  It is known that two Banach space operators that are Schur coupled are also
equivalent after extension, or equivalently, matricially coupled. The converse
implication, that operators which are equivalent after extension or matricially
coupled are also Schur coupled, was only known for Fredholm Hilbert space
operators and Fredholm Banach space operators with index 0. We prove that this
implication also holds for Hilbert space operators with closed range,
generalizing the result for Fredholm operators, and Banach space operators that
can be approximated in operator norm by invertible operators. The combination
of these two results enables us to prove that the implication holds for all
operators on separable Hilbert spaces.
