Spectral theorem for unbounded normal operators in quaternionic Hilbert spaces (1509.03007v4)

Abstract: In this article, we prove the following spectral theorem for right linear normal operators (need not to be bounded) in quaternionic Hilbert spaces: Let $T$ be an unbounded right quaternionic linear normal operator in a quaternionic Hilbert space $H$ with domain $\mathcal{D}(T)$, a right linear subspace of $H$ and fix a unit imaginary quaternion, say $m$. Then there exists a Hilbert basis $\mathcal{N}$ of $H$ and a unique quaternionic spectral measure $F$ on the $\sigma$- algebra of $\mathbb C_m^{+}$ (upper half plane of the slice complex plane $\mathbb C_m$) associated to $T$ such that \begin{equation*} \left\langle x | Ty \right\rangle = \int\limits_{\sigma_{S}(T) \cap \mathbb{C}{m}^{+}}\lambda \ dF{x,y}(\lambda),\; \text{ for all}\; y \in \mathcal{D}(T),\ x \in H, \end{equation*} where $F_{x,y}$ is a quaternion valued measure on the $\sigma$- algebra of $\mathbb{C}{m}^{+}$, for any $x,y\in H$ and $\sigma{S}(T)$ is the spherical spectrum of $T$. Here the representation of $T$ is established with respect to the Hilbert basis $\mathcal{N}$. To prove this result, we reduce the problem to the complex case and obtain the result by using the classical result.