Spectral theorem for quaternionic normal operators: Multiplication form

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