×

A problem on the structure of Fréchet spaces. (English) Zbl 1262.46001

S. A. Argyros and R. G. Haydon [Acta Math. 206, No. 1, 1–54 (2011; Zbl 1223.46007)] have recently shown that there exists a complex Banach space \(X\) such that each operator \(T\in L(X)\) is a perturbation of a multiple of the identity by a compact operator, i.e., for each \(T\in L(X)\) there exist \(\lambda\in \mathbb{C}\) and \(K\in K(X)\) such that \(T=\lambda I+K\). The author states the following problem that arises naturally in the structure of Fréchet spaces:
{Problem}. Is there a non-normable Fréchet space such that every continuous linear operator \(T\in L(E)\) has the form \(T=\lambda I+S\), where \(\lambda\in\mathbb{C}\) and \(S\) is a bounded operator, i.e., there exists a 0-neighbourhood \(U\) in \(E\) such that \(T(U)\) is bounded.
The author shows that a positive solution of this problem would have consequences in several open problems about the existence of hypercyclic, chaotic and topologizable operators in Fréchet spaces (we refer to the paper for the definitions). More precisely, if there exists a non-normable Fréchet space \(E\) which is a positive solution to the problem, then the following results hold:
(a) If \(E\) can be selected Montel (every closed and bounded subset of \(E\) is compact) then \(E\) does not admit a chaotic operator.
(b) There exists \(T\in L(E)\) hypercyclic and \(\lambda>0\) such that \(\lambda T\) is not hypercyclic.
(c) There exists \(T\in L(E)\) which is not topologizable.

MSC:

46A04 Locally convex Fréchet spaces and (DF)-spaces
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46H35 Topological algebras of operators
47A16 Cyclic vectors, hypercyclic and chaotic operators

Citations:

Zbl 1223.46007
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Androulakis, G. and Schlumprecht, Th., (2001). Strictly singular, non-compact operators exist on the Gowers-Maurey example, J. Lond. Math. Soc. (2), 64, 655–674. · Zbl 1015.46007 · doi:10.1112/S0024610701002769
[2] Ansari, I., (1997). Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal., 148, 384–390. DOI: 10.1006/jfan.1996.3093 · Zbl 0898.47019 · doi:10.1006/jfan.1996.3093
[3] Argyros, S. A. and Haydon, R., (2009). A hereditarily indecomposable L1-space that solves the scalarplus-compact problem, Preprint, arXiv:0903.3921v2
[4] Argyros, S. A.; López-Abad, J. and Todorcevic, S., (2005). A class of Banach spaces with few nonstrictly singular operators, J. Funct. Anal., 222, 306–384. DOI: 0.1016/j.jfa.2004.11.001 · Zbl 1086.46005 · doi:10.1016/j.jfa.2004.11.001
[5] Bayart, F. and Matheron, E., (2009). Dynamics of linear operators. Cambridge Tracts in Mathematics, 179. Cambridge University Press, Cambridge. · Zbl 1187.47001
[6] Bernal, L., (1999). On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc., 127, 1003–1010. · Zbl 0911.47020 · doi:10.1090/S0002-9939-99-04657-2
[7] Bessaga, C. and Pelczyński, A., (1957). On a class of B 0-spaces, Bull. Acad. Polon. Sci., 4, 375–377. · Zbl 0077.31002
[8] Bessaga, C.; Pelczyń Ski, A. and Rolewicz, S., (1961). On diametral approximative dimension and linear homogeeity of F-spaces, Bull. Acad. Polon. Sci., 9, 678–683.
[9] Bonet, J., (2007). Topologizable operators in locally convex spaces, in Topological algebras and applications, A. Mallios and M. Haralampidou, eds., Contemp. Math., 427, 103–108, Amer. Math. Soc., Providence, RI. · Zbl 1124.46031
[10] Bonet, J., (2006). Two questions on hypercyclic operators on Fréchet spaces, in Mini-Wokshop: Hypercyclicity and Linear Chaos, Report No. 37/2006, T. Bermúdez, G. Godefroy, K. G. Grosse-Erdmann, and A. Peris eds., pp. 2271.
[11] Bonet, J.; Frerick, L.; Peris, A. and Wengenroth, J., (2005). Transitive and hypercyclic operators on locally convex spaces, Bull. Lond. Math. Soc., 37, 2, 254–264. DOI: 10.1112/S0024609304003698 · Zbl 1150.47005 · doi:10.1112/S0024609304003698
[12] Bonet, J. and Lindströ M., (1994). Spaces of operators between Fréchet spaces, Math. Proc. Cambridge Philos. Soc., 115, 1, 133–144. DOI: 10.1017/S0305004100071978 · Zbl 0804.46011 · doi:10.1017/S0305004100071978
[13] Bonet, J.; Martí Nez-Giménez, F. and Peris, A., (2001). Banach space which admits no chaotic operator, Bull. Lond. Math. Soc., 33, 196–198. DOI: 10.1112/blms/33.2.196 · Zbl 1046.47008 · doi:10.1112/blms/33.2.196
[14] Bonet, J. and Peris, A., (1998). Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal., 159, 2, 587–595. DOI: 10.1006/jfan.1998.3315 · Zbl 0926.47011 · doi:10.1006/jfan.1998.3315
[15] De La Rosa, M.; Frerick, L.; Grivaux, S. and Peris, A., (2010). Frequent hypercyclicity, chaos, and unconditional Schauder decompositions, Preprint. · Zbl 1258.47012
[16] Dubinski, E. and Vogt, D., (1989). Complemented subspaces in tame power series spaces, Studia Math., 93, 71–85. · Zbl 0694.46003
[17] Godefroy, G. and Shapiro, J. H., (1990). Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98, 2, 229–269. DOI: 10.1016/0022-1236(91)90078-J · Zbl 0732.47016 · doi:10.1016/0022-1236(91)90078-J
[18] Gowers, W. T., (1996). A new dichotomy for Banach spaces, Geom. Funct. Anal., 6, 6, 1083–1093. DOI: 10.1007/BF02246998 · Zbl 0868.46007 · doi:10.1007/BF02246998
[19] Gowers, W. T. and Maurey, B., (1993). The unconditional basic sequence problem, J. Amer. Math. Soc., 6, 851–874. DOI: 10.1090/S0894-0347-1993-1201238-0. DOI: 10.2307/2152743 · Zbl 0827.46008 · doi:10.1090/S0894-0347-1993-1201238-0
[20] Gowers, W. T. and Maurey, B., (1997). Banach spaces with small spaces of operators, Math. Ann., 307, 4, 543–568. DOI: 10.1007/s002080050050 · Zbl 0876.46006 · doi:10.1007/s002080050050
[21] Grosse-Erdmann, K. G., (1999). Universal families and hypercyclic operators, Bull. Amer. Math Soc., 36, 3, 345–381. DOI: 10.1090/S0273-0979-99-00788-0 · Zbl 0933.47003 · doi:10.1090/S0273-0979-99-00788-0
[22] Grosse-Erdmann, K. G., (2003). Recent developments in hypercyclicity, RACSAM Rev. R. Acad. Cien. Ser. A Mat., 97, 273–286. http://www.rac.es/ficheros/doc/00127.pdf · Zbl 1076.47005
[23] Grosse-Erdmann, K. G. and Peris, A., (2010). Linear Chaos, Springer, Berlin. (To appear). · Zbl 1246.47004
[24] Komorowski, R. A. and Tomczak-Jaegermann, N., (1995). Banach spaces without local unconditional structure, Israel J. Math., 89, 205–226. DOI: 10.1007/BF02808201 · Zbl 0830.46008 · doi:10.1007/BF02808201
[25] Köthe, G., (1969, 1979). Topological Vector Spaces I and II, Springer, Berlin. · Zbl 0179.17001
[26] Laursen, K. B. and Neumann, M. M., (2000). An Introduction to Local Spectral Theory, London Mathematical Society Monographs, New Series, 20, Clarendon Press, oxford University Press, New York. · Zbl 0957.47004
[27] León-Saavedra, F. and Müller, V., (2004). Rotations of hypercyclic and supercyclic operators, Integral Equations Operator Theory, 50, 385–391. DOI: 10.1007/s00020-003-1299-8 · Zbl 1079.47013 · doi:10.1007/s00020-003-1299-8
[28] Martínez-Giménez, F. and Peris, A., (2002). Chaos for backward shift operators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12, 1703–1715. · Zbl 1070.47024 · doi:10.1142/S0218127402005418
[29] Maurey, B., (2003). Banach spaces with few operators, in Handbook of the Geometry of Banach Spaces, Vol. 2, W. B. Johnson and J. Lindenstrauss, eds., 1247–1297, Elsevier, Amsterdam. · Zbl 1044.46011
[30] Meise, R. and Vogt, D., (1997). Introduction to Functional Analysis, Clarendon Press, Oxford. · Zbl 0924.46002
[31] Miñarro, M. A., (1994). Every non-normable Fréchet space contains a non-trivial direct sum, Turkish J. Math., 18, 165–167. · Zbl 0865.46002
[32] Valdivia, M., (1977). Sur certains hyperplanes qui ne sont pas ultrabornologiques dans les espaces ultrabornologiques, C. R. Math. Acad. Sci. Paris, 284, 935–937. · Zbl 0344.46006
[33] Zelazko, W., (2007). Operator algebras on locally convex spaces, in Topological algebras and applications, A. Mallios and M. Haralampidou, eds., Contemp. Math., 427, 431–442, Amer. Math. Soc., Providence, RI. · Zbl 1123.46034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.