Extending the Spectral Theorem of Unbounded Self-Adjoint Operators on Infinite-Dimensional Hilbert Spaces

I'm a physics student trying to do the maths of the Hilbert space in quantum mechanics with a bit more rigour than I'm accustomed to. I am trying to find ways to extend the spectral theorem for unbounded self-adjoint operators on an infinite dimensional Hilbert space to other operators. The form of the spectral theorem I'm working with is from Hall, which says that a self-adjoint operator is unitarily equivalent to a multiplication operator on some other Hilbert space. (Page 207, Theorem 10.10.) Specifically, I have an operator $$\hat{H}$$ on the infinite dimensional Hilbert space, which is not itself self-adjoint but is related to self-adjoint operators by at least two simple relations. First, both

$$\hat{H}\hat{\gamma}^0 \; \text{and} \; \hat{\gamma}^0\hat{H}$$

are self-adjoint, where $$\hat{\gamma}^0$$ can be written in block-diagonal form as

$$\hat{\gamma}^0 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$

It therefore has all sorts of useful properties, such as being bounded, Hermitian, invertible, unitary, and $$(\hat{\gamma^0})^2=1$$. I also have the knowledge that

$$(\hat{H})^2$$

is self-adjoint. Is there any way of proving that $$\hat{H}$$ must be isomorphic (though presumably not in general by a unitary transformation) to a multiplication operator on a Hilbert space, from the fact that $$\hat{H}^2$$ and/or $$\hat{H}\hat{\gamma}^0$$ are? It feels like there might or might not be something obvious I'm missing. Would any further properties of $$\hat{H}$$ be needed?