  In a right quaternionic Hilbert space, following the complex formalism,
decomposable operators, the so-called Bishop's property and the single valued
extension property are defined and the connections between them are studied to
certain extent. In particular, for a decomposable operator, it is shown that
the S-spectrum, approximate S-point spectrum, surjectivity S-spectrum and the
union of all local S-spectra coincide. Using continuous right slice-regular
functions we have also studied certain properties of local S-spectrum and local
S-spectral subspaces.
