Spectral theorem for quaternionic normal operators: Multiplication form
Bulletin des Sciences Mathematiques
Let H be a right quaternionic Hilbert space and let T be a quaternionic normal operator with domain D(T)⊂H. We prove that there exists a Hilbert basis N of H, a measure space (Ω0,ν), a unitary operator U:H→L2(Ω0;H;ν) and a ν-measurable function η:Ω0→C such that Tx=U⁎MηUx,for allx∈D(T) where Mη is the multiplication operator on L2(Ω0;H;ν) induced by η with U(D(T))⊆D(Mη). We show that every complex Hilbert space can be seen as a slice Hilbert space of some quaternionic Hilbert space and establish the main result by reducing the problem to the complex case then lift it to the quaternion case.
Ramesh, G. and Santhosh Kumar, P., "Spectral theorem for quaternionic normal operators: Multiplication form" (2020). Journal Articles. 376.