Advances in Operator Theory

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

Na Liu‎, Wei Luo, and Qingxiang Xu

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‎‎‎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

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

Received: 27 June 2018
Accepted: 12 July 2018
First available in Project Euclid: 27 July 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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

Hilbert $C^*$-module‎ ‎polar decomposition ‎centered operator


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.

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  • M. Embry-Wardrop and A. Lambert, Measurable transformations and centered composition operators, Proc. Royal Irish Acad. 90A (1990), no. 2, 165–172.
  • M. Frank and K. Sharifi, Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules, J. Operator Theory 64 (2010), no. 2, 377–386.
  • T. Furuta, On the polar decomposition of an operator, Acta Sci. Math. 46 (1983), no. 1-4, 261–268.
  • R. Gebhardt and K. Schm$\ddot{u}$dgen, Unbounded operators on Hilbert $C^*$-modules, Internat. J. Math. 26 (2015), no. 11, 197–255.
  • F. Gesztesy, M. Malamud, M. Mitrea, and S. Naboko, Generalized polar decompositions for closed operators in Hilbert spaces and some applications, Integral Equations Operator Theory 64 (2009), no. 1, 83–113.
  • P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton, N.J., 1967.
  • W. Ichinose and K. Iwashita, On the uniqueness of the polar decomposition of bounded operators in Hilbert spaces, J. Operator Theory 70 (2013), no. 1, 175–180.
  • M. Ito, T. Yamazaki, and M. Yanagida, On the polar decomposition of the Aluthge transformation and related results, J. Operator Theory 51 (2004), no. 2, 303–319.
  • M. Ito, T. Yamazaki, and M. Yanagida, On the polar decomposition of the product of two operators and its applications, Integral Equations Operator Theory 49 (2004), no. 4, 461–472.
  • M. M. Karizaki, M. Hassani, and M. Amyari, Moore-Penrose inverse of product operators in Hilbert $C^*$-modules, Filomat 30 (2016), no. 13, 3397–3402.
  • E. C. Lance, Hilbert $C^*$-Modules-A Toolkit for Operator Algebraists, Cambridge University Press, Cambridge, 1995.
  • B. B. Morrel and P. S. Muhly, Centered operators, Studia Math. 51 (1974), 251–263.
  • V. Paulsen, C. Pearcy, and S. Petrovi$\acute{c}$, On centered and weakly centered operators, J. Funct. Anal. 128 (1995), no. 1, 87–101.
  • G. K. Pedersen, $C^*$-Algebras and Their Automorphism Groups, Academic Press, New York, 1979.
  • J. Stochel and F. H. Szafraniec, The complex moment problem and subnormality: A polar decomposition approach, J. Funct. Anal. 159 (1998), no. 2, 432–491.
  • N. E. Wegge-Olsen, $K$-Theory and $C^*$-Algebras: A Friendly Approach, Oxford Univ. Press, Oxford, England, 1993.
  • Q. Xu and X. Fang, A note on majorization and range inclusion of adjointable operators on Hilbert $C^*$-modules, Linear Algebra Appl. 516 (2017), 118–125.