Deficiency Indices of Some Classes of Unbounded $\mathbb{H}$-Operators

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.