  The Calder\'on formulas (i.e., the combination of single-layer and
hyper-singular boundary integral operators) have been widely utilized in the
process of constructing valid boundary integral equation systems which could
possess highly favorable spectral properties. This work is devoted to studying
the theoretical properties of elastodynamic Calder\'on formulas which provide
us with a solid basis for the design of fast boundary integral equation methods
solving elastic wave problems defined on a close-surface or an open-surface in
two dimensions. For the closed-surface case, it is proved that the Calder\'on
formula is a Fredholm operator of second-kind except for certain circumstances.
Regarding to the open-surface case, we investigate weighted integral operators
instead of the original integral operators which are resulted from dealing with
edge singularities of potentials corresponding to the elastic scattering
problems by open-surfaces, and show that the Calder\'on formula is a compact
perturbation of a bounded and invertible operator. To complete the proof, we
need to use the well-posedness result of the elastic scattering problem, the
analysis of the zero-frequency integral operators defined on the straight arc,
the singularity decompositions of the kernels of integral operators, and a new
representation formula of the hyper-singular operator. Moreover, it can be
demonstrated that the accumulation point of the spectrum of the invertible
operator is the same as that of the eigenvalues of the Calder\'on formula in
the closed-surface case.
