For compact operator K on Hilbert space H: equation (I - K)x = y has unique solution iff homogeneous (I - K)z = 0 has only trivial solution. Otherwise solutions exist iff y ⊥ ker(I - K*). Foundational duality for integral equations.
For compact operator K on Hilbert space H: equation (I - K)x = y has unique solution iff homogeneous (I - K)z = 0 has only trivial solution. Otherwise solutions exist iff y ⊥ ker(I - K*). Foundational duality for integral equations.