An image problem for compact operators
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- by Isabelle Chalendar and Jonathan R. Partington PDF
- Proc. Amer. Math. Soc. 134 (2006), 1391-1396 Request permission
Abstract:
Let $\mathcal {X}$ be a separable Banach space and $(\mathcal {X}_n)_n$ a sequence of closed subspaces of $\mathcal {X}$ satisfying $\mathcal {X}_n\subset \mathcal {X}_{n+1}$ for all $n$. We first prove the existence of a dense-range and injective compact operator $K$ such that each $K\mathcal {X}_n$ is a dense subset of $\mathcal {X}_n$, solving a problem of Yahaghi (2004). Our second main result concerns isomorphic and dense-range injective compact mappings between dense sets of linearly independent vectors, extending a result of Grivaux (2003).References
- Sophie Grivaux, Construction of operators with prescribed behaviour, Arch. Math. (Basel) 81 (2003), no. 3, 291–299. MR 2013260, DOI 10.1007/s00013-003-0544-3
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
- R. I. Ovsepian and A. Pełczyński, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $L^{2}$, Studia Math. 54 (1975), no. 2, 149–159. MR 394137, DOI 10.4064/sm-54-2-149-159
- Ivan Singer, Bases in Banach spaces. II, Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. MR 610799, DOI 10.1007/978-3-642-67844-8
- Paolo Terenzi, Extension of uniformly minimal $M$-basic sequences in Banach spaces, J. London Math. Soc. (2) 27 (1983), no. 3, 500–506. MR 697142, DOI 10.1112/jlms/s2-27.3.500
- Bamdad R. Yahaghi, On injective or dense-range operators leaving a given chain of subspaces invariant, Proc. Amer. Math. Soc. 132 (2004), no. 4, 1059–1066. MR 2045421, DOI 10.1090/S0002-9939-03-07139-9
Additional Information
- Isabelle Chalendar
- Affiliation: Institut Girard Desargues, UFR de Mathématiques, Université Claude Bernard Lyon 1, 69622 Villeurbanne Cedex, France
- MR Author ID: 612759
- Email: chalenda@igd.univ-lyon1.fr
- Jonathan R. Partington
- Affiliation: School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
- Email: J.R.Partington@leeds.ac.uk
- Received by editor(s): November 3, 2004
- Received by editor(s) in revised form: December 9, 2004
- Published electronically: October 7, 2005
- Communicated by: Joseph A. Ball
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1391-1396
- MSC (2000): Primary 47A15, 47A46, 47B07
- DOI: https://doi.org/10.1090/S0002-9939-05-08084-6
- MathSciNet review: 2199185