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The purpose of this paper is to push forward the theory of operator-valued Riemann–Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method  á la  Deift–Zhou. In this paper, we demonstrate that the operator-valued Riemann–Hilbert problem arising in the characterization of so-called   $c$  -shifted integrable integral operators allows one to extract the large-  $x$   asymptotics of the Fredholm determinant associated with such operators.

Large- x Analysis of an Operator-Valued Riemann–Hilbert Problem