  Indicial operators are model operators associated to an elliptic differential
operator near a corner singularity on a stratified manifold. These model
operators are defined on generalized tangent cone configurations and exhibit a
natural scaling invariance property with respect to dilations of the radial
variable. In this paper we discuss extensions of symmetric indicial operators
from a functional analytic point of view. In the first, purely abstract part of
this paper, we consider a general unbounded symmetric operator that exhibits
invariance with respect to an abstract scaling action on a Hilbert space, and
we describe its extensions in terms of generalized eigenspaces of the
infinitesimal generator of this action. Among others, we obtain a Green formula
for the adjoint pairing, an algebraic formula for the signature, and in the
semibounded case explicit descriptions of the Friedrichs and Krein extensions.
In the second part we consider differential operators of Fuchs type on the half
axis with unbounded operator coefficients that are invariant under dilation,
and show that under suitable ellipticity assumptions on the indicial family
these operators fit into the abstract framework of the first part, which in
this case furnishes a description of extensions in terms of polyhomogeneous
asymptotic expansions. We also obtain an analytic formula for the signature of
the adjoint pairing in terms of the spectral flow of the indicial family for
such operators.
