抄録
By definition the cosine of the angle between the two subspaces M and N is sup \left{ {\left| {\left. {‹ {f, g} \
ight.} › } \
ight|:f \in M, g \in N\left| {\left| f \
ight|} \
ight| = 1 = \left| {\left. {\left| {\left. g \
ight|} \
ight.} \
ight|} \
ight.} \
ight}. Assuming that A and B are both Hilbert space operators with closed range, AB has closed range if and only if the angle between BH and ker A \cap {\left[ {\ker A \cap BH} \
ight]^ ⊥ } is positive.
