Three-operator Problems in Banach Spaces
doi:10.17398/2605-5686.33.2.149
Keywords:
Three-space property, extending operators, lifting operators, semigroup, operator idealAbstract
We study the analogue of 3-space problems for classes of operators acting on Banach spaces. We show examples of classes of operators having or failing the 3-operator property, and give several methods to obtain classes with this property.
Downloads
References
P. Aiena, M. González, A. Martı́nez-Abejón, Operator semigroups in Banach space theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 4 (1) (2001), 157 – 205.
F. Albiac, N.J. Kalton, “Topics in Banach Space Theory”, Graduate Texts in Math. 233, Springer, New York, 2006.
T. Alvarez, M. González, Some examples of tauberian operators, Proc. Amer. Math. Soc. 111 (4) (1991), 1023 – 1027.
K. Astala, H.-O. Tylli, Seminorms related to weak compactness and to tauberian operators, Math. Proc. Cambridge Phil. Soc. 107 (2) (1990), 365 – 375.
A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González, Y. Moreno, “Separably Injective Banach Spaces”, Lecture Notes in Math. 2132, Springer, 2016.
A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González, Y. Moreno, On separably injective Banach spaces, Adv. Math. 234 (2013), 192 – 216.
A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González, Y. Moreno, Corrigendum to “On separably injective Banach spaces [Adv. Math. 234 (2013), 192 – 216]”, Adv. Math. 318 (2017), 737 – 747.
S.F. Bellenot, The J-sum of Banach spaces, J. Funct. Anal. 48 (1) (1982), 95 – 106.
F. Cabello Sánchez, J.M.F. Castillo, The long homology sequence for quasi-Banach spaces, with applications, Positivity 8 (4) (2004), 379 – 394.
F. Cabello Sánchez, J.M.F. Castillo, N.J. Kalton, Complex interpolation and twisted twisted Hilbert spaces, Pacific J. Math. 276 (2) (2015), 287 – 307.
J.M.F. Castillo, On Banach spaces X such that L(Lp , X) = K(Lp , X), Extracta Math. 10 (1) (1995), 27 – 36.
J.M.F. Castillo, M. González, “Three-space Problems in Banach Space Theory”, Lecture Notes in Math. 1667, Springer-Verlag, Berlin, 1997.
J.M.F. Castillo, M. González, A. Martínez-Abejón, Classes of operators preserved by extension and lifting, J. Math. Anal. Appl. 462 (1) (2018), 471 – 482.
J.M.F. Castillo, M. Simões, J. Suárez, On a question of Pelczyński about the duality problem for weakly compact strictly singular operators, Bull. Pol. Acad. Sci. Math. 60 (1) (2012), 27 – 36.
P. Domański, “Lp-spaces and Injective Locally Convex Spaces”, Dissertationes Math. 298, 1990.
P. Domański, Ideals of extendable and liftable operators, RACSAM, Rev. R. Acad. Cienc. Exactas Fı́s. Nat. Ser. A Mat. 97 (2) (2003), 229 – 241.
M. González, A. Martínez-Abejón, Ultrapowers and semi-Fredholm operators, Boll. Un. Mat. Ital. B (7) 11 (2) (1997), 415 – 433.
M. González, A. Martínez-Abejón, “Tauberian Operators”, Operator Theory: Advances and Applications 194, Birkhäuser Verlag, Basel, 2010.
M. González, V. M. Onieva, Lifting results for sequences in Banach spaces, Math. Proc. Cambridge Philos. Soc. 105 (1) (1989), 117 – 121.
M. González, V. M. Onieva, Characterizations of tauberian operators and other semigroups of operators, Proc. Amer. Math. Soc. 108 (2) (1990), 399 – 405.
M. González, E. Saksman, H.-O. Tylli, Representing non-weakly compact operators, Studia Math. 113 (3) (1995), 265 – 282.
N. Kalton, A. Wilansky, Tauberian operators on Banach spaces, Proc. Amer. Math. Soc. 57 (2) (1976), 251 – 255.
J. Lindenstrauss and L. Tzafriri, “Classical Banach spaces II, Function Spaces”, Ergebnisse der Math. und ihrer Grenzgebiete 97, Springer- Verlag, Berlin-New York, 1979.
A. Pietsch, “Operator Ideals”, North-Holland Publishing Co., Amsterdam- New York, 1980.
A. Pietsch, “History of Banach Spaces and Linear Operators”, Birkhauser, Boston, MA, 2007.
H. Rosenthal, On wide-(s) sequences and their applications to certain classes of operators, Pacific J. Math. 189 (2) (1999), 311 – 338.
D.G. Tacon, Generalized semi-Fredholm transformations, J. Austral. Math. Soc. 34 (1) (1983), 60 – 70.
Q. Zeng, Five short lemmas in Banach spaces, Carpathian J. Math. 32 (1) (2016), 131 – 140.
Q. Zeng, H. Zhong, Three-space theorem for semi-Fredholmness, Arch. Math. (Basel) 100 (1) (2013), 55 – 61.
