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 and 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.

[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].

[3]

J.-P. Demailly, Analyse Numérique et Equations Différentielles, Grenoble Sciences. EDP Sciences, Les Ulis, 2016.

[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.

[5]

N. J. Higham, Computing the polar decomposition with applications, SIAM. J. Stat. Comput., 7 (1986), 1160-1174.  doi: 10.1137/0907079.

[6]

N. J. Higham, Functions of Matrices Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA (2008). doi: 10.1137/1.9780898717778.

[7]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1980.

[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.

[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.

[10]

J. von Neumann, $\ddot{U}$ber adjungierte Funktionaloperatoren, Ann. of Math., 33 (1932), 294-310.  doi: 10.2307/1968331.

[11]

G. K. Pedersen, $C^*$-Algebras and their Automorphism Groups, Academic Press INC. (London) 1979.

[12]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 2nd Edition, Springer, Berlin, 2007. doi: 10.1007/b98885.

[13]

S. Sakai, $C^*$-algebras and $W^*$-algebras, Springer Verlag. Berlin 1971. doi: 10.1007/978-3-642-61993-9.

show all references

References:
[1]

C. Apostol, The reduced minimum modulus, Michigan Math. J., 32 (1985), 279-294.  doi: 10.1307/mmj/1029003239.

[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].

[3]

J.-P. Demailly, Analyse Numérique et Equations Différentielles, Grenoble Sciences. EDP Sciences, Les Ulis, 2016.

[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.

[5]

N. J. Higham, Computing the polar decomposition with applications, SIAM. J. Stat. Comput., 7 (1986), 1160-1174.  doi: 10.1137/0907079.

[6]

N. J. Higham, Functions of Matrices Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA (2008). doi: 10.1137/1.9780898717778.

[7]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1980.

[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.

[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.

[10]

J. von Neumann, $\ddot{U}$ber adjungierte Funktionaloperatoren, Ann. of Math., 33 (1932), 294-310.  doi: 10.2307/1968331.

[11]

G. K. Pedersen, $C^*$-Algebras and their Automorphism Groups, Academic Press INC. (London) 1979.

[12]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 2nd Edition, Springer, Berlin, 2007. doi: 10.1007/b98885.

[13]

S. Sakai, $C^*$-algebras and $W^*$-algebras, Springer Verlag. Berlin 1971. doi: 10.1007/978-3-642-61993-9.

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