  In this paper we define the deficiency indices of a closed symmetric right
$\mathbb{H}$-linear operator and formulate a general theory of deficiency
indices in a right quaternionic Hilbert space. This study provides a necessary
and sufficient condition in terms of deficiency indices and in terms of
S-spectrum, parallel to their complex counterparts, for a symmetric right
$\mathbb{H}$-linear operators to be self-adjoint.
