  The relation between thermal fluctuations and the mechanical response of a
free membrane has been explored in great detail, both theoretically and
experimentally. However, understanding this relationship for membranes, locally
pinned by proteins, is significantly more challenging. Given that the coupling
of the membrane to the cell cytoskeleton, the extracellular matrix and to other
internal structures is crucial for the regulation of a number of cellular
processes, understanding the role of the pinning is of great interest. In this
manuscript we consider a single protein (elastic spring of a finite rest
length) pinning a membrane modelled in the Monge gauge. First, we determine the
Green$'$s function for the system and complement this approach by the
calculation of the mode coupling coefficients for the plane wave expansion, and
the orthonormal fluctuation modes, in turn building a set of tools for
numerical and analytic studies of a pinned membrane. Furthermore, we explore
static correlations of the free and the pinned membrane, as well as the
membrane shape, showing that all three are mutually interdependent and have an
identical long-range behaviour characterised by the correlation length.
Interestingly, the latter displays a non-monotonic behaviour as a function of
membrane tension. Importantly, exploiting these relations allows for the
experimental determination of the elastic parameters of the pinning. Last but
not least, we calculate the interaction potential between two pinning sites and
show that, even in the absence of the membrane deformation, the pinnings will
be subject to an attractive force due to changes in membrane fluctuations.
