Quaternionic Hilbert spaces and operator theory - Lim, Zhi Xing (Nanyang Technological University, 2024)
This report serves to study Hilbert spaces and operator theory over skew fields. In particular, we will study these topics over the quaternions since it has been well-developed, and explore the properties of quaternionic linear operators, adjoint and normal operators. Operators over quaternionic Hilbert spaces are important when exploring certain topics, which include functional analysis, Fourier transform, and quantum mechanics. An application of operators over the quaternions is to rotate vectors in 3-dimensional space by conjugating said vectors with the axis of rotation.