### The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators

#### Abstract

‎‎‎Let $T$ be an adjointable operator between two Hilbert $C^*$-modules‎, ‎and let $T^*$ be the adjoint operator of $T$‎. ‎The polar decomposition of $T$ is characterized as $T=U(T^*T)^\frac{1}{2}$ 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.

#### Article information

Source
Adv. Oper. Theory, Volume 3, Number 4 (2018), 855-867.

Dates
Accepted: 12 July 2018
First available in Project Euclid: 27 July 2018

https://projecteuclid.org/euclid.aot/1532656920

Digital Object Identifier
doi:10.15352/aot.1807-1393

Mathematical Reviews number (MathSciNet)
MR3856178

Subjects
Primary: 46L08: $C^*$-modules
Secondary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)

#### Citation

Liu‎, Na; Luo, Wei; Xu, Qingxiang. The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators. Adv. Oper. Theory 3 (2018), no. 4, 855--867. doi:10.15352/aot.1807-1393. https://projecteuclid.org/euclid.aot/1532656920

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