Spectral theorem for unbounded normal operators in quaternionic Hilbert spaces

TL;DR

This paper establishes a spectral theorem for unbounded normal operators in quaternionic Hilbert spaces, extending classical spectral theory to the quaternionic setting by reducing the problem to the complex case.

Contribution

It introduces a spectral theorem for unbounded normal operators in quaternionic Hilbert spaces, including the construction of a quaternionic spectral measure.

Findings

Spectral measure exists for unbounded quaternionic normal operators.

Representation of operators via spectral integrals is established.

Reduction to complex case enables classical spectral theorem application.

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.