Publication: Operator ranges and endomorphisms with a prescribed behaviour on Banach spaces
Loading...
Official URL
Full text at PDC
Publication Date
2022-06
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
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
UCM subjects
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.