Representation for bounded linear operator on Hilbert spaces

Luo Lvlin

Published 2016-11-07Version 1

In this paper we construct some $C^{*}$-algebra induced by polar decomposition $T=U|T|$. We get that $T$ is unitary equivalent to $\sqrt{|\eta|}M_{z\phi}$ on $\mathcal{L}^{2}(\sigma(|T|),\mu_{|T|})$, where $\phi\in\mathcal{L}^{\infty}(\sigma(|T|),\mu_{|T|})$ and $\eta\in\mathcal{L}^{1}(\sigma(|T|),\mu_{|T|})$. Also, we get that $T$ is normal if and only if $T$ is unitary equivalent to $M_{z\phi}$ on $\mathcal{L}^{2}(\sigma(|T|),\mu_{|T|})$, and if and only if $T\in\mathcal{A}^{'}(|T|)$, where $\phi\in\mathcal{L}^{\infty}(\sigma(|T|),\mu_{|T|})$ and $\mathcal{A}^{'}(|T|)$ is the commutant of $\mathcal{A}(|T|)$.