On Determinants of Integrable Operators with Shifts

Integrable integral operator can be studied by means of a matrix Riemann-Hilbert problem. However, in the case of so-called integrable operators with shifts, the associated Riemann-Hilbert problem becomes operator-valued and this complicates strongly the analysis. In this note, we show how to circumvent, in a very simple way, the use of such a setting while still being able to characterize the large-x asymptotic behavior of the determinant associated with the operator.