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

TL;DR

This paper develops a theory of deficiency indices for symmetric right quaternionic linear operators, establishing criteria for self-adjointness in quaternionic Hilbert spaces analogous to complex cases.

Contribution

It introduces deficiency indices for quaternionic operators and provides necessary and sufficient conditions for self-adjointness based on these indices and the S-spectrum.

Findings

Defined deficiency indices for quaternionic operators

Established criteria for self-adjointness in quaternionic Hilbert spaces

Linked deficiency indices to the S-spectrum

Abstract

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.