Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An upper bound on the dimension of the reflexivity closure


Authors: Calin Ambrozie, Bojan Kuzma and Vladimir Müller
Journal: Proc. Amer. Math. Soc. 138 (2010), 1721-1731
MSC (2010): Primary 47L05; Secondary 15A03
DOI: https://doi.org/10.1090/S0002-9939-09-10184-3
Published electronically: November 18, 2009
MathSciNet review: 2587457
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathcal V},{\mathcal W}$ be linear spaces over an algebraically closed field, and let $ \mathscr{S}$ be an $ n$-dimensional subspace of linear operators that maps $ {\mathcal V}$ into $ {\mathcal W}$. We give a sharp upper bound for the dimension of the intersection of all images of nonzero operators from $ \mathscr{S}$, namely $ \dim ( \bigcap_{A\in\mathscr{S}\setminus\{0\}}\mathrm{Im} A ) \leq \dim{\mathcal V}-n+1$. As an application, we also give a sharp upper bound for the dimension of the reflexivity closure $ \operatorname{Ref}\mathscr{S}$ of $ \mathscr{S}$, namely $ \dim ( \operatorname{Ref}\mathscr{S} ) \leq n(n+1)/2$.


References [Enhancements On Off] (What's this?)

  • 1. J. F. Adams, Vector fields on spheres, Ann. Math. (2) 75 (1962), 603-632. MR 0139178 (25:2614)
  • 2. J. F. Adams, P. D. Lax, R. S. Phillips, On matrices whose real linear combinations are nonsingular, Proc. Amer. Math. Soc. 16 (1965), 318-322; Correction. ibid. 17 (1966), 945-947.
  • 3. E. A. Azoff, M. Ptak, On rank two linear transformations and reflexivity, J. London Math. Soc. (2) 53 (1996), 383-396. MR 1373068 (97b:47003)
  • 4. J. Bernik, R. Drnovšek, D. Kokol-Bukovšek, T. Košir, M. Omladič, Reducibility and triangularizability of semitransitive spaces of operators, Houston J. Math. 34 (2008), 235-247. MR 2383705 (2009e:47113)
  • 5. J. Bračič, B. Kuzma, Reflexivity defect of spaces of linear operators, Lin. Algebra Appl. 430 (2009), 344-359. MR 2460522 (2009j:47152)
  • 6. M. B. Delai, Extensions d'opérateurs auto-adjoints et défaut de réflexivité, Lin. Alg. Appl. 297 (1999), 81-85. MR 1723840 (2000f:47014)
  • 7. J. A. Deddens, F. A. Fillmore, Reflexive linear transformations, Lin. Alg. Appl. 10 (1975), 89-93. MR 0358390 (50:10856)
  • 8. J. Harris, Algebraic geometry, Springer-Verlag, New York, 1995. MR 1416564 (97e:14001)
  • 9. K.Y. Lam, P. Yiu, Linear spaces of real matrices of constant rank, Lin. Alg. Appl. 195 (1993), 69-79. MR 1253270 (95e:15001)
  • 10. H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1994. MR 879273 (88h:13001)
  • 11. R. Meshulam, P. Šemrl, Locally linearly dependent operators and reflexivity of operator spaces, Lin. Alg. Appl. 383 (2004), 143-150. MR 2073900 (2005g:47138)
  • 12. R. Meshulam, P. Šemrl, Minimal rank and reflexivity of operator spaces, Proc. Amer. Math. Soc. 135 (2007), 1839-1842. MR 2286094 (2007j:47136)
  • 13. J. Sylvester, On the dimension of spaces of linear transformations satisfying rank conditions, Lin. Alg. Appl. 78 (1986), 1-10. MR 840165 (87e:15005)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47L05, 15A03

Retrieve articles in all journals with MSC (2010): 47L05, 15A03


Additional Information

Calin Ambrozie
Affiliation: Mathematical Institute of the Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Prague 1, Czech Republic – and – Mathematical Institute, Bucharest, P.O. Box 1-764, RO-014700 Romania
Email: ambrozie@math.cas.cz

Bojan Kuzma
Affiliation: University of Primorska, Cankarjeva 5, SI-6000 Koper, Slovenia – and – Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia
Email: bojan.kuzma@pef.upr.si

Vladimir Müller
Affiliation: Mathematical Institute of the Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Prague 1, Czech Republic
Email: muller@math.cas.cz

DOI: https://doi.org/10.1090/S0002-9939-09-10184-3
Keywords: Determinantal varieties, space of linear operators, intersection of images, reflexivity defect
Received by editor(s): January 20, 2009
Received by editor(s) in revised form: August 26, 2009
Published electronically: November 18, 2009
Additional Notes: The first author was supported by grants IAA 100190903 of GA AV, Cncsis 54Gr/07, Ancs CEx23-05, MEB 090905
The second author was supported by a joint Czech-Slovene grant, MEB 090905.
The third author was supported by grants No. 201/09/0473 of GA ČR and IRP AV OZ 10190503
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society