Schur coupling and associated operator relations

The concept now known as Schur complement goes back to a 1917 paper of Issai Schur, with earlier implicit traces in the work of Laplace (1812) and Sylvester (1851). In this paper, Schur proved his determinant formula for a 2 x 2 block matrix \[M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \qquad \text{ with $D$ invertible}\]

namely, \(\det(M) = \det(D)\det(M/D)\) with \(M/D := A-BD^{−1}C\) being the Schur complement of \(M\) with respect to \(D\). The identity is easily derived from Aitken’s 1939 diagonalization formula: \[ \begin{bmatrix} A & B\\ C & D \end{bmatrix} = \begin{bmatrix} I & BD^{-1}\\ 0 & I \end{bmatrix} \begin{bmatrix} M/D & 0 \\ 0 & D \end{bmatrix} \begin{bmatrix} I & 0 \\ D^{-}C & I \end{bmatrix} \] which led to many applications in matrix analysis and numerical linear algebra. With the 1947 paper of M.G. Krein, the Schur complement also made an entrance in operator theory, and from the 1980s it appeared in the form of Schur Coupling (where both \(A\) and \(D\) are invertible, and the coupling is between the two associated Schur complements) as a tool to study integral operators. Essential to this approach is the connection with other operator relations, such as Equivalence After Extension (EAE) and Matricial Coupling (MC). This led to the question, posed in 1994, whether these operator relations coincide at the level of Banach space operators. By then it was known that EAE and MC indeed coincide and were implied by Schur Coupling (SC), and all proofs can be done in an explicit and algebraic way, but the last implication “EAE/MC ⇒ SC” remained unresolved. In this talk we provide an answer to the question whether SC is implied by EAE/MC, in which Banach space geometry plays a significant role.

The talk is based on joint work with M. Messerschmidt, A.C.M. Ran, M. Roeland and M. Wortel.