Banach Journal of Mathematical Analysis

Geometry of the left action of the p-Schatten groups

Maria Eugenia Di Iorio y Lucero

Full-text: Open access

Abstract

Let $\mathcal{H}$ be an infinite dimensional Hilbert space, $\mathcal{B}_{p}\left(\mathcal{H}\right)$ the $p$-Schatten class of $\mathcal{H}$ and $U_{p}\left(\mathcal{H}\right)$ be the Banach-Lie group of unitary operators which are $p$-Schatten perturbations of the identity. Let $A$ be a bounded selfadjoint operator in $\mathcal{H}$. We show that $$\mathcal{O}_A:=\left\{UA : U \in U_{p}\left(\mathcal{H}\right) \right\}$$ is a smooth submanifold of the affine space $A + \mathcal{B}_{p}\left(\mathcal{H}\right)$ if only if $A$ has closed range. Furthermore, it is a homogeneous reductive space of $U_{p}\left(\mathcal{H}\right)$. We introduce two metrics: one via the ambient Finsler metric induced as a submanifold of $A + \mathcal{B}_{p}\left(\mathcal{H}\right)$, the other, by means of the quotient Finsler metric provided by the homogeneous space structure. We show that $\mathcal{O}_A$ is a complete metric space with the rectifiable distance of these metrics.

Article information

Source
Banach J. Math. Anal., Volume 7, Number 1 (2013), 73-87.

Dates
First available in Project Euclid: 22 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1358864549

Digital Object Identifier
doi:10.15352/bjma/1358864549

Mathematical Reviews number (MathSciNet)
MR3004267

Zentralblatt MATH identifier
1279.47035

Subjects
Primary: 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]
Secondary: 46T05: Infinite-dimensional manifolds [See also 53Axx, 57N20, 58Bxx, 58Dxx] 2057N 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]

Keywords
Analytic submanifold Finsler metric Riemannian metric Schatten operator

Citation

Di Iorio y Lucero, Maria Eugenia. Geometry of the left action of the p-Schatten groups. Banach J. Math. Anal. 7 (2013), no. 1, 73--87. doi:10.15352/bjma/1358864549. https://projecteuclid.org/euclid.bjma/1358864549


Export citation

References

  • E. Andruchow and G. Larotonda, The rectifiable distance in the unitary Fredholm group, Studia Math. 196 (2010), 151–178.
  • E. Andruchow and D. Stojanoff, Geometry of unitary orbits, J. Operator Theory 26 (1991), no. 1, 25–41.
  • D. Beltiţ$\breve{\text{a}}$, Smooth homogeneous structures in operator theory, Monographs and Surveys in Pure and Applied Mathematics 137, 2005.
  • D. Beltiţ$\breve{\text{a}}$, T.S. Ratiu, and A.B. Tumpach, The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits, J. Funct. Anal. 247 (2007), no. 1, 138–168.
  • P. Bóna, Some considerations on topologies of infinite dimensional unitary coadjoint orbits, J. Geom. Phys. 51 (2004), no. 2, 256–268.
  • A.L. Carey, Some homogeneous spaces and representations of the Hilbert Lie group ${U}_{2}\left(\mathcal{H}\right)$, Rev. Roumaine Math. Pures Appl. 30 (1985), no. 7, 505–520.
  • G. Corach, H. Porta and L. Recht, The geometry of spaces of projections in $C^*$-algebras, Adv. Math. 101 (1993), no. 1, 59–77.
  • P. de la Harpe, Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space, Lectures Notes in Mathematics, 285, Springer-Verlag, Berlin, 1972.
  • J. Dieudonne, Foundations of modern analysis. enlarged and corrected printing, Pure and Applied Mathematics 10–I, Academic Press, New York-London, 1969.
  • C.W. Groetsch, Generalized inverses of linear operators Marcel Dekker, Inc., New York-Basel, 1977.
  • Tosio Kato, Perturbation theory for linear operators, Springer-Verlang, New York Inc., 1966.
  • G. Larotonda, Unitary orbits in a full matrix algebra, Integral Equations Operator Theory 54 (2006), no. 4, 511–523.
  • L.E. Mata Lorenzo and L.Recht, Infinite dimensional homogeneous reductive spaces, Acta Cient. Venezolana 43 (1992), 76–90.
  • R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philo. Soc. 51(1955), 406–413.
  • I. Raeburn, The relationship between a conmutative Banach Algebra and it's maximal ideal space, J. Funct. Anal. 25 (1977), no. 4, 366–390.
  • R.W. Sharpe, Differential geometry. Cartan's generalization of Klein's Erlangen program with a foreword by S. S. Chern, Graduate Texts in Mathematics,166, Springer-Veralang, New York, 1997.
  • M. Takesaki, Theory of operator algebras III, Springer-Verlag, 1979.