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) subset of H. We prove that there exists a Hilbert basis N of H, a measure space (Omega(0), v), a unitary operator U : H -> L-2 (Omega(0); H; v) and a v-measurable function eta : Omega(0) -> C such that Tx = U* M(eta)Ux , for all x is an element of D(T) where M-eta is the multiplication operator on L-2(Omega(0); H; v) induced by eta with U (D(T)) subset of D(M-eta). 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. (C) 2020 Elsevier Masson SAS. All rights reserved.