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- W2201223722 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*} leftlangle x | Ty rightrangle = intlimits_{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 $x,yin H$ 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." @default.
- W2201223722 created "2016-06-24" @default.
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- W2201223722 date "2015-09-10" @default.
- W2201223722 modified "2023-09-23" @default.
- W2201223722 title "Spectral theorem for unbounded normal operators in quaternionic Hilbert spaces" @default.
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- W2201223722 doi "https://doi.org/10.48550/arxiv.1509.03007" @default.
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