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It is known that two Banach space operators that are Schur coupled are alsoequivalent after extension, or equivalently, matricially coupled. The converseimplication, that operators which are equivalent after extension or matriciallycoupled are also Schur coupled, was only known for Fredholm Hilbert spaceoperators and Fredholm Banach space operators with index 0. We prove that thisimplication also holds for Hilbert space operators with closed range,generalizing the result for Fredholm operators, and Banach space operators thatcan be approximated in operator norm by invertible operators. The combinationof these two results enables us to prove that the implication holds for alloperators on separable Hilbert spaces.

Equivalence after extension and matricial coupling coincide with Schur coupling, on separable Hilbert spaces