# American Institute of Mathematical Sciences

August  2021, 14(8): 3043-3054. doi: 10.3934/dcdss.2020463

## Representation and approximation of the polar factor of an operator on a Hilbert space

 Université de Lille, Département de Mathématiques, UMR-CNRS 8524, Laboratoire P. Painlevé, 59655 Villeneuve d'Ascq Cedex. France

In memory of our friend Ezzeddine Zahrouni, who left us very early

Received  February 2020 Revised  August 2020 Published  August 2021 Early access  November 2020

Fund Project: This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01)

Let $H$ be a complex Hilbert space and let $\mathcal{B}(H)$ be the algebra of all bounded linear operators on $H$. The polar decomposition theorem asserts that every operator $T \in \mathcal{B}(H)$ can be written as the product $T = V P$ of a partial isometry $V\in \mathcal{B}(H)$ and a positive operator $P \in \mathcal{B}(H)$ such that the kernels of $V$ and $P$ coincide. Then this decomposition is unique. $V$ is called the polar factor of $T$. Moreover, we have automatically $P = \vert T\vert = (T^*T)^{\frac{1}{2}}$. Unlike $P$, we have no representation formula that is required for $V$.

In this paper, we introduce, for $T\in \mathcal{B}(H)$, a family of functions called a "polar function" for $T$, such that the polar factor of $T$ is obtained as a limit of a net built via continuous functional calculus from this family of functions. We derive several explicit formulas representing different polar factors. These formulas allow new for methods of approximations of the polar factor of $T$.

Citation: Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3043-3054. doi: 10.3934/dcdss.2020463
##### References:
 [1] C. Apostol, The reduced minimum modulus, Michigan Math. J., 32 (1985), 279-294.  doi: 10.1307/mmj/1029003239.  Google Scholar [2] F. Chabbabi and M. Mbekhta, Polar decomposition, Aluthge and mean transforms, Linear and Multilinear Algebra and Function Spaces, 89–107, Contemp. Math., 750, Centre Rech. Math. Proc., Amer. Math. Soc., Providence, RI, [2020].  Google Scholar [3] J.-P. Demailly, Analyse Numérique et Equations Différentielles, Grenoble Sciences. EDP Sciences, Les Ulis, 2016.  Google Scholar [4] R. Duong and F. Philipp, The effect of perturbations of linear operators on their polar decomposition, Proc. Amer. Math. Soc., 145 (2017), 779-790.  doi: 10.1090/proc/13252.  Google Scholar [5] N. J. Higham, Computing the polar decomposition with applications, SIAM. J. Stat. Comput., 7 (1986), 1160-1174.  doi: 10.1137/0907079.  Google Scholar [6] N. J. Higham, Functions of Matrices Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA (2008). doi: 10.1137/1.9780898717778.  Google Scholar [7] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1980. Google Scholar [8] R. C. Li, New perturbation bounds for the unitary polar factor, SIAM J. Matrix Anal. Appli., 16 (1995), 327-332.  doi: 10.1137/S0895479893256359.  Google Scholar [9] M. Mbekhta, Approximation of the polar factor of an operator acting on a Hilbert space, J. Math. Anal. Appl., 487 (2020), 123954, 12 pp. doi: 10.1016/j.jmaa.2020.123954.  Google Scholar [10] J. von Neumann, $\ddot{U}$ber adjungierte Funktionaloperatoren, Ann. of Math., 33 (1932), 294-310.  doi: 10.2307/1968331.  Google Scholar [11] G. K. Pedersen, $C^*$-Algebras and their Automorphism Groups, Academic Press INC. (London) 1979.  Google Scholar [12] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 2nd Edition, Springer, Berlin, 2007. doi: 10.1007/b98885.  Google Scholar [13] S. Sakai, $C^*$-algebras and $W^*$-algebras, Springer Verlag. Berlin 1971. doi: 10.1007/978-3-642-61993-9.  Google Scholar

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##### References:
 [1] C. Apostol, The reduced minimum modulus, Michigan Math. J., 32 (1985), 279-294.  doi: 10.1307/mmj/1029003239.  Google Scholar [2] F. Chabbabi and M. Mbekhta, Polar decomposition, Aluthge and mean transforms, Linear and Multilinear Algebra and Function Spaces, 89–107, Contemp. Math., 750, Centre Rech. Math. Proc., Amer. Math. Soc., Providence, RI, [2020].  Google Scholar [3] J.-P. Demailly, Analyse Numérique et Equations Différentielles, Grenoble Sciences. EDP Sciences, Les Ulis, 2016.  Google Scholar [4] R. Duong and F. Philipp, The effect of perturbations of linear operators on their polar decomposition, Proc. Amer. Math. Soc., 145 (2017), 779-790.  doi: 10.1090/proc/13252.  Google Scholar [5] N. J. Higham, Computing the polar decomposition with applications, SIAM. J. Stat. Comput., 7 (1986), 1160-1174.  doi: 10.1137/0907079.  Google Scholar [6] N. J. Higham, Functions of Matrices Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA (2008). doi: 10.1137/1.9780898717778.  Google Scholar [7] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1980. Google Scholar [8] R. C. Li, New perturbation bounds for the unitary polar factor, SIAM J. Matrix Anal. Appli., 16 (1995), 327-332.  doi: 10.1137/S0895479893256359.  Google Scholar [9] M. Mbekhta, Approximation of the polar factor of an operator acting on a Hilbert space, J. Math. Anal. Appl., 487 (2020), 123954, 12 pp. doi: 10.1016/j.jmaa.2020.123954.  Google Scholar [10] J. von Neumann, $\ddot{U}$ber adjungierte Funktionaloperatoren, Ann. of Math., 33 (1932), 294-310.  doi: 10.2307/1968331.  Google Scholar [11] G. K. Pedersen, $C^*$-Algebras and their Automorphism Groups, Academic Press INC. (London) 1979.  Google Scholar [12] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 2nd Edition, Springer, Berlin, 2007. doi: 10.1007/b98885.  Google Scholar [13] S. Sakai, $C^*$-algebras and $W^*$-algebras, Springer Verlag. Berlin 1971. doi: 10.1007/978-3-642-61993-9.  Google Scholar
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