  The Bayesian approach to ill-posed operator equations in Hilbert space
recently gained attraction. In this context, and when the prior distribution is
Gaussian, then two operators play a significant role, the one which governs the
operator equation, and the one which describes the prior covariance. Typically
it is assumed that these operators commute. Here we extend this analysis to
non-commuting operators, replacing the commutativity assumption by a link
condition. We discuss its relation to the commuting case, and we indicate that
this allows to use interpolation type results to obtain tight bounds for the
contraction of the posterior Gaussian distribution towards the data generating
element.
