Publication:
Operator ranges and endomorphisms with a prescribed behaviour on Banach spaces

No Thumbnail Available
Official URL
Full text at PDC
Publication Date
2022-06
Authors
Jiménez Sevilla, María del Mar
Lajara, Sebastián
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
We obtain several extensions of a theorem of Shevchik which asserts that if R is a proper dense operator range in a separable Banach space E, then there exists a compact, one-to-one and dense-range operator T : E → E such that T(E) ∩ R = {0}, and some results of Chalendar and Partington concerning the existence of compact, one-to-one and dense-range endomorphisms on a separable Banach space E which leave invariant a given closed subspace Y ⊂ E, or more generally, a countable increasing chain of closed subspaces of E.
Description
Unesco subjects
Keywords
Citation
[1] Y. A. Abramovich and C. D. Aliprantis, An invitation to Operator Theory. Graduate Studies in Mathematics vol. 50, Amer. Math. Soc., Providence, Rhode Island (2002). [2] G. Bennet and N. Kalton, Inclusion theorems for K-spaces, Canadian J. of Math. 25 (1973), 511-524. [3] J. M. Borwein and D. W. Tingley, On supportless convex sets, Proc. Amer. Math. Soc. 94 (1985), 471-476. [4] J. B. Conway, A Course in Functional Analysis. Graduate Texts in Mathematics, Springer-Verlag, New York (1990). [5] R. W. Cross, On the continuous linear image of a Banach space, J. Austral. Math. Soc. (Series A) 29 (1980), 219-234. [6] R. W. Cross and V. Shevchik, Disjointness of operator ranges, Quaest. Math. 21 (1998), 247-260. [7] I. Chalendar and J. Partington, An image problem for compact operators, Proc. Amer. Math. Soc. 134 (5) (2005), 1391-1396. [8] L. Drewnowski, A solution to a problem of De Wilde and Tsirulnikov, Manuscripta Math. 37 (1) (1982), 61-64. [9] A. F. M. ter Elst and M. Sauter, Nonseparability and von Neumann's theorem for domains of unbounded operators, J. Operator Theory 75 (2016), 367-386. [10] M. Fabian, P. Habala, P. Hajek, V. Montesinos, V. Zizler, Banach Space Theory: The basis for linear and nonlinear analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York, 2011 [11] P. A. Fillmore and J. P. Williams, On operator ranges, Adv. Math. 7 (1971), 254-281. [12] V. P. Fonf, On a property of families of imbedded Banach spaces, Teor. Funktsii Funktionsal Anal. i Prilozen 55 (1991), 140-145. [13] V. P. Fonf, S. Lajara, S. Troyanski and C. Zanco, Operator ranges and quasicomplemented subspaces of Banach spaces, Studia Math. 248 (2) (2019), 203-216. [14] S. Grivaux, Construction of operators with a prescribed behaviour, Arch. Math. 81 (2003), 291-299. [15] P. Hájek, V. Montesinos, J. Vanderwerff and V. Zizler, Biorthogonal systems in Banach spaces. CMS Books in Mathematics, Springer (2008). [16] D. Kitson and R.M. Timoney, Operator ranges and spaceability, J. Math. Anal. Appl. 378 (2) (2011) 680-686. [17] G. W. Mackey, Note on a theorem of Murray, Bull. Amer. Math. Soc. 52 (1946), 322-325. [18] A. N. Plichko, Some remarks on operator ranges (Russian), Teor. Funktsii Funktsional. Anal. i Prilozen. 53 (1990), 69-70. (English translation in J. Soviet Math. 58 (1992), 540-541). [19] V. Shevchik, On subspaces of a Banach space that coincide with the ranges of continuous linear operators (Russian), Dokl. Akad. Nauk SSSR 263 (1982), 817-819. [20] I. Singer, On biorthogonal systems and total sequences of functionals, Math. Ann. 193 (1971), 183-188. [21] I. Singer, On biorthogonal systems and total sequences of functionals II, Math. Ann. 201 (1973), 1-8. [22] I. Singer, Bases in Banach spaces II, Springer-Verlag, Berlin (1981). [23] B. R. Yahaghi, On injective or dense-range operators leaving a given chain of subspaces invariant, Proc. Amer. Math. Soc. 132 (4) (2004), 1059-1066.
Collections