Liu, N., Luo, W., Xu, Q. (2018). The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators. Advances in Operator Theory, 3(4), 855-867. doi: 10.15352/aot.1807-1393
Na Liu; Wei Luo; Qingxiang Xu. "The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators". Advances in Operator Theory, 3, 4, 2018, 855-867. doi: 10.15352/aot.1807-1393
Liu, N., Luo, W., Xu, Q. (2018). 'The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators', Advances in Operator Theory, 3(4), pp. 855-867. doi: 10.15352/aot.1807-1393
Liu, N., Luo, W., Xu, Q. The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators. Advances in Operator Theory, 2018; 3(4): 855-867. doi: 10.15352/aot.1807-1393
The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators
Receive Date: 27 June 2018,
Revise Date: 12 July 2018,
Accept Date: 12 July 2018
Abstract
Let $T$ be an adjointable operator between two Hilbert $C^*$-modules and $T^*$ be the adjoint operator of $T$. The polar decomposition of $T$ is characterized as $T=U(T^*T)^\frac12$ and $\mathcal{R}(U^*)=\overline{\mathcal{R}(T^*)}$, where $U$ is a partial isometry, $\mathcal{R}(U^*)$ and $\overline{\mathcal{R}(T^*)}$ denote the range of $U^*$ and the norm closure of the range of $T^*$, respectively. Based on this new characterization of the polar decomposition, an application to the study of centered operators is carried out.
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