Two geometric constants for operators acting on a separable Banach space

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dc.contributor.authorMartín Peinador, Elena
dc.contributor.authorIndurain, E.
dc.contributor.authorPlans, A.
dc.contributor.authorRodés-Usán, Álvaro
dc.date.accessioned2023-06-20T18:46:22Z
dc.date.available2023-06-20T18:46:22Z
dc.date.issued1988
dc.description.abstractLet A be an operator from a separable Banach space X into another Banach space Y. For every M-basis (an) of X the authors define two numbers: hA = hA,(an) = infn||A|[an,...]||and HA = HA,(an) = supnm(A|[an,...]), where [ ] stands for closed linear span and m stands for minimum modulus, i.e. m(A)=inf||x||=1||Ax||. First they prove that reflexivity of X can be characterized by the stability of HA,(an) under changes of the M-basis. In the case X is a separable reflexive Banach space these constants are related with s-numbers. The authors show that HA is the infimum of the Gelʹfand numbers of A and hA is a lower bound of the Bernstein numbers of A defined by J. Zemánek [Studia Math. 80 (1984), no. 3, 219–234]. They prove that a separable Banach space X is reflexive if and only if the infimum of the Gelʹfand numbers of every operator A from X into a Banach space Y can be computed in terms of one sequence of closed, nested, finite codimensional subspaces with null intersection. Several relationships between these numbers and the spectral theory are discussed. Finally, in the framework of a separable Hilbert space X and a selfadjoint operator A on X, it is shown that HA and hA are respectively the maximum and the minimum of the limit points of the spectrum of A. If the operator is not selfadjoint, HA and hA are exactly the maximum and minimum of the limit points of the spectrum of (A*A)1/2.
dc.description.departmentDepto. de Álgebra, Geometría y Topología
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.refereedTRUE
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/21869
dc.identifier.citationBessaga, C. and Pelczynsky, A.: On bases and unconditional convergence of series in Banach spaces. Studia Math. T. XVII, 151-164 (1958) Diestel, J.: Sequences and series in Banach spaces. Springer Verlag, 1984 Kato, T.: Perturbation theory for linear operators. Springer Verlag, 1966 Mil`man, V.: Geometric theory of Banach spaces. Russian Math. Survery, 25, 111-170 (1970) Pietsch, A.: s-numbers of operators in Banach spaces. Studia Math. LI, 201-223 (1974) Plans, A.: Zerlegung von Folgen in Hilbertraum in Heterogonalsysteme. Archiv der Math., X, 304-306 (1954) Plans, A.: Resultados acerca de una generalización de la semejanza en el espacio de Hilbert. Collect. Math. V, XIII, 3º, 241-248 (1961) Plans, A.: Los operadores acotados en relación con los sistemas asintóticamente ortogonales. Collect. Math. V, XV, 104-110 (1963) Riesz, F., and Sz-Nagy, B.: Lecons d’analyse fonctionnelle. Gauthier-Villars, 1974 Singer, I.: Basic sequences and reflexivity of Banach spaces. Ann of Math. 52, 512-527 (1950) Zemanek, J.: Geometric characteristics of semi-Fredholm operators and their asymptotic behaviour. Studia Math. LXXX, 219-234 (1984) Zippin, M.: A remark on bases and reflexivity in Banach spaces. Israel J. Math. Vol. 6, 74-79 (1968)
dc.identifier.issn0214-3577
dc.identifier.officialurlhttp://revistas.ucm.es/index.php/REMA/article/view/REMA8888110023A/17319
dc.identifier.relatedurlhttp://revistas.ucm.es
dc.identifier.urihttps://hdl.handle.net/20.500.14352/58571
dc.issue.number1-3
dc.journal.titleRevista matemática de la Universidad Complutense de Madrid
dc.language.isoeng
dc.page.final30
dc.page.initial23
dc.publisherEditorial de la Universidad Complutense
dc.rights.accessRightsrestricted access
dc.subject.cdu517.98
dc.subject.ucmTopología
dc.subject.unesco1210 Topología
dc.titleTwo geometric constants for operators acting on a separable Banach space
dc.typejournal article
dc.volume.number1
dspace.entity.typePublication
relation.isAuthorOfPublication0074400c-5caa-43fa-9c45-61c4b6f02093
relation.isAuthorOfPublication.latestForDiscovery0074400c-5caa-43fa-9c45-61c4b6f02093
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