Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals

Eduardo Chiumiento; Pedro Massey

doi:10.1515/forum-2024-0010

Article

Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals

Eduardo Chiumiento and Pedro Massey

Published/Copyright: May 15, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Forum Mathematicum Volume 37 Issue 3

Abstract

We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

Keywords: Moore–Penrose inverse; polar decomposition; essential codimension; symmetrically-normed ideal; homogeneous space; real analytic map

MSC 2020: 22E65; 58B10; 47A55; 47B10

Communicated by Maria Gordina

Funding statement: This research was partially supported by CONICET (PIP 2021/2023 11220200103209CO), ANPCyT (2015 1505/ 2017 0883) and FCE-UNLP (11X974).

A Appendix

In this section we present a proof of Lemma 2. Hence, we let 𝔖 be a symmetrically-normed ideal, ν a positive Borel measure on ( 0 , ∞ ) and we let h : [ 0 , ∞ ) → 𝔖 sa be a function satisfying:

∥ h ⁢ ( t ) ∥ 𝔖 ≤ q ⁢ ( t ) , where q ⁢ ( t ) is a bounded, continuous, positive and non-increasing function such that ∫ 0 ∞ q ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) < ∞ .
For δ > 0 we have ∥ h ⁢ ( t + δ ) - h ⁢ ( t ) ∥ 𝔖 ≤ α ⁢ δ ⁢ r ⁢ ( t ) , where α > 0 and r : [ 0 , ∞ ) → [ 0 , ∞ ) is a continuous and non-increasing function such that

lim t → ∞ ⁡ r ⁢ ( t ) = 0 and ∫ 0 ∞ r ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) < ∞ .

Recall that the operator ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ∈ ℬ ⁢ ( ℋ ) sa is determined by the identity

〈 ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ⁢ x , y 〉 = ∫ 0 ∞ 〈 h ⁢ ( t ) ⁢ x , y 〉 ⁢ 𝑑 ν ⁢ ( t ) , x , y ∈ ℋ .

Since ∥ ⋅ ∥ ≤ ∥ ⋅ ∥ 𝔖 , item (i) above shows that ∫ 0 ∞ ∥ h ⁢ ( t ) ∥ ⁢ 𝑑 ν ⁢ ( t ) < ∞ . Then the previous facts imply that for every p ≥ 1 ,

∥ ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ∥ ≤ ∫ 0 ∞ ∥ h ⁢ ( t ) ∥ ⁢ 𝑑 ν ⁢ ( t ) , ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) = ∑ m = 1 ∞ ∫ [ m - 1 2 p , m 2 p ) h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ,

where the series converges in the operator norm. We now introduce the sequence

R p = ∑ m = 1 ∞ ν ⁢ ( [ m - 1 2 p , m 2 p ) ) ⁢ h ⁢ ( m 2 p ) ∈ 𝔖 sa , p ≥ 1 ,

where we have used that ∥ h ⁢ ( t ) ∥ 𝔖 ≤ q ⁢ ( t ) is a decreasing function,

∑ m = 1 ∞ ν ⁢ ( [ m - 1 2 p , m 2 p ) ) ⁢ ∥ h ⁢ ( m 2 p ) ∥ 𝔖 ≤ ∑ m = 1 ∞ ν ⁢ ( [ m - 1 2 p , m 2 p ) ) ⁢ q ⁢ ( m 2 p ) ≤ ∫ 0 ∞ q ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) < ∞ ,

so that the series defining R p is absolutely convergent in 𝔖 (and hence determines an element in 𝔖 sa ). By item (ii) above we get that for m , p ≥ 1 :

(A.1) ∥ h ⁢ ( m 2 p ) - h ⁢ ( m - 1 2 p + t ) ∥ ≤ ∥ h ⁢ ( m 2 p ) - h ⁢ ( m - 1 2 p + t ) ∥ 𝔖 ≤ α 2 p ⁢ r ⁢ ( m - 1 2 p ) for ⁢ t ∈ [ 0 , 1 2 p ) ,

where we have used that m 2 p = ( m - 1 2 p + t ) + δ for some δ ∈ [ 0 , 2 - p ] , and that r ⁢ ( m - 1 2 p + t ) ≤ r ⁢ ( m - 1 2 p ) for t ∈ [ 0 , 2 - p ) . By the same item we also get that there exists η ≥ 1 such that for p ≥ 1 we have that

(A.2) ∑ m = 1 ∞ r ⁢ ( m - 1 2 p ) ⁢ ν ⁢ ( [ m - 1 2 p , m 2 p ) ) ≤ η ⁢ ∫ 0 ∞ r ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) .

Therefore, we get that

(A.3) ∥ R p - ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ∥ ≤ α ⋅ η 2 p ⁢ ∫ 0 ∞ r ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) → p → ∞ 0 .

We now show that { R p } p ≥ 1 is a Cauchy sequence in 𝔖 : indeed, for p ≥ 1 we have that

R p = ∑ m = 1 ∞ ( ν ⁢ ( [ 2 ⁢ m - 2 2 p + 1 , 2 ⁢ m - 1 2 p + 1 ) ) + ν ⁢ ( [ 2 ⁢ m - 1 2 p + 1 , 2 ⁢ m 2 p + 1 ) ) ) ⁢ h ⁢ ( 2 ⁢ m 2 p + 1 )

and hence,

R p + 1 - R p = ∑ m = 1 ∞ ν ⁢ ( [ 2 ⁢ m - 2 2 p + 1 , 2 ⁢ m - 1 2 p + 1 ) ) ⁢ ( h ⁢ ( 2 ⁢ m - 1 2 p + 1 ) - h ⁢ ( 2 ⁢ m 2 p + 1 ) ) .

The previous identity implies that

∥ R p + 1 - R p ∥ 𝔖 ≤ ∑ m = 1 ∞ ν ⁢ ( [ 2 ⁢ m - 2 2 p + 1 , 2 ⁢ m - 1 2 p + 1 ) ) ⁢ α 2 p + 1 ⁢ r ⁢ ( 2 ⁢ m - 1 2 p + 1 ) ≤ α ⋅ η 2 p + 1 ⁢ ∫ 0 ∞ r ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) .

This last fact shows that { R p } p ≥ 1 is a Cauchy sequence in 𝔖 so that it converges to an operator R ∈ 𝔖 sa ; but then, { R p } p ≥ 1 also converges to R in the operator norm. The previous comments together with (A.3) imply that R = ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ∈ 𝔖 sa . On the other hand, using (A.1), we have that

| ∥ h ⁢ ( m - 1 2 p + δ ) ∥ 𝔖 - ∥ h ⁢ ( m 2 p ) ∥ 𝔖 | ≤ ∥ h ⁢ ( m - 1 2 p + δ ) - h ⁢ ( m 2 p ) ∥ 𝔖 ≤ α 2 p ⁢ r ⁢ ( m - 1 2 p ) for ⁢ δ ∈ [ 0 , 1 2 p ) .

Hence, using the previous fact and (A.2), we get that

(A.4) | ∫ 0 ∞ ∥ ⁢ h ⁢ ( t ) ⁢ ∥ d ⁢ ν ⁢ ( t ) - ∑ m = 1 ∞ ν ⁢ ( [ m - 1 2 p , m 2 p ) ) ⁢ ∥ h ⁢ ( m 2 p ) ∥ 𝔖 | ≤ α ⋅ η 2 p ⁢ ∫ 0 ∞ r ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) .

Since ∥ R p - ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ∥ 𝔖 → 0 when p → ∞ , given ϵ > 0 there exists p 0 ≥ 1 such that if p ≥ p 0 , then

∥ ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ∥ 𝔖 ≤ ∥ R p ∥ 𝔖 + ϵ .

On the other hand, (A.4) shows that there exists p 1 ≥ p 0 such that for p ≥ p 1 we get that

∥ ∫ 0 ∞ h ⁢ ( t ) ⁢ 𝑑 ν ⁢ ( t ) ∥ 𝔖 ≤ ∥ R p ∥ 𝔖 + ϵ ≤ ∑ m = 1 ∞ ν ⁢ ( [ m - 1 2 p , m 2 p ) ) ⁢ ∥ h ⁢ ( m 2 p ) ∥ 𝔖 + ϵ ≤ ∫ 0 ∞ ∥ h ⁢ ( t ) ∥ 𝔖 ⁢ 𝑑 ν ⁢ ( t ) + 2 ⁢ ϵ .

Acknowledgements

We thank the reviewer for his/her careful reading of this work and helpful comments.

References

[1] W. O. Amrein and K. B. Sinha, On pairs of projections in a Hilbert space, Linear Algebra Appl. 208/209 (1994), 425–435. 10.1016/0024-3795(94)90454-5Search in Google Scholar

[2] E. Andruchow, Pairs of projections: Geodesics, Fredholm and compact pairs, Complex Anal. Oper. Theory 8 (2014), no. 7, 1435–1453. 10.1007/s11785-013-0327-1Search in Google Scholar

[3] E. Andruchow, E. Chiumiento and G. Larotonda, Geometric significance of Toeplitz kernels, J. Funct. Anal. 275 (2018), no. 2, 329–355. 10.1016/j.jfa.2018.02.015Search in Google Scholar

[4] E. Andruchow, G. Corach and M. Mbekhta, On the geometry of generalized inverses, Math. Nachr. 278 (2005), no. 7–8, 756–770. 10.1002/mana.200310270Search in Google Scholar

[5] E. Andruchow and G. Larotonda, Hopf–Rinow theorem in the Sato Grassmannian, J. Funct. Anal. 255 (2008), no. 7, 1692–1712. 10.1016/j.jfa.2008.07.027Search in Google Scholar

[6] E. Andruchow, G. Larotonda and L. Recht, Finsler geometry and actions of the p-Schatten unitary groups, Trans. Amer. Math. Soc. 362 (2010), no. 1, 319–344. 10.1090/S0002-9947-09-04877-6Search in Google Scholar

[7] C. Apostol, The reduced minimum modulus, Michigan Math. J. 32 (1985), no. 3, 279–294. 10.1307/mmj/1029003239Search in Google Scholar

[8] J. Avron, R. Seiler and B. Simon, The index of a pair of projections, J. Funct. Anal. 120 (1994), no. 1, 220–237. 10.1006/jfan.1994.1031Search in Google Scholar

[9] D. Beltiţă, Smooth Homogeneous Structures in Operator Theory, Chapman & Hall/CRC Monogr. Surveys Pure Appl. Math. 137, Chapman & Hall/CRC, Boca Raton, 2006. Search in Google Scholar

[10] D. Beltiţă, T. Goliński, G. Jakimowicz and F. Pelletier, Banach–Lie groupoids and generalized inversion, J. Funct. Anal. 276 (2019), no. 5, 1528–1574. 10.1016/j.jfa.2018.12.002Search in Google Scholar

[11] D. Beltiţă and G. Larotonda, Unitary group orbits versus groupoid orbits of normal operators, J. Geom. Anal. 33 (2023), no. 3, Paper No. 95. 10.1007/s12220-022-01187-5Search in Google Scholar

[12] D. Beltiţă and T. S. Ratiu, Symplectic leaves in real Banach Lie–Poisson spaces, Geom. Funct. Anal. 15 (2005), no. 4, 753–779. 10.1007/s00039-005-0524-9Search in Google Scholar

[13] E. Boasso, On the Moore–Penrose inverse in C ∗ -algebras, Extracta Math. 21 (2006), no. 2, 93–106. Search in Google Scholar

[14] L. G. Brown, R. G. Douglas and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of C ∗ -algebras, Proceedings of a Conference on Operator Theory, Lecture Notes in Math. 345, Springer, Berlin (1973), 58–128. 10.1007/BFb0058917Search in Google Scholar

[15] J. R. Cardoso, Computation of the matrix pth root and its Fréchet derivative by integrals, Electron. Trans. Numer. Anal. 39 (2012), 414–436. Search in Google Scholar

[16] A. L. Carey, Some homogeneous spaces and representations of the Hilbert Lie group U ⁢ ( H ) 2 , Rev. Roumaine Math. Pures Appl. 30 (1985), no. 7, 505–520. Search in Google Scholar

[17] G. Chen, M. Wei and Y. Xue, Perturbation analysis of the least squares solution in Hilbert spaces, Linear Algebra Appl. 244 (1996), 69–80. 10.1016/0024-3795(94)00210-XSearch in Google Scholar

[18] E. Chiumiento, Geometry of ℑ -Stiefel manifolds, Proc. Amer. Math. Soc. 138 (2010), no. 1, 341–353. 10.1090/S0002-9939-09-10080-1Search in Google Scholar

[19] E. Chiumiento, Global symmetric approximation of frames, J. Fourier Anal. Appl. 25 (2019), no. 4, 1395–1423. 10.1007/s00041-018-9632-4Search in Google Scholar

[20] E. Chiumiento and P. Massey, On restricted diagonalization, J. Funct. Anal. 282 (2022), no. 4, Paper No. 109342. 10.1016/j.jfa.2021.109342Search in Google Scholar

[21] E. Chiumiento and P. Massey, Restricted orbits of closed range operators and equivalences of frames for subspaces, Studia Math., to appear. Search in Google Scholar

[22] G. Corach, A. Maestripieri and M. Mbekhta, Metric and homogeneous structure of closed range operators, J. Operator Theory 61 (2009), no. 1, 171–190. Search in Google Scholar

[23] G. Corach, A. Maestripieri and D. Stojanoff, Orbits of positive operators from a differentiable viewpoint, Positivity 8 (2004), no. 1, 31–48. 10.1023/B:POST.0000023202.40428.a8Search in Google Scholar

[24] P. de la Harpe, Classical Banach–Lie Algebras and Banach–Lie Groups of Operators in Hilbert Space, Lecture Notes in Math. 285, Springer, New York, 1972. 10.1007/BFb0071306Search in Google Scholar

[25] P. Del Moral and A. Niclas, A Taylor expansion of the square root matrix function, J. Math. Anal. Appl. 465 (2018), no. 1, 259–266. 10.1016/j.jmaa.2018.05.005Search in Google Scholar

[26] K. Dykema, T. Figiel, G. Weiss and M. Wodzicki, Commutator structure of operator ideals, Adv. Math. 185 (2004), no. 1, 1–79. 10.1016/S0001-8708(03)00141-5Search in Google Scholar

[27] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators, American Mathematical Society, Providence, 1960. Search in Google Scholar

[28] G. H. Golub and V. Pereyra, The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate, SIAM J. Numer. Anal. 10 (1973), 413–432. 10.1137/0710036Search in Google Scholar

[29] M. C. Gouveia and R. Puystjens, About the group inverse and Moore–Penrose inverse of a product, Linear Algebra Appl. 150 (1991), 361–369. 10.1016/0024-3795(91)90180-5Search in Google Scholar

[30] S. Izumino, Convergence of generalized inverses and spline projectors, J. Approx. Theory 38 (1983), no. 3, 269–278. 10.1016/0021-9045(83)90133-8Search in Google Scholar

[31] B. Jia, Compact perturbations of Moore–Penrose invertible operators, J. Math. Anal. Appl. 523 (2023), no. 2, Paper No. 127048. 10.1016/j.jmaa.2023.127048Search in Google Scholar

[32] V. Kaftal and J. Loreaux, Kadison’s Pythagorean theorem and essential codimension, Integral Equations Operator Theory 87 (2017), no. 4, 565–580. 10.1007/s00020-017-2365-ySearch in Google Scholar

[33] J. J. Koliha, Continuity and differentiability of the Moore–Penrose inverse in C ∗ -algebras, Math. Scand. 88 (2001), no. 1, 154–160. 10.7146/math.scand.a-14320Search in Google Scholar

[34] J.-P. Labrousse and M. Mbekhta, Les opérateurs points de continuité pour la conorme et l’inverse de Moore–Penrose, Houston J. Math. 18 (1992), no. 1, 7–23. Search in Google Scholar

[35] G. Larotonda, Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces, Forum Math. 31 (2019), no. 6, 1567–1605. 10.1515/forum-2019-0127Search in Google Scholar

[36] J. Leiterer and L. Rodman, Smoothness of generalized inverses, Indag. Math. (N. S.) 23 (2012), no. 3, 487–521. 10.1016/j.indag.2012.03.001Search in Google Scholar

[37] J. Loreaux, Restricted diagonalization of finite spectrum normal operators and a theorem of Arveson, J. Operator Theory 81 (2019), no. 2, 257–272. 10.7900/jot.2017dec01.2221Search in Google Scholar

[38] K.-H. Neeb, Holomorphic highest weight representations of infinite-dimensional complex classical groups, J. Reine Angew. Math. 497 (1998), 171–222. 10.1515/crll.1998.038Search in Google Scholar

[39] K.-H. Neeb, Infinite-dimensional groups and their representations, Infinite dimensional Kähler manifolds (Oberwolfach 1995), DMV Sem. 31, Birkhäuser, Basel (2001), 131–178. 10.1007/978-3-0348-8227-9_2Search in Google Scholar

[40] T. Sano, Fréchet derivatives for operator monotone functions, Linear Algebra Appl. 456 (2014), 88–92. 10.1016/j.laa.2013.05.018Search in Google Scholar

[41] B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Note Ser. 35, Cambridge University, Cambridge, 1979. Search in Google Scholar

[42] B. Simon, Unitaries permuting two orthogonal projections, Linear Algebra Appl. 528 (2017), 436–441. 10.1016/j.laa.2017.03.026Search in Google Scholar

[43] G. W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev. 19 (1977), no. 4, 634–662. 10.1137/1019104Search in Google Scholar

[44] Ş. Strătilă and D. Voiculescu, On a class of KMS states for the unitary group U ⁢ ( ∞ ) , Math. Ann. 235 (1978), no. 1, 87–110. 10.1007/BF01421594Search in Google Scholar

[45] H. Upmeier, Symmetric Banach Manifolds and Jordan C ∗ -Algebras, North-Holland Math. Stud. 104, North-Holland, Amsterdam, 1985. Search in Google Scholar

[46] J. L. van Hemmen and T. Ando, An inequality for trace ideals, Comm. Math. Phys. 76 (1980), no. 2, 143–148. 10.1007/BF01212822Search in Google Scholar

[47] P.-A. Wedin, Perturbation theory for pseudo-inverses, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 217–232. 10.1007/BF01933494Search in Google Scholar

Received: 2024-01-03

Revised: 2024-04-30

Published Online: 2024-05-15

Published in Print: 2025-02-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/forum-2024-0010

Keywords for this article

Moore–Penrose inverse; polar decomposition; essential codimension; symmetrically-normed ideal; homogeneous space; real analytic map