On a conjecture by Mbekhta about best approximation by polar factors
HTML articles powered by AMS MathViewer
- by Eduardo Chiumiento PDF
- Proc. Amer. Math. Soc. 149 (2021), 3913-3922 Request permission
Abstract:
The polar factor of a bounded operator acting on a Hilbert space is the unique partial isometry arising in the polar decomposition. It is well known that the polar factor might not be a best approximant to its associated operator in the set of all partial isometries, when the distance is measured in the operator norm. We show that the polar factor of an arbitrary operator $T$ is a best approximant to $T$ in the set of all partial isometries $X$ such that $\dim (\ker (X)\cap \ker (T)^\perp )\leq \dim (\ker (X)^\perp \cap \ker (T))$. We also provide a characterization of best approximations. This work is motivated by a recent conjecture by M. Mbekhta, which can be answered using our results.References
- W. O. Amrein and Kalyan B. Sinha, On pairs of projections in a Hilbert space, Linear Algebra Appl. 208/209 (1994), 425–435. MR 1287363, DOI 10.1016/0024-3795(94)90454-5
- Jorge Antezana and Eduardo Chiumiento, Approximation by partial isometries and symmetric approximation of finite frames, J. Fourier Anal. Appl. 24 (2018), no. 4, 1098–1118. MR 3843851, DOI 10.1007/s00041-017-9547-5
- Constantin Apostol, The reduced minimum modulus, Michigan Math. J. 32 (1985), no. 3, 279–294. MR 803833, DOI 10.1307/mmj/1029003239
- J. Avron, R. Seiler, and B. Simon, The index of a pair of projections, J. Funct. Anal. 120 (1994), no. 1, 220–237. MR 1262254, DOI 10.1006/jfan.1994.1031
- Mohamed Barraa and Mohamed Boumazgour, Inner derivations and norm equality, Proc. Amer. Math. Soc. 130 (2002), no. 2, 471–476. MR 1862127, DOI 10.1090/S0002-9939-01-06053-1
- Eduardo Chiumiento, Global symmetric approximation of frames, J. Fourier Anal. Appl. 25 (2019), no. 4, 1395–1423. MR 3977122, DOI 10.1007/s00041-018-9632-4
- G. Corach and A. Maestripieri, Products of orthogonal projections and polar decompositions, Linear Algebra Appl. 434 (2011), no. 6, 1594–1609. MR 2775769, DOI 10.1016/j.laa.2010.11.033
- Michael Frank, Vern I. Paulsen, and Terry R. Tiballi, Symmetric approximation of frames and bases in Hilbert spaces, Trans. Amer. Math. Soc. 354 (2002), no. 2, 777–793. MR 1862567, DOI 10.1090/S0002-9947-01-02838-0
- B. Laszkiewicz and K. Ziȩtak, Approximation of matrices and a family of Gander methods for polar decomposition, BIT 46 (2006), no. 2, 345–366. MR 2238678, DOI 10.1007/s10543-006-0053-4
- C.-S. Lin, The unilateral shift and a norm equality for bounded linear operators, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1693–1696. MR 1487321, DOI 10.1090/S0002-9939-99-04743-7
- P. J. Maher, Partially isometric approximation of positive operators, Illinois J. Math. 33 (1989), no. 2, 227–243. MR 987820
- Mostafa Mbekhta, Approximation of the polar factor of an operator acting on a Hilbert space, J. Math. Anal. Appl. 487 (2020), no. 1, 123954, 12. MR 4066111, DOI 10.1016/j.jmaa.2020.123954
- Barry Simon, Unitaries permuting two orthogonal projections, Linear Algebra Appl. 528 (2017), 436–441. MR 3652853, DOI 10.1016/j.laa.2017.03.026
- Yue Qing Wang, Hong Ke Du, and Yan Ni Dou, On the index of Fredholm pairs of idempotents, Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 4, 679–686. MR 2495517, DOI 10.1007/s10114-009-7067-1
- Pei Yuan Wu, Approximation by partial isometries, Proc. Edinburgh Math. Soc. (2) 29 (1986), no. 2, 255–261. MR 847878, DOI 10.1017/S0013091500017624
Additional Information
- Eduardo Chiumiento
- Affiliation: Departamento de Matemática & Centro de Matemática La Plata, FCE-UNLP, Calles 50 y 115, (1900) La Plata, Argentina; and Instituto Argentino de Matemática, ‘Alberto P. Calderón’, CONICET, Saavedra 15 3er. piso, (1083) Buenos Aires, Argentina
- MR Author ID: 855072
- Email: eduardo@mate.unlp.edu.ar
- Received by editor(s): December 10, 2020
- Received by editor(s) in revised form: February 2, 2021
- Published electronically: June 23, 2021
- Additional Notes: This research was supported by Grants CONICET (PIP 2016 0525), ANPCyT (2015 1505/ 2017 0883) and FCE-UNLP (11X829)
- Communicated by: Javad Mashreghi
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 3913-3922
- MSC (2020): Primary 47A05, 47A46, 47A53
- DOI: https://doi.org/10.1090/proc/15537
- MathSciNet review: 4291589