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

Decomposable operators, local S-spectrum and S-spectrum in the quaternionic setting